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Subset systems and generalized distributive lattices

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Subset systems and generalized distributive lattices
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SUBSET SYSTEMS AND GENERALIZED DISTRIBUTIVE LATTICES


By
ERIC R. ZENK


















A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL
OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT
OF THE REQUIREMENTS FOR THE DEGREE OF
DOCTOR OF PHILOSOPHY

UNIVERSITY OF FLORIDA


2004




SUBSET SYSTEMS AND GENERALIZED DISTRIBUTIVE LATTICES
By
ERIC R. ZENK
A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL
OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT
OF THE REQUIREMENTS FOR THE DEGREE OF
DOCTOR OF PHILOSOPHY
UNIVERSITY OF FLORIDA
2004


ACKNOWLEDGMENTS
I would like to thank -
Jorge Martinez. Sometimes words are inadaquate. He has been kind, generous, patient,
and interesting to work with. I have grown under his guidance.
The people who inhabited the University of Florida math department from 1999 to
2004. They make the department the wonderful place it is.
My friend Cielo: earth is wonderful, when one can see into the sky.
My M. L.
My mathematical siblings other graduate students who worked with Jorge, especially
Ricardo Carrera.
My family Wayne, Phyllis, Margie, Jeff, and Rob.
Those teachers who, by challenging me, caused me to improve.
Participants in the f-Rings and Ordered Algebraic Structures conferences in Gaines
ville and Nashville during my time in graduate school. Because of these conferences, I
feel like I am joining a family of researchers, rather than just getting a degree. It is
an honor to know them.
Those who helped proofread this document: constructive comments were made by
Jorge Martinez, Scott McCullough, and Pham Tiep.
Anyone taking the time to read these words: a dissertation, like any book, is meant to
be read.
u


TABLE OF CONTENTS
ACKNOWLEDGEMENTS
ABSTRACT v
CHAPTERS
1 INTRODUCTION 1
1.1 Distributive Lattices 1
1.2 Subset Systems
1.3 Methods and Results 5
2 PRIMER ON CATEGORIES AND POSETS 8
2.1 Distinguished Maps
2.2 Bounds 12
2.3 Natural Transformations 17
2.4 (Co)Limits 19
2.5 Adjoint Functors 23
3 ALGEBRAS OF A MONAD 28
3.1 Categories of Algebras 28
3.2 Adjoint Connections induce Monads 32
3.3 Detecting Categories of Algebras 34
3.4 Distributive Laws 37
4 GENERATING SUBMONADS 42
4.1 Subfunctors 42
4.2 Meseguers Lemmas 50
4.3 A Partial Algebra Which Does Not Extend 59
5 FREE ALGEBRAS 63
5.1 Complete semilattices 64
5.2 Completely Distributive Complete Lattices 68
5.3 Some categories of algebras 71
6 COEQUALIZERS 82
6.1 Epis and Equalizers in P 82
iii


6.2 Factorization of Maps Using Preorders 85
6.3 Factorization of Meetsemilattice maps 88
6.4 Coequalizers in DU, 90
7 (j,m)-SPACES 93
7.1 Spatial/Sober Functorial Galois Connection 94
7.2 The Skula Topology and Extremal Monos 97
7.3 Computing Limits 100
7.4 Quotients, Extremal and Regular Epis 101
7.5 Flat Spectra 104
7.6 Epicomplete objects in 105
108
REFERENCES
BIOGRAPHICAL SKETCH
no


Abstract of Dissertation Presented to the Graduate School
of the University of Florida in Partial Fulfillment of the
Requirements for the Degree of Doctor of Philosophy
SUBSET SYSTEMS AND GENERALIZED DISTRIBUTIVE LATTICES
By
Eric R. Zenk
August 2004
Chairman: Jorge Martinez
Major Department: Mathematics
Distributive lattices alone, or with enriched structure are mathematical objects
of fundamental importance. This text studies generalized distributive lattices: the general
ization is that certain infinite meets and joins are required to exist. Subset systems (natural
rules which select a family of subsets of each poset) j and m label which sets have joins and
meets, respectively.
A calculus of subfunctors is developed: using this calculus, it is shown that any
subfunctor F of a monad (containing the image of the unit) generates a submonad F. Under
suitable conditions, any partial F-algebra extends to an F-algebra. The monad F for the free
distributive (j. m)-complete lattice is the submonad of the completely distributive complete
lattice monad generated by a subfunctor obtained from j and m.
The category DP^ of (j. m)-complete lattices which can be embedded in a completely
distributive complete lattice is a full subcategory of F-algebras. DP(n is complete and has
coequalizers.
(j, m)-complete families of subsets of a set (generalized topological spaces) are inves
tigated in analogy to classical point-set topology. Assuming suitable restrictions on j and


m, subspaces can be defined. Assuming these restrictions, there are well-behaved categories
corresponding to To- and sober- spaces.


CHAPTER 1
INTRODUCTION
1.1 Distributive Lattices
A semilattice is a set with an operation A satisfying the following universally quantified
equations
a A a = a,
a A b b A a, and
(a A b) A c = a A (b A c).
A and V, which are related by the condition
a V b b.
< b to mean a A b = a; with this order a A b
is the largest thing smaller than both a and b and a V b is the smallest thing larger than both
a and b.
A lattice is distributive if either of the following, equivalent, universally quantified
equations hold
a A (b V c) (a A b) V (a A c),
a V (b A c) = (a V b) A (a V c).
Let us consider the concept and its relevance.
A distributive lattice bears some resemblance to ordinary arithemetic where A and
V correspond to addition and multiplication; the principal difficulty with this view is that
a A a = a V a = a, which does not hold in arithmetic. There is more symmetry in the
A lattice is a set with two semilattice operations
that
a A b = a 4=>-
A lattice may be partially ordered by defining a
1


2
equations defining distributive lattices than in ordinary arithmetic; formally A and V are
interchangable, and switching them reverses the order. Distributive lattices are interesting
algebraic structures in the same right as rings structures with + and sensibly defined.
Another perspective is that distributive lattices are models of logic, with A repre
senting and and V representing or. Obviously, the connectives and and or both
satisfy the semilattice rules. In this interpretation, the distributive laws are tautologies and
a < b means the proposition a implies b. Conventional logics are often described by dis
tributive lattices obeying extra equations, which correspond to additional tautologies to be
modeled.
A perspective particularly revelant to the author is that distributive lattices (with
some additional structure) describe topological situations. Intuitively, a topological space
is an amorphous blob, from which certain pieces can be cleanly removed. The removable
pieces are called closed parts and the complements (i.e, things left over after a closed part
has been removed) are called open parts. The usual definition of a topological space is a set
X, together with a designated family of open subsets, such that
X and the empty subset are open,
if U and V are open, so is the intersection U ft V, and
if (Ui) is a family of open subsets, then the union |Jt f/ is also open.
The lattice of open sets encodes how a space is woven together. Real analysis provides some
justification for the usual definition of a topological space. However, the author wondered
how the concept of a topology changes if one varies the definition by requiring either fewer
unions of open sets be open, or more intersections of open sets be open.
1.2 Subset Systems
A category is an abstract class of objects with structure, and maps (or, homomor-
phisms) which preserve the relevant structure. The maps in a category allow comparisons


3
between objects. Category theory allows formal comparisons between various theories of
algebraic objects: e.g., one can compare the category of all rings with the category of all
distributive lattices.
Many categories of partially ordered sets (with additional structure) fit into a simple,
general pattern. The objects are partially ordered sets in which certain intentionally distin
guished subsets have suprema or nfima while the maps are order preserving functions which
preserve said nfima and suprema. P denotes the category of all posets and order preserving
maps.
The challenge is how one selects subsets which have nfima and suprema. We follow
Thatcher, Wright and Wagner [31], who introduced the useful (but blandly named) concept
of a subset system.
Definition and Remarks 1.2.1. A subset system Z, is a rule which assigns a family Z(A)
of subsets to any poset A, such that for any order preserving map / : A > B,
{,f(S) : 5 G Z(A)} C Z(B).
Z-complete posets are posets A in which each set S G Z(A) has a supremum. (See Erne [7]
for more extensive bibliography regarding subset systems.)
The following examples of subset systems may convey the generality and usefulness
of the concept.
1. If k is a cardinal, we use n to denote the subset system which selects all subsets
with cardinality less than k. We use oo for the functor which places no restriction on
cardinality. This is a subset system because for any function / : A > B, S C A, with
|A| < k implies |/(A)| < n.
2. A subset S C A is (upward) directed if for each x, y G S, there exists u G S such that
x < u and y < it. The rule dir which selects all directed subsets of a poset is a subset
system: if f{x),f(y) G /(S') and S is directed, then there is u G S such that x < u
and y < u. Thus f(u) is a common upper bound for f(x),f(y).


4
3. A subset S C A is (upward) compatible if for any x,y 6 S there is u A such that
x < u and y < u. Compatible differs from directed because for the former u 6 S,
while for the later we only require u G A. Similar arguments show that compat, which
selects compatible subsets is a subset system.
4. A subset of S C A is a chain if x, y S implies x < y or y < x. The rule ch which
selects all chains in a poset is a subset system.
5.We say a subset 5 C A is (upward) self-bounded if there is s 6 S such that for all
x S, x < s; a self-bounded set contains a maximum element. The rule sb, which
selects all self-bounded sets, is a subset system. Note that any order preserving map
preserves joins of upward self-bounded sets.
6. A non-example: An anti-chain is a set of pairwise incomparable elements. The rule
ac which selects anti-chains is not a subset system, because there is an order preserving
surjection / : D > 2, where D is the two-point anti-chain and 2 is the two point chain.
7. Generating examples: Let Q be any class of posets closed under order preserving
surjections. One may define a subset system Zq by
Za(A) = {S C A : S e Q}.
This construction shows there is a great multitude of subset systems. The subset sys
tem, compat, described above, is not generated this way, because one cannot determine
if S C A is compatible merely by looking at the poset S with its induced order.
We use subset systems, which we generically call j and m, to select which subsets
have joins and meets, respectively. Now we enumerate some categories of interest in this
discussion.
- the category of all (j, m)-complete posets: that is, posets in which j-suprema and
TO-infima exist and are preserved by all maps.


5
DP^ the category of all completely distributive complete lattices.
DP^ the full subcategory of P^ containing objects which can be P^-embedded in a
completely distributive complete lattice. DP^ is discussed in Section 5.3.
M, the full subcategory of P^ containing posets with an F structure, where F is
the monad defined in Section 5.3.
SpFJm the full subcategory of P, containing spatial posets. See Chapter 7.
- the category of generalized spaces with (j, m)-complete families of distinguished
subsets: see Chapter 7.
F£, the full subcategory of SpH?Jm containing spatial posets with flat spectrum: A is
defined to have flat spectrum if the maps A + 2 are trivially ordered: see Section 7.5.
1.3 Methods and Results
The discussion of DP^ and Mjn, which offer generalizations of distributive lattices,
uses the language of category theory. A quick introduction occurs in Chapter 2. The crucial
notion of a free object is formalized by monads, which are introduced in Chapter 3.
Chapter 4 describes a theory of subfunctors. The class of subfunctors of a functor
F bears a strong similarity to the power set lattice of a set X. Given a monad, with
functor part T, Meseguers Lemmas 4.2.1, 4.2.2, 4.2.6, and 4.2.7, show that any subfunctor
F of T containing all constants has a monadic closure, i.e., a smallest submonad F of T
which exceeds F. (Meseguers Lemmas were formulated and proved by the author, but the
technique is similar to one in Meseguer [25].) Intuitively, TX is the full set of polynomials (in
the sense of universal algebra) with variables in X, FX is a natural subset of polynomials,
and FX is the smallest natural subset of polynomials which is closed under composition
and contains FX. An algebra structure for T is a way of evaluating all polynomials; a
partial algebra structure for F is a way of evaluating polynomials in F." Under suitable
conditions, partial algebras extend to F-algebras.


6
In Chapter 5, Meseguers Lemmas are brought to bear upon monads for free complete
semilattices and free completely distributive complete lattices. Given any subset systems j
and m, there is a submonad F of the free completely distributive complete lattice monad.
The category M£, of F-algebras offers a (somewhat mysterious) generalization of the category
of distributive lattices. The subcategory BP^, containing all (j, m)-complete posets which
may be embedded in a completely distributive lattice is somewhat easier to understand and
still well behaved.
The existence of free objects in OP^ contrasts with the nonexistence of free objects
in P££ [9] and the category of complete Boolean algebras [8, 9], A fundamental difference
between these categories and DP, is the requirement that joins and meets obey a distributive
law.
The power of category theory comes as much from what it ignores as what it examines.
Significant conclusions are often obtained without examining the grubby details of what
is going on. But this innocence of grubby details limits the scope of investigation. In
the case of this document, several nicely behaved categories M?m, PI,, and Sp¥Jm are
introduced. For general subset systems j and m, the author does not even know if these
categories differ! The end of Section 5.3 from Corollary 5.3.10 onwards describes most
of the authors knowledge on the relationship between these categories.
Chapter 6 explores congruences, quotients and coequalizers in P, P3m and OH,. Much
classical algebra (ring theory, lattice theory, group theory, etc.) is simplified by the fact that
any surjection is a regular epimorphism. For the categories introduced here, the situation is
not so simple. Example 6.2.5 shows that a j-join preserving surjective image of a j-complete
poset need not be j-complete.
The results of Chapter 7 pre-date the other results presented here. Herrlich [12]
contains a detailed examination of reflections (and coreflections) in categories of topological
spaces. This dissertation aimed to generalize results summarized in Herrlich [12], for (j,m)-
spaces. A (j, m)-space consists of an underlying set and a family of open subsets which are


7
closed under j-unions and m-intersections. Continuous maps of (j, m)-spaces are functions
such that preimages of open sets are open. The initial aim was to find reflections and
coreflections of the category of (j, m)-spaces (obeying a To-style separation axiom), and
study how the existence and properties of reflections and coreflections varied depending
upon the subset systems j and m. An obvious prerequisite to such a project is knowledge
of factorizations of continuous maps.
The chapter contains a description of (j, m)-subspaces and (j, m)-quotients. In addi
tion, Section 7.5 describes a reflection of (j, m)-spaces that corresponds to the 7) reflection
of topological spaces. Lastly, Section 7.6 describes epicomplete (j, m)-spaces.


CHAPTER 2
PRIMER ON CATEGORIES AND POSETS
The text assumes a familiarity with the theory of sets typically used in mathematical
arguments. So familiar constructions unions, intersections, cartesian products, quotients
by equivalence classes, functions, Zorns Lemma, and transfinite induction are used without
further comment. (See Halmos [10] if this background is needed.) A basic familarity with
general topology is helpful.
Also some comfort with category theory is assumed. Namely, the reader can fill in
the blanks in the following informal definitions.
A category A consists of a class of objects Obj(A) and maps Map(A), such that each
object has an identity map, and there is an associative notion of composition of maps.
The set of A-maps from A\ to A2 is denoted A(Ai, A2).
A functor F : A > B assigns each A G Obj(A) an object F(A) Obj(B) and
each A-map / : Aj A2 a B-map Ff : F(A1) > F(A2). The assignment respects
composition and identity arrows.
If A is a category, Aop is the category with the same objects as A, but all arrows
reversed. For a category theoretic concept C, the dual is obtained by applying C to
Ap.
A contravariant functor A + B is a functor A Bop.
Diagrams are used to display the behavior of a collection of maps; a diagram commutes
if any composites with the same domain and codomain are equal. For example, the
8


9
diagram
W
f
h
Y-
commutes if and only if gf ih.
i
Recall the following properties of functors:
9
Definition 2.0.1. Let F : A > B be a functor. For each A\,A2 E A, F gives a function
from the hom-set A(Ai, A?) into rB(FAi, FA2) by
(/ : At A2) ~ (Ff : Ax A2).
If, for each Ai and A2 this map is onto, then F is said to be full. If, for each A\ and A2 this
map is one-to-one, then F is said to be faithful.
A full subcategory of a category A is a category B such that Obj(B) C Obj(A) and
all / : A > B with A, B G Obj(B) are B-maps. B C A is full if, and only if, the inclusion
functor is full.
Good general references for category theory are MacLane [21], Borceux [6], and Her-
rlich and Strecker [13]. MacLane [21] gives a concise, high level summary of most category
theory and includes a chapter on monads. Herrlich and Strecker [13] is quite user friendly and
concretely describes many examples of adjoint functors. Borceux [6] covers a large amount
of material; the exposition is clear and very detailed.
2.1 Distinguished Maps
Definition and Remarks 2.1.1. Begin by defining a dual pair of concepts which coincide
with the notions injective and surjective in the category Set.
1. A map / : A\ > A2 is epi, a.k.a epic (in noun form, an epimorphism) if
whenever g and h are maps A2 A3 such that gf = hf, then g = h.


10
2. A map / : A2 > A3 is mono, a.k.a. monic (in noun form, a monomorphism)
if whenever g and h are maps A\ > A2 and such that fg = fh, then g h.
One may verify that a composition of epimorphisms (resp. monomorphisms) is epi (resp.
mono). Moreover, if f = ab is epi (resp. mono), then a is also epi (resp. b is also mono).
Definition 2.1.2. A map / : A > B is an isomorphism, if there is g : B A such that
id^ = gf and idB = fg.
In most categories of sets with structure: a map is mono if and only if it is injective,
surjective maps are epi, but epimorphisms may not be surjective.
Example 2.1.3. Consider tfAb the category of torsion-free abelian groups, i.e., abelian
groups such that
na = 0 => a = 0
for any natural number n and group element a, together with group homomorphisms. The
inclusion i : Z Q of the integers in the rational numbers is epi, but not onto.
In categories of sets with relational structure, bijective maps are not necessarily
isomorphisms.
Example 2.1.4. Consider Top the category of topological spaces and continuous maps.
The identity function i : > R from the reals (with discrete topology) to the reals (with
the usual topology), is a continuous bijection. However, the inverse function i_1 is not
continuous.
Example 2.1.5. Consider P the category of partially ordered sets and order preserving
maps. Either bijection two elements is order preserving. But the inverse function 4>_1 is not order preserving.
For further discussion and more examples of epimorphisms and monomorphisms see
Herrlich and Strecker [13, Section 6] and Borceux [6, Volume 1, Sections 1.7 and 1.8].


11
Definition and Remarks 2.1.6. Consider a pair of maps /, 5 : Ai > A2. A map i : Aq >
A\ right-identifies f and g if fi = gi. A map i : Aq > A\ is called the equalizer of / and g
if:
(eql) i right-identifies / and g, and
(eq2) i has the feature that whenever j : B0 > A\ right-identifies / and g, there
is a unique map e : Bo > Aq such that j ie.
The definite article is used for equalizers, because (eq2) implies that if i : Ao A and
i' : A'0 > A axe equalizers for / and g, then there is an isomorphism j : Aq Aq such that
i' ij. Notation: i = eq(f,g).
If there are / and g such that i : A0 > A\ is the equalizer of / and g, then i is regular
mono. Regular monomorphisms are monomorphisms. If / is epi and regular mono, then / is
an isomorphism. For proofs of the assertions in this paragraph and a discussion of examples,
see Borceux [6, Volume 1, Section 2] or Herrlich and Strecker [13, Section 16].
The definitions of coequalizer and regular epi are dual to equalizer and regular
mono, but are repeated for emphasis, e : A2 > A3 left-identifies f and g if ef eg. A
map e : A2 A3 is the coequalizer of / and g if
(coeql) e left-identifies / and g, and
(coeq2) e has the feature that whenever d : A2 B3 left-identifies / and g, there is
a unique map c : A3 B3 such that e = cd.
Notation: e = coeq(/, 5). If e is the coequalizer of some pair of maps, then e is called
regular epi The duals of all basic properties of regular monomorphisms hold for regular
epimorphisms.
Definition 2.1.7. An epimorphism / is extremal if whenever f = gh and g is mono, then g
is an isomorphism. Dually, a monomorphism / is extremal if whenever / = gh and h is epi,
then h is an isomorphism. For more detailed discussions, see Borceux [6, Volume 1, Section
4.3] and Herrlich and Strecker [13, Section 17].


12
Definition 2.1.8. The map / : A B is split mono if there exists g : B A such that
id = gf. The map g : B > A is split epi if there exists / : A > B such that id^ = gf. For
more information, see Herrlich and Strecker [13, Section 5]. Note that our terminology differs
slightly from the reference; section and split mono are synonyms, as are retraction
and split epi.
Lemma 2.1.9. For A either mono or epi, consider the following statements.
1. f is split-A.
2. f is regular-A.
3. f is extremal-A.
4. f is A.
The implications 1 => 2 => 3 => 4 always hold. None of the converses generally hold.
For proof see: (1 2) Herrlich and Strecker [13, 16.15], (2 => 3) Herrlich and Strecker
[13, 17.11] or Borceux [6, Volume 1, 4.3.3(1)], (3 => 4) holds by definition.
2.2 Bounds
Recall that a partial order on a set A is a relation < satisfying:
(pol) For all a 6 A, a < a.
(po2) Whenever a < b and b < c, a < c.
(po3) Whenever a < b and b < a, a = b.
A preorder is a relation that satisfies (pol) and (po2). If ^ is a preorder on A, define an
equivalence relation ~ on A by
a ~ b <==> a ^b and b < a.
The relation is a partial order on A/ ~. A set A with a partial order (resp. preorder) is
called a partially ordered set (resp. preordered set).


13
Definition and Remarks 2.2.1. Let A be a preordered set. Define the up- and down-
closures of x 6 A by
| x = {a A : a < x}
and
\ x = {a A : x < a}.
More generally, if 5 C A define
I 5 U{| i:iGS} = {ayl:3sGS,aXs}
and
| 5 = U{f i:iGS} = {a6A:3iS|S^ a}.
If 5 C A an upper (resp. lower) bound for 5 is a £ A such that for all s G 5, s < a (resp.
a < s). Use the following notation for the set of upper bounds of 5,
1(5) = n{f x : x 5} = {a G A : Vs 6 5, s < a}
and similar notation for the set of lower bounds
1(5) = n{j x : x 5} = (a G A : Vs G 5, a < s}.
If a G A, S C A and | a = 1(5), then a is a least upper bound, a.k.a. join, a.k.a.
supremum of 5. An equivalent way to say this is
a < x <=> V s 5, s < x.
If both a and a' are suprema of 5, then a,a' 1(5). Thus a ~ a'. In a partially ordered
set, the (unique) supremum of 5 is written \/ 5. If a A, S C A and j a 1(5), then a is
a greatest lower bound, a.k.a. meet, a.k.a. infimum of 5. An equivalent way to say a is a
least upper bound is
x < a V s 5, x < s.


14
In a partially ordered set, the (unique) infimum of 5 is denoted /\ 5. (In using this notion
with a preordered set (A, :<), one refers to the associated poset A/ For example, \J S
then denotes the equivalence class containing all suprema of 5.) If a = V 5 or a = f\ 5, a is
an optimum bound for 5.
The following are elementary properties of the bound-operators and optimum bounds.
Lemma 2.2.2. For any preordered set A (in particular any poset), the following properties
hold. Use IB to denote either operator or B.
1. For x,y A, x ^ y [ x C j y <==> ] y C] x.
2. For S,T C A, S CT => B(T) C 1(5).
3. For SC A, 1(5) = I(| 5), 1(5) 1(T 5).
4- For (5j)j6/ a family of subsets of A, B(u5) = nB(5).
5. Iff: A^B is order preserving, then /(B,4(5)) C Bs(/(5)).
Definition and Remarks 2.2.3. Let A and B be preordered sets. A Galois connection
between A and B is a pair of functions / : A > B and g : B > A such that
(gel) / and g are order reversing.
(gc2) For all a A and b G B, a ^ g(f{a)) and b < f(g(b)).
Below, basic properties of Galois connections are listed. Symmetry in the definition
allows symmetry in proofs. For any true statement about Galois connections, then the
statement obtained by switching the roles of / and g, along with A and B is also a true
statement about Galois connections.
Note that / and g respect ~. The notation / ~ g means for all a A, f(a) ~ g(b)."
1 9 gfg Suppose a £ A. By (gc2), a r< f(g{a)) and g(a) < g(f(g(a))). Using
a ^ f(g{a)) and (gel), g(f(g(a))) ^ g(a).


15
2. f(A/ ~) is dually order isomorphic to {6 E B : b ~ fg(b)}: This follows from the
preceding statement. Since / ~ fgf, any member f(a) E f(A/ ~) is equivalent to
f(g(f(a))). Moreover, if b E B and b ~ f(g(b)), then b E f(A/ ~) because b ~ f(g(b)).
3. a < g{b) b < /(a): if a ^ g(b), then b < f(g(b)) < f{a). The converse is proved
similarly.
4. For any S C A, Bf(S) = ^_1B(5). Calculate
x E B(f(S)) Vsg5,iS/(s)
<=^- Vs S,s ^ g(x)
*=> g(x)eB(S)
x p_1(5)
5.If A and B are posets, the previous item implies /\/(5) = f(\/ S).
It is helpful to rephrase condition 3 for posets: ug(b) is the largest a with b < /(a).
In symbols,
9(b) = \/{a E A:b -< /(a)}.
Basic information about Galois connections has been well-known sincethe 1940s;
Raney [28] contains a bibliography of this early literature. The basic properties and def
inition are listed in Herrlich and Strecker [13, Exercise 27Q]. The particular summary here
is by the author.
The concept of a Galois connection is symmetric, and allows one to transfer a great
deal of information between preordered sets. However, the fact that the functions involved
are order reversing is sometimes inconvenient. The concept of an adjoint connection between
posets is obtained by formally reversing one of the posets involved.


16
Definition and Remarks 2.2.4. Suppose A and B are preordered sets. A pair / : A > B,
g : B > A of order preserving functions is an adjoint connection between A and B if
(adl) For all a 6 A, a < g(f(a)).
(ad2) For all b e B, f(g(b)) < b.
Each basic property for Galois connections corresponds to a basic property of adjoint
connections. The basic properties of adjoint connections are listed below; proofs are omitted.
1 / fgf and g ~ gfg
2. f(A/ ~) = {b G B : b ~ g(f{b))} and g{B/ ~) = {a 6 A : a ~ f(g(a))}.
3. /(a) ^ b <=> a X g(b)
4. For any SC A, Bg(S) /_1B(5) and B/(S) = g~1M(S)
5. If A and B are posets, f(/\ S) = f\ f(S) and g(\J S) = \/ g(S).
Because of the asymmetry between / and g, and property 3, / is called the left adjoint
and g the right adjoint. Again there is an interpretation of 3 in words, /(a) is the smallest
b such that a < g{b)\ g(b) is the largest a such that /(a) ^ 6.
The definition and basic properties of adjoint connections are folklore. For another
discussion of them see Johnstone [16, Chapter I, Paragraph 3].
Definition 2.2.5. A poset A is complete if each subset SC A (including the empty set)
has a supremum. Since the supremum of set of lower bounds for S is a lower bound for S,
A is complete if and only if each subset S C A (including the empty set) has an infimum.
There is a criterion for testing when a given order preserving (resp. reversing) map
between posets is part of an adjoint (resp. Galois) connection.
Theorem 2.2.6. Adjoint Existence Suppose A and B are posets.


17
1. Suppose f : A > B is order preserving and A is complete. Then f is a left adjoint if
and only if f(\J S) = \f f(S) for all S C A.
2. Suppose g : A > B is order preserving and A is complete. Then g is a right adjoint if
and only if g(f\ S) = /\ g(S) for all S C A.
3. Suppose f : A > B is order reversing and A is complete. Then f is part of a Galois
connection if and only if f(\J S) = f\ f(S) for all SC A.
This theorem is the poset version of the adjoint functor theorem. Folklore: see Johnstone
[16, Chapter I, Section 4, Paragraph 2] or Joyal and Tierney [17, Chapter 1, Section 1].
Proof. A proof for 1 follows; the other items are similar. Define g : B > A by
9P) = \J{aeA: f(a) < b}.
Since / preserves suprema, f(g(b)) < b. Thus, g(b) is the largest a such that /(a) In adjoint (resp. Galois) connections, /(a) and g(b) can be defined as suprema or
intima. There is a sort of converse to this; one can view the supremum as an adjoint to a
particular map. See Lemma 5.1.1.
2.3 Natural Transformations
Maps compare objects in a category, functors compare categories, and natural trans
formations compare functors. This section contains no new results; results and some expo
sition are paraphrased from Herrlich and Strecker [13, Section 13].
Definition 2.3.1. Let F, G : A > 3 be functors. A natural transformation a : F > G is
a rule which assigns a map aA : FA > GA to each A A such that if / : A > B is an
A-map the following diagram commutes.
Ff
FA >FB
r\A
r]B
GA
Gf
GB


18
Construction 2.3.2. Let F, G, H : X > A be functors, a : F > G and f3 : G > H natural
transformations. (The situation is drawn in the diagram below.)
F
G
Q
H
' J
0
Then the assignment (/3a)A = (/3A)(aA) is a natural transformation.
Proof. Let / : A > B be a map. Since a and (3 are natural, each square below commutes.
FA
Ff
*FB
GA-
Gf
GB
HA
Hf
HB
Therefore, the outside rectangle commutes; hence, (3a is natural.
Call /3a the vertical composition of a and (3.
Construction 2.3.3. If F : A > B and G, H : 3 G are functors and a : G > H is a
natural transformation, then (aF)A := a(FA) is a natural transformation aF : GF > HF.
Construction 2.3.4. If F : H > C and G, H : A H are functors and a : G > H is a
natural transformation, then (Fa)A := F(aA) is a natural transformation Fa : FG > FH.
Construction 2.3.5. Suppose F, G : A > B and H,J:T>*G are functors, and a : F *
G, (3 : H > J are natural transformations. (The situation is drawn below.)
F H
0 e


19
Then for each object A Obj(.A) the following square commutes.
FHA FPA > FJA
aHA
aJA
GHAG0A >GJA
The assignment (/3 a)A := (aJA)(F/3A) (Gf3A)(aHA) is a natural transformation.
Proof. The square commutes because a is a natural transformation; to see this, one applies
the natural property at the map (3A : HA + JA. To prove the natural property of 0a,
we use the squares which define 0a A and 0aB. Compare the corners of the squares using
maps obtained from / by application of the functors FH, GH, FJ, and GJ. The resulting
commutative cube shows /3a is a natural transformation.
Call 0 a the horizontal composition of a and 0.
Definition 2.3.6. A natural equivalence a : F G is a natural transformation such that
each component aA is an isomorphism. Given functors F and G are naturally equivalent if
there is a natural equivalence a : F G.
Categories A and B are equivalent if there are functors F : A > T and G : T> > A
such that FG is naturally equivalent to ids and GF is naturally equivalent to id^.
A and are dual if there exist contravariant functors F : A B and G : B > A
such that FG is naturally equivalent to ids and GF is naturally equivalent to id^.
2.4 (Co)Limits
For intuition, it is useful to view the objects of a category as a preordered class, with
the preorder A < B if and only if there is a map f : A B. Categories are more complex
than preordered classes, because there may be many maps / which manifest A < B.
Definition and Remarks 2.4.1. Let A be a category. A diagram D in A (a.k.a. a small
subcategory of A) is a set of objects Obj(D) C Obj(A) and maps Map(D) C Map(A)


20
between them; for technical reasons one requires that if f,g G Map(D) and fg is defined
then fg G Map(D), and that for all A G Obj(D), idA Map(D). This definition of a diagram
is equivalent to, but differs from the one in the majority of the literature; see MacLane [21],
Borceux [6], or Herrlich and Strecker [13] for the standard definition.
Suppose D is a diagram. A source (S, {s^ : S A : A G Obj(£))}) for D consists of
S G Obj(A) and maps sA : S -> A such that if / : A -> A' is a map in D, fsA = sA>. A source
for D is a lower bound compatible with all maps in D. A source (L, {£A : A G Obj(Z))})
for D is the limit of D if whenever (5, : A G Obj(D)}) is a source, there is a unique map
c : S L such that for each A G Obj(D), sA iAc.
Dually, define a sink, or cosource for D to be an object S together with maps iA :
A > S (for A G Obj(D)) such that for each f : A> A' in Map(D), iA = iA'f.
A colimit (C, {¡jla : A G Obj(D)}) for D is a sink for D, such that if (5, {m : A G
Obj(D)}) is any sink there is a unique map c : C > S such that for each A G Obj(D),
Ha = ciA.
Note that limits and colimits of D are unique up to compatible isomorphism. To
prove this (for limits), suppose (L,£A) and (L',£'A) are both limits for D. The limit property
guarantees that there are unique compatible maps c : L > L' and d : L' > L. But
cd : L' > L' and dc : L -> L are both compatible maps. By uniqueness, cd = id'£ and
dc = idL. From now on, we shall use the definite article when writing about limits and
colimits.
Example 2.4.2. Let us consider Set. If X is a set of sets, form a diagram containing
all members of X and no functions. The limit for this diagram is the Cartesian product
(nx,7rx).
Recall that ¡Q X contains all functions / : X > |J X,with the feature that f(X) G X
for all X G X. Such / are called choice functions, because they choose one member of each
X G X. The functions nx 'WX X are defined by nx(f) = f(X).


21
If there is a source (S, {s* : S -> X : X X}), there is a unique function c : S -> f] ^
making
commute for each X. It is defined by c(a)(X) sx(a).
Motivated by this example, one defines the category-theoretic product of a set X of
objects (in any category) as the limit of the diagram containing all members of X and no
maps. In most categories of sets with structure products (exist and) look like products in
Set, with suitable structure added.
In Set, the colimit of the diagram containing all members of X and no functions,
is the disjoint union ]J X, with inclusion maps px X i JJ X. The colimit property is
satisfied by (JJ X, px) because if there is a sink (C, ix), the function c : ]J X C defined
by c(x) = ix(x) for the unique X X containing x is the unique compatible map.
In categories other than Set, coproducts are defined identically. Usually the coprod
uct of a set X of ^.-objects is the *A-object freely generated by JJ X.
Example 2.4.3. The [co]equalizer of /, g : A > B is the [co]limit of the diagram containing
objects A and B along with maps / and g. The notation for equalizers is customarily
simplified by omitting the source map to B. Previously, i : E > A was defined to be the
equalizer if fi = gi and i factors through any other map which right identifies / and g. i
is the source map to A. The source map to B is redundant: it must be fi = gi : E B.
Similar notational economy is applied to coequalizers.


22
Example 2.4.4. Let us consider the diagram
A
f
B *C.
(The diagram also contains identities for all objects, but for brevity these are omitted.) The
limit (B Xc A, nA, ttb) of this diagram is called the pullback of / along g, or the pullback of g
along /. To be explicit, fnA = nBg and if (Q,qA : Q A,qB : Q > B) satisfies fqA = qBg,
then there is a unique map i : Q > B Xp A.
In Set,
B xc A = {(b, c) e B x A: g(b) = /(c)},
and the projection maps n are the restrictions of the projections from the cartesian product.
Pullbacks in any concrete category A equipped with a limit preserving faithful functor
U : A Set are computed identically. Two particular instances of pullbacks deserve
special attention.
First, let / : A C be any map and g : S > C be a subset inclusion of S C C.
Note that (s, a) S xc A if and only if g(s) = /(a); suppressing mention of g, this reads
S xc A {(s,6) : f(b) = s}. Thus, this pullback is canonically isomorphic to the preimage
of S under /. This example partially motivates the name pullback.
Second, let / : A > B be any map and consider the pullback of the diagram,
A
f
A *B
i.e., the pullback of / along itself. Using the computation for pullbacks in Set, given above,
A xB A {(a, a') A x A : f(a) = /(a')}.


23
This relation on A is often called the kernel of /. Thus, one calls (A x# A,nleft,nright) the
kernel pair of /. More in-depth discussion of pullbacks is given in Borceux [6, Volume 1,
Section 2.5] and Herrlich and Strecker [13, Section 21].
Definition and Remarks 2.4.5. A category is said to be complete if each diagram has a
limit; it is said to be cocomplete if each diagram has a colimit.
Unlike the situation for posets, a category may be complete without being cocomplete
and vice versa. (See Herrlich and Strecher [13, Section 23, pl61ff] for a detailed discussion
of this and related issues.)
Consider a functor F : A > B. If D is a diagram in A, there is a diagram FD with
Obj (FD) = {FA : A e Obj(D)} and Map (FD) = {Ff : f G Map(D)}.
Let D be a diagram in A. A functor F : A > B preserves the limit of D, if whenever
the limit (L,£a) exists in A, (DL, D(iA)) is the limit of FD. F preserves limits if for any
diagram D, F preserves the limit of D.
If G : A B is contravariant, D is a diagram in A, and (L, {£A : A G Obj(D)}) is
the limit of D, then (GL, {GiA : GA > GL}) is a sink for GD. If (GL, {G£A : GA GL})
is the colimit for GD, then G takes limits to colimits. Analogously, may G take colimits to
limits.
2.5 Adjoint Functors
There are several useful concepts of adjoint connections between categories. There is
a strong analogy between preordered sets and categories; any category may be preordered
by
A^B <=> 3/ : A -> B.
Categories are more complex, because many maps could manifest A A B. We begin with
the concept analogous to 2.2.3.
Definition and Remarks 2.5.1. A (functorial) Galois connection between categories A
and 'B consists of contravariant functors T : A > G : B > A, together with natural


24
transformations
77 : id* GF
and
e:W3 > FG
such that (Fr]A)(eFA) = id^ and (GeB)(r]GB) = ides for each A 6 Obj(A) and B
Obj(33); these equations are the so-called triangle identities. (Alternate terminology: if
(F,G,r],e) is a functorial Galois connection, then F and G are adjoint on the right.)
1. The categories fix(r¡) and fix(e) containing all objects such that r]A (resp eB) is an
isomorphism are dual. [3, Section 4, Lemma 1]
2. There is a natural bijection A(A, GB) (5, FA) given by
(f : A GB) H-> F(f)(eB) : B > FA.
The inverse map is
(g : B > FA) >-> G(f)(vA) : A -> GB.
There are two (identical) calculations required to check that the functions are mutually
inverse. One is summarized by the diagram below.
GFA (GFf) > GFGB GB
The square commutes because 77 is natural. The triangle commutes because of the
identity ides = (GeB)(r¡GB). The reader may formulate and check what is meant by
naturality of the bijection.
3.F and G both take colimits to limits.


25
4. Each rjA has the following universal property: if / : A GB, there is a unique map
/ = F(f)(eB) : B EM which makes the following diagram commute:
A GFA
Each eB has the analogous universal property.
Much of the literature just deals with adjoint connections, where both functors are covariant.
By duality, any such result can be translated in terms of Galois connections. Banaschewski
and Bruns [3] includes a reasonably thorough expository section on functors which are adjoint
on the right.
The (functorial) Galois connection concept is symmetric, but the contravariance of
the functors involved is sometimes awkward. An analogous asymmetric concept, with the
functors both covariant is described below.
Definition and Remarks 2.5.2. A (functorial) adjoint connection between catego-ries A
and 3 consists of functors F : A > 3 and G : 3 A and natural transformations
r] : idyi GF and e : FG > id such that (GeB)(r¡GB) = ides and (eFA)(Fr¡A) = id^;
these equations are the so-called triangle identities. In this situation, one also says F and
G are adjoint functors, F is the left adjoint and G is the right adjoint. The basic
properties of functorial adjoint connections closely correspond to the basic properties of
functorial Galois connections.
1. The categories fix(r/) and fix(e) containing all objects such that 7/.4 (resp eB) is an
isomorphism are equivalent.
2. There is a natural bijection A(A, GB) 3(FA, B) given by
(f : A* GB) t-> (eB)F(f) : FA > B.


26
The inverse map is
(g : B FA) G(/)foi4) :A^GB.
3. F preserves colimits; G preserves limits. Borceux [6, Volume 1, 3.2.2]
4. Each rjA has the following universal property: if / : A > GB, there is a unique map
/ = (eB)F(f) : FA B which makes the following diagram commute:
A >GFA
Each eB has the analogous universal property: if / : FA > B, there is a unique
/ = G(f)(rjA) : A > GB such that
commutes. Any functor for which there is a natural transformation rj with the above
universal property is part of an adjoint connection.
For discussions of adjoint functors, see MacLane [21, Chapter IV], Herrlich and Strecker [13,
Sections 26, 27, 28], or Borceux [6, Volume 1, Chapter 3].
There is a criterion for determining when functors are adjoints, which corresponds to
Theorem 2.2.6.
Theorem 2.5.3. (Adjoint Functor Theorem, [21, V.6.2],) Let A be a complete category.
A functor G : A > B has a left adjoint if and only if
1. G preserves limits, and


27
2. (solution set condition) for each B 3 there is a set I and an I-indexed family of
maps fi'.B> GAi such that any map f : B > GA can be written as h = (Gt)fi for
some i 6 I and t : A¡ > A.


CHAPTER 3
ALGEBRAS OF A MONAD
Monads (a.k.a. triples, a.k.a. standard constructions) and their (Eilenberg-Moore)
algebras provide a concise formulation of many important categorical aspects of universal
algebra. The category of algebras for a monad has special properties, which are summarized
in Section 3.1; a category of algebras is often complete and cocomplete, and always has
free objects. Each (functorial) adjoint connection induces a monad; the correspondence
between adjunctions and monads is discussed in Section 3.2. In Section 3.3, the question of
when an adjoint connection connects A to a category of algebras is addressed. In Section
3.4, the question when does the composite of two monadic adjunctions yield a monadic
adjunction? is discussed.
The results in this chapter are reasonably well known. MacLane [21, Chapter VI],
Borceux [6, Volume 2, Chapter 4], Barr and Wells [4, Chapters 3 and 9], Manes [23] and
the introduction to the seminar notes [1] contain good expositions of monads from various
perspectives.
3.1 Categories of Algebras
Definition 3.1.1. A monad T = (T, rj, /i) on A consists of a functor
T:A^A,
a natural transformation
V i T,
and a natural transformation
/i: T2 > T,
28


29
such that the following identities (expressed by commutative diagrams) hold: the unit laws
T2
(In these diagrams, Tn denotes the n-fold composite of T with itself.)
Intuitively, TA is the free object on A; r]A is the insertion of variables map; fjA
is the semantic composition, i.e., a map which allows one to view a polynomial with
polynomial variables as a polynomial. See Example 3.2.2 for a concrete illustration of the
roles of T, 77 and ¡1.
Definition and Remarks 3.1.2. Let T be a monad on A. A T-algebra (A, a) consists of
A Obj(A) and a : TA > A (the so-called structure map) such that a(r)A) kU (unit
law) and
T2A *-TA
M a
TA


30
(associative law) commutes. A homomorphism / : (A, a) > (B, b) of T-algebras is an A-map
/ such that
Tf
TA *-TB
a b
commutes. The category of all T-algebras and T-algebra homomorphisms is denoted AT.
There is a forgetful functor UT : AT + A; it is defined by UT(A,a) = A and
UT(f) = f. UT has a left adjoint FT : A AT defined by
Ft(A) = (TA,pA)
and
FT(f) = Tf.
The associated natural transformations are
r]T = r,: id^ UtFt = T
and
eT : FtUt id^r : eT(A, a) := a.
[21, VI.2.Theorem 1]
Limits in Ar are computed in A in the following sense.
Proposition 3.1.3. [4, 3.3.4], [6, Volume 2, 4.3.1] Suppose (T,r/,/x) is a monad on A. If D
is a diagram in Ar such that UT(D) has a limit (L,pu(A)), then there is a unique structure
map I : TL L such that each pu(A) a T-algebra homomorphism.
Proof. Let D be a diagram in Ar, such that UTD has a limit (L, IA). For each A G Obj(T),
name the structure map sA : TA -+ A. The goal is to produce a structure map s :TL > L
such that each tA : L A is a T-algebra map.


31
The requirement that each is T-algebra map amounts to: for each A, the diagram
below commutes.
FL
SA
Since (L,£a) is a source for UD, (FL, sa(FIa)) is a source for UD. Thus, there is a unique
A map s : FL > L making each diagram above commute.
Checking the unit law, s(iL) = id: the diagram
commutes using the definition of s, and that i is a natural transformation. Thus, for each
A, a as(L). Prom the uniqueness of the map from the limit of D to any other source
for D, it follows that id = s(iL).
A similar comparision of squares diagram can be used to verify that the algebra
associative law holds.
Corollary 3.1.4. If A is complete and T is a monad on A, then AT is complete.
If A is cocomplete, AT is often also cocomplete. The following theorem was originally
proved with fewer hypotheses in Linton [20], Other expositions are given in Borceux [6,
Volume 2, 4.3.4] and Barr and Wells [4, Section 9.3].
Theorem 3.1.5. Let A be cocomplete and T be a monad on T. AT is cocomplete if and
only if AJ has coequalizers.


32
3.2 Adjoint Connections induce Monads
The following proposition gives a correspondence between monads and adjoint con
nections. It should be emphasized that the correspondence is not bijective. Each monad
gives rise to a unique adjoint connection, but in general many adjunctions induce the same
monad.
Proposition 3.2.1. Correspondence between monads and adjoint connections
1. Let F : A > 3, G : CB > A, r¡ : kU > GF, e : FG > id be an adjoint connection.
Then (GF,r¡,GeF) is a monad on A.
2. If T = (T, T], /i) is a monad, then
GtFt = T,
T
T1 =1h
and
GTeTFT = [i.
The proof of the preceding Proposition consists of verifying identities: the triangle
identities for adjoint connections imply the unit laws; the associative law holds because
the square defining horizontal compositions commutes. Details are given in MacLane [21,
VI. 2. Theorem 1],
This correspondence allows construction of many examples of monads. Often, but not
always, a naturally occurring adjoint connection corresponds to the category of algebras
over the induced monads.
Example 3.2.2. Consider the category Grp of groups. The forgetful functor Ucrp Grp
Set and the free group functor FgtP : Grp Set form an adjoint connection between Grp
and Set. Recall that ForpiX) is the set of all reduced words


33
where x 6 X, s 6 Z; a word is reduced if for all i with 2 < i < n, x,_i / x. The operation
on TGrp(A') is concatenation of words, followed by reduction. T = ^Grp-^Grp is the functor
part of the induced monad; 77 : idset > T is the natural transformation whose component at
X sends x X to word x1; if w*1 u>n 6 T2(X) and for each z, = x^1 then
(ma-)K' ...<) = (*5y~*Ksr-i
_6i,m(m
x
n.m(n)\a,
n,m(n)
Each group G is an T-algebra; the group multiplication and inversion give a map from the free
group on the underlying set of G to G. Conversely, each T-algebra structure gives a group
multiplication and inversion. Grp is equivalent to the category of Setr. (This example is
typical in the sense that any category of finitary algebras in the sense of universal algebra
- is also a category of monad algebras. The parts T, rj, and ¡1 of a monad generally have the
same roles as in this example.)
Example 3.2.3. Some (non-trivial) adjunctions involving Set induce the trivial monad.
For example, consider the category Top of topological spaces and continuous maps. The
forgetful functor
i/rbp : Top > Set
has a left adjoint
Ftop : Set -> Top,
which sends a set X to the discrete topological space with underlying set X. One easily
checks that T = GTopFTop is the identity functor on Set, and that all associated natural
transformations are identities.
Definition 3.2.4. Suppose F : A > G:B* A, rj : id^ -7 GF, and e : FG > id
forms an adjoint connection. By Proposition 3.2.1, the adjoint connection induces a monad
(T = GF, 77, GfiF). If B is equivalent to AT, then we say the adjoint connection (sometimes
just the right adjoint G) is monadic. Often a concrete category B has a canonical forgetful
functor G : B > A\ in this case, one may even say B is monadic over A omitting mention
of G.


34
These examples illustrate the qualitatively different behaviors of monadic and non-
monadic adjoint connections. Monadic categories are determined by the combinatorial struc
ture of the free algebra functor FT. Most relational categories like Top, P, the category
of graphs, etc. have free functors, but these free functors do not add any structure to the
underlying set; they merely attach the most discrete possible relation to the given set.
3.3 Detecting Categories of Algebras
Because of the special properties of AT, the question of when an adjoint connection
is monadic has great practical importance.
Remark 3.3.1. Let (A, a) be a T-algebra. Because of the associative and unit laws, and
because 77 is natural, the following equations hold:
(nat) (Ta)(r[TA) (rp4)a,
(unitl) (jA)(t¡TA) = idTA,
(unit2) a(r)A) = id, and
(assoc) a(Ta) = a(fiA).
These equations imply that a coeq(¡iA,Ta)\ for if b : TA B right-identifies ¡1A and Ta,
then b(r]A) : A B and
b(r¡A)a = b(Ta){T]TA) by (nat)
= b(fiA)(r]TA) since b right identifies
b by (unitl).
One verifies that / = b(r¡A) the unique map / : A > B with fa = 6; for if fa = b, then
/ = fa(rjA) = b{r¡A).
These equations give a great deal of information about a. Thus, the key hypothesis
in Becks theorem the criterion for when an adjunction is monadic is the preservation
and reflection coequalizers obeying the equations described above.
Using the above described equations for motivation we offer the following definitions.


35
Definition and Remarks 3.3.2. Consider maps f,g:A>B.
1. If e : B + C has the feature that for any functor F, Fe = coeq(Ff, Fg) then we say e
is an absolute coequalizer.
2. If there are maps e : B > C, sc C > B and sb : B > A such that
fsB
kU = gsB,
idc = esc,
ef = eg,
then we say e is a split coequalizer.
Note that for every T-algebra {A, a), a is a split coequalizer of Ta and pA. It is also easy to
see that every split coequalizer is absolute.
The following theorem is due to Beck (unpublished). Linton [19] contains a detailed
discussion of variations on the theorem. The theorem was originally phrased in terms of
split coequalizers only; Pare [26] refined the theorem to include the absolute coequalizer
condition. Many variations on the hypotheses exist; the version stated here is found in
MacLane [21, VI.7.1], Other expositions of the theorem may be found in Barr and Wells [4,
Section 3.3] and Borceux [6, Volume 2, Section 4.4].
Theorem 3.3.3. Let (F : A > 25, G : 3 > A, r?, e) be an adjoint connection, and T = GF,
rj = g, and g = GeF be the associated monad. The following conditions are equivalent:
1. The adjunction (F,G,r],e) is monadic.
2. If f, g : A B Map(!B) and the pair (Gf,Gg) has an absolute coequalizer e!, then
e = coeq(f,g) exists and Ge e'.


36
3. If f,g : A > B e Map(!B) and the pair (Gf,Gg) has a split coequalizer e', then
e = coeq(/, g) exists and Ge = e'.
Recognizing monadic adjunctions is complex because compositions of monadic func
tors are not generally monadic.
Example 3.3.4. The category Ab of abelian groups is monadic over Set, and tfAb (see
Example 2.1.3) is monadic over Ab, because any reflection is monadic. Each free abelian
group is torsion free, so the monad on Set induced by the free abelian group functor is the
same as the monad induced by the free torsion free abelian group functor. Thus, if tfAb
were monadic over set, then tfAb and Ab would be equivalent categories both equivalent
to a suitable category of monad algebras. The categories tfAb and Ab are obviously not
equivalent. (The preceding example is summarized from Borceux [6, Volume 2, Example
4.6.4].)
If G : A Set, the hypotheses of Becks theorem can be reformulated to make it
easier to check whether G is monadic.
Theorem 3.3.5. [11, Theorem 4.2] Let G : A > Set be a functor with a left adjoint.
Suppose that A is complete and has coequalizers. The following are equivalent:
1. G is monadic.
2. G satisfies the following conditions
G preserves and reflects regular epimorphisms;
if f : GA > X is an isomorphism, then there is a unique map g : A > B such
that Gg f;
G reflects kernel pairs.
The following lemma provides useful information concerning when a functor between
two categories of algebras is monadic.


37
Lemma 3.3.6. [6, Volume 2, Corollary 4.5.7] Suppose U : 23 > A, V : G > A, and
Q : B > C are functors. IfU VQ, U and V are monadic, and 23 has coequalizers, then Q
is monadic. In particular, Q has a left adjoint.
3.4 Distributive Laws
Compositions of monadic adjunctions are not generally monadic; when a composition
of monadic adjunctions is monadic, it indicates a distributive law between the two structures.
The following section summarizes the results later needed from Becks [5]; this exposition
follows Becks notation, except that composition of functions here will read right-to-left.
For the duration of the section, assume there are two monads T = (T, t]t, pT) and S =
(S, r]s, ps) over some base category A. For any monad, we use F and U, with superscripts for
the name of the monad, to denote the free algebra functor and forgetful functor associated
with a monad, respectively. See 3.1.2 for the definitions of F and U.
This material is rather abstract, so it helps to have an example in mind: after each
definition and theorem, we will illustrate what it means using S the free monoid monad
over Set and T the free abelian group monad over Set; the composite monoid TS gives
the free ring. For a set X, SX consists of all strings from X, with concatenation as the
operation: 5 acts on functions by
Sf(xix2 xn) = f(x1)f(x2) f(xn).
TX consists of all formal (finite) linear combinations of elements of X, with integer coeffi
cients. T acts on functions in the expected way.
Despite the concrete illustrations in terms of these monads, all theorems and defintions
will apply to any monads S and T.
Definition and Remarks 3.4.1. A distributive law of S over T is a natural transformation
£ : ST->TS


38
satisfying the compatibility conditions:
T S
If there is a distributive law £ : ST TS of S over T, then the composite monad is TS =
(TS,r]TS := r]TT]s,pTS := pT ps(T£S)). In the definitions of r)TS and pTS, juxtaposition
denotes horizontal composition of natural transformations. The verification that TS actually
defines a monad is omitted; for more information see Beck [5],
In the case when S=free monoid and T=free abelian group, the natural trans
formation £ : ST > TS expresses a product-of-sums as a sum-of-products in the usual
way:
n 5^ aik n ac(k),k
keK iei c keK
where c ranges over all choice functions c : K > I. It is a somewhat enlightening exercise
to check that defined this way is a natural transformation ST TS.
A distributive law may of T over S also be viewed as a way of lifting T to a monad
over S.


39
Definition 3.4.2. T has a lifting into ./Is, if there is a monad T such that TUS = UST,
UsrjT r]TUs and Usp^ = pTUs. To sketch the situation, T is a lifting of T onto if
commutes.
Theorem 3.4.3. There is a bijective correspondence between liftings of T to /Is and dis
tributive laws of S over T.
The proof is outlined in Beck [5]. The correspondence is defined as follows: if i is a
distributive law, and (A, a) is an 5-algebra, then T is defined as follows:
T(A,a) = (TA, (Ta){iA)),
r)T{A, a) = t]TA : (A, a) -+ T(A, a),
and
fiT(A, a) = pTA : TT(A, a) T(A, a).
Consider the following diagram:
the square commutes because r¡T is a natural transformation; the triangle commutes because
l is compatible with rf. Since the perimeter of the diagram commutes, rf A is an 5-algebra


40
map, A similar diagram, which uses the compatibility between I and pT, shows pF is an
5-algebra map. Since the underlying maps are defined in A and obey the monad laws, tjt
and pF also obey the monad laws. If T is a lifting of T over As, I is defined to be the
following composition:
ST STS = STUsFs = USFSUSTFS uSfFfs USTFS = ST.
(In the above equation, es is the counit of the adjunction (Fs, Us, r/5, es); es(A. a) = a.)
After some detailed computation, one verfies that this is a distributive law, and that the
correspondences described are mutually inverse.
Corollary 3.4.4. If T has a lifting to As, then there is a composite monad TS.
Theorem 3.4.5. Suppose I: ST > TS is a distributive law. The categories ATS and (AS)T
are equivalent.
Let us consider this in more detail. An object in (AS)T consists of an A-object,
A, along with an 5-structure as : SA > A, T-structure ax : TA + A, and 5-structure
t: ST A > A for TA such that the following diagram commutes.
SciT
ST A 1- SA
t aS
TA >A
Given a T5-algbra {A, a), A has a T-structure aT map defined by
aT = a(Tr]SA) :TA->A
and an 5-structure map as defined by
as = a(r]TSA) : SA > A.


41
The following diagram which expresses the distributivity of the structure maps commutes.
Thus we have a map ATS > (AS)T. given by,
(A, a) e ATS i (A, ar, as,Tas(£A))
which is functorial, because both rj1' and rf are natural.
The inverse functor {AS)T > ATS, maps (A,ar,as,t) to asS(ar) art. The verifi
cations for these assertions is given in Beck [5],


CHAPTER 4
GENERATING SUBMONADS
In this chapter, a technique for creating monads is laid out; before proceeding formally,
let us consider a rough outline of the technique. Suppose (T, r], /r) is a monad on a reasonable
category, and that / : F T is a subfunctor of T. The goal is to extend F to a submonad
of T.
Intuitively speaking, F is a natural collection of polynomials. One should therefore
require that for all A, (r/A)(A) C F(A) i.e., that for each A and a G A, FA contains the
constant polynomial with value a. F is not generally closed under composition; i.e., it
is not generally true that (^,A)(F2A) C A. To correct this problem, we start with F and
iteratively add polynomials obtained by composing members of FA.
The purpose of the chapter is to formalize the preceding vague outline. The author
abstracted and clarified [25, Proposition 3.4], which may be viewed as a special case of this
result. Thus, the main results of this chapter are called Meseguers Lemmas 4.2.1, 4.2.2,
4.2.6, and 4.2.7 to acknowledge the analogy which prompted the technique. The author
believes the general formulation of the technique is new.
Section 4.1 details requirements on a reasonable category, explains precisely what
is meant by subfunctor, and gives methods for constructing subfunctors. Section 4.2 gives
a proof of Meseguers Lemmas. Section 4.3 describes an example showing the necessity of a
technical hypothesis of the lemmas.
4.1 Subfunctors
This section describes a general theory of subfunctors. A subfunctor of F : X > Set
is a natural transformation r] : E + F such that each component is a subset inclusion. The
42


43
concept subfunctor is quite useful, but in some categories such as P, Top, and Locale
(explained in Example 4.1.2), there is either
no obvious meaning for subset inclusion (in Locale), or
more than one possible structure on each subset (in P and Top).
The following section expounds a theory in which subfunctor means natural transforma
tion whose components are all extremal mono.
Axioms 4.1.1. Throughout, the base category A is assumed to have the following features:
1.A is complete.
2.A is an (epi, extremal mono) category. Recall that an (epi, extremal mono) category
is a category with the features that
For each map /, it is possible to factor f as f me where e is epi and m is
extremal mono. This factorization is unique, in the sense that if / = m'e' is
another way of writing / as an epi followed by an extremal mono, there is an
isomorphism i such that
e
commutes.
if a and b are extremal mono and ab is defined, then ab is extremal mono.
3.A is extremally-wellpowered. This means: for each A, there is a set
{/a S\-> A}
of extremal monomorphisms, such that if m : T A is any extremal monomorphism,
there is an index A and isomorphism i such that f\ mi.


44
If A has these features, then it is an SF-category. (SF stands for subfunctor.)
Example 4.1.1. By Herrlich and Strecker [13, 34.5], any wellpowered category with in
tersections and equalizers is an (epi, extremal mono) category. So if A is complete and
wellpowered then A is an SF-category.
It follows that P, Set, Top and practically any reasonable category of topological
spaces, is an SF-category.
Example 4.1.2. The category of frames i.e., complete lattices in which
a a\J S = \J {a A s : s E S}
holds for all elements a and subsets S is complete and cocomplete. This category also has
(regular epi, mono) factorizations. Thus, Locale, the dual category to the category of frames,
is an SF-category. Locale is extremally-wellpowered, but not wellpowered. The authors
interest in locales motivated him to use the given definition (which only requires extremal
wellpoweredness) for SF-category rather than defining -SF-category to mean complete and
wellpowered, so that the theory of subfunctors would apply to localic subfunctors in addition
to the previous examples. (For background on frames and their relation to point-set topology,
see Isbell [15], Johnstone [16], Joyal and Tierney[17] and Madden [22].)
Lemma 4.1.3. Assume that A is an (epi, extremal mono) category. Then
1. (diagonalization) [13, 33.3] If ge = mf, where e is epi and m is extremal mono, then
there exists k such that mk = g and ke = f.
2. [13, 34.2(2)] Any intersection of extremal subobjects is extremal.
3.[13, 34.2(3)] Pullbacks of extremal monomorphisms are extremal mono.
Definition and Remarks 4.1.4. Defining Sub(A) the lattice of extremal subob
jects: In an SF-category A, the lattice of (equivalence classes) of extremal subobjects has
particularly nice structural features.


45
An extremal subobject of A is an extremal monomorphism s : S > A. Define a
preorder on the class of extremal subobjects by
(s : S A) C (t : T + A) 4= 3c : S T, s tc.
For brevity, one often only mentions the map or object part of an extremal subobject. To
avoid confusion, the same letter will be used to denote both parts, with the lower case letter
used for the function.
Since t is mono, there is at most one map c which manifests s C t. If s C t and t C s,
then there are cx and c2 such that s tc\ and t sc2. So t = fc!c2 and s = sc2cj. Since
c2ci shows s C s, c2Ci = idg. Similarly, cic2 = idy. One identifies extremal subobjects s and
t if s C t and f C s, or equivalently, when there is an isomorphism c such that s tc. The
set of equi valance classes under this relation is denoted Sub (A).
Any map c, which exhibits s C t, is extremal mono. For if ca cb, then sa =
tea = teb = sb, so (since s is mono) a = 6; so c is mono. If c = me, where e is epi, then
s = tc = tme. Since s is extremal mono, e must be an isomorphism. Thus c is extremal
mono.
Recall that the intersection of a set M of monomorphisms with a common codomain
is the limit of the diagram generated by M. Since A is complete and intersections of extremal
monomorphisms are extremal, Sub(A) is a complete lattice, with meet operation f) and
join operation |J. In general, joins in Sub(A) are not disjoint unions; in general, (JA is
computed using
(jAi = f){A,:Vi, ACA'}.
(See Borceux [6, Volume 1, 4.2.2, 4.2.3, 4.2.4].)
A map f : A B induces an adjoint connection between Sub(A) and Sub(B). (See
Borceux [6, Volume 1, 4.4.6]; Borceuxs results are phrased in terms of strong monomor
phisms. Under our assumptions a map is strong mono if and only if it is extremal mono.)
If s : S A Sub(A), use the (epi, extremal mono) factorization to obtain a unique


46
s' 6 Sub(£?) such that
A
f
B
S
S *f+1(S)
commutes. The notation /+1(S) is used for the image of S, to remind the reader of the
analogy to ordinary set theoretic images of subsets under maps. The map (s : S > A) i
(s' : S' = f+1(S) > B) is the left adjoint. The right adjoint f~l : Sub(S) > Sub(A) is
obtained by taking the pullback of s : S > B along /. Thus,
f+1{S) CT ^ SCf-\T),
/+l(Us<)=lbw(s<>'
and
r1(f]si) = f]r1(si).
Definition and Remarks 4.1.5. Let F : X A be a functor. A subfunctor E of F is a
rule that selects an extremal subobject eX : EX > FX for each X G Obj(X) such that if
f : X >Y, then
(Ff)+1(EX) C EY.
Any such assignment E of F gives rise to a functor E : X > A. Ef : EX > EY is defined
to be the composition EX (Ff)+1(EX) C EY.
Since eY : EY > FY is mono, Ef is uniquely determined by the condition that
Ff
FX >-FY
eX
eY
EX *EY
commutes. Thus, (Ff)+1(EX) C EY implies that Ef can be defined to make the above
square commute.


47
The converse also holds: if there is a map Ef such that the square commutes, then
(Ff)+1(EX) C EY. To prove this, suppose there is a map Ef which makes the square
commute. Use the unique factorization Ef = ab where a : A EY is extremal mono and b
is epi. Note that (Ff)(eX) = (eY)(Ef) (eY)ab gives a factorization of (Ff)(eX) into an
epi b followed by an extremal mono (eY)a. Thus,
[{eY)a : A > FT] [(Ff)+1(EX) -* FT]
and (Ff)+1(EX) C EY. Thus, a subfunctor is exactly a natural transformation whose
components are all extremal monomorphisms.
Let Subfun(F) denote the class of subfuctors of F : X > A. Define a preorder on
Subfun(F) by
E\ C F2 <=> W1 Obj(.A), Ei{A) C E2(A).
If Ei C E2, then, for each A, there is a unique extremal monomorphism cA such that
e\A (e2A)(cA). One easily verifies that cA are the components of a natural transformation
c : E\ > E2; in fact, c : E\ > E2 is a subfunctor.
The following constructions show that the class Subfun(F) behaves very much like
Sub(j4).
Construction 4.1.6. Subfun(F) is complete. If {Ei : i 1} is any class of sub functors,
there is a supremum (JF and infimum P) Et. The supremum is given by (IJEi)(A)
UFj(j4). The infimum is given by (PF)(A) = P| Ei(A).
Proof. Since A is extremally wellpowered, for each A, the class {EfA) : i 1} has a
representative set. Thus, the object-by-object definitions for |J F, and fj Ex make sense. It
is obvious that if U F and p) F are subfunctors that they are the optimum bounds in the
subfunctor lattice.
Let f : X *Y be any map. A h-> P| EX(A) is a subfunctor because for each i I,
(F/)+1(P| E(A)) C (Ff)+\Ei(A)) C Ei(B),


48
so (F/nai^c a £(£?).
>11 Ei(A) is a subfunctor because,
(F/)+1(LJf(A)) = U(Ff)+l(EfiA)) C U^(5).
i i
a
Construction 4.1.7. Suppose a : F + G is a natural transformation. Then there is an
(order theoretic) adjoint conection
a+1 : Subfun(F) -> Subfun(G) : a-1.
The left adjoint a+1 is defined by (a+1E)(A) = (aA)+1(EA). The right adjoint a-1 is defined
by (a~1E)(A) = a~1(EA).
Proof The order on Subfun(F) is defined object-by-object, and whenever
f : A B,
f+1 and f~l form an adjoint connection between Sub(.4) and Sub(F). So it suffices to check
that a+1E and a~lE define subfunctors.


Let E Subfun(F), / : X > Y be a map, and consider the following diagram.
(aA)+1(EA) (aB)+1(EB)
The trapezoids (i) and (iii) are obtained by factoring (aB)(eB) and (aA)(eA), respectively;
in each case 2 is the epi part and e' is the extremal mono part. The square (iv) expresses
the naturality of a. The trapezoid (ii) expresses that e : E F is a subfunctor.
To show a+1E is a subfunctor of G, it suffices to show there is a map
k : (aA)+\EA) -* (aB)+1(EB)
that makes the top trapezoid commute. For this, use the diagonalization property from
Lemma 4.1.3. Define / = zB(Ef), g = (Gf)e'A, e = zA and m = e'B; note that ge mf
with e epi and m extremal mono. Thus, the diagonalization property guarantees the desired
k exists.
Assume / : A B. A similar diagram is used to verify that a_1E is a subfunctor
of F whenever E e Subfun(G). The missing map (aA) 1{EA) > (aB) X{EB) is obtained
from the pullback property of (aB)~1(EB).


50
Corollary 4.1.8. Let a : F > G be a natural transformation, and e : E > F be a subfunc
tor of F. Let e' : a+1E > F be the subfunctor described in Construction 4.1.7. There is a
natural transformation z : a+1E > F such that for each object A, (aA) (e'A)(zA) is the
(epi, extremal mono)-factorization of aA.
Proof. One defines e' and 2 as in the preceding proof. Examining the comparison of squares
diagram used to produce a+1E/, shows that 2 and e! are natural.
Construction 4.1.9. Assume E,F,G,H are functors A > A. Suppose that
1. a : E > F and (3 : G > H are sub functors,
2. Either E or F preserves extremal monomorphisms.
then the horizontal composition (3a : EG FH is a subfunctor.
Proof. The horizontal composition of natural transformations is always natural. Hypothesis
2 implies that, for any A, (3aA is an extremal monomorphism, because the class of extremal
monomorphisms is closed under composition and
((3a)A := (aHA)(E(3A) = (F0A)(aGA),
by definition of horizontal composition (see Construction 2.3.5).
4.2 Meseguers Lemmas
Lemma 4.2.1. Suppose A is an SF-category, {T,p,rf) is a monad on A, and F is a sub
functor ofT, such that (r]A)+1{A) C FA. There is a smallest subfunctor F of T such that
F C F C T and for all A Obj(yi)
(pA)(F2A) C FA.
If the equation
(fj,A)(G2A) C GA
holds for G, we say G is closed under p.


51
Proof. Let T denote the class of all subfunctors of T, that are larger than F and closed
under p. S' is nonempty because it contains T. By Construction 4.1.6, F = P|T exists.
Since S (pA)(S) is an order preserving map Sub(T2v4) Sub(2L4), so for each A
(pA)(F2A) C FA.
The preceding gives an easy candidate for the functor part of the monad generated
by F. One needs a more complex argument if one wants detailed information about the
natural transformations related to r¡ and /i which make F into a monad.
The proofs of the following lemmas require detailed computation. For clarity, the
goal of each paragraph in the proof is written in bold-face.
Lemma 4.2.2. (Assume notation and hypotheses from Lemma If.2.1.) There are natural
transformations : id^ F and m : F2 > F which make (F, n, m) a monad.
Proof. One defines four sequences of natural transformations:
1. f\ : F\ > T the subfunctor generated at stage A,
2. n\ : id^ F\ a natural transformation obtained from rj by suitably modifying the
domain and codomain,
3. m\ : Ff > F\+1 a natural transformation obtained from p by suitably modifying
domain and codomain, and
4. c\ : F\ > F\+i the inclusion.
Define F0 = F; use the notation f0 : F0 T. m0 and Co are defined according to the same
pattern that defines later ms and cs, which is described below.


52
Defining fX'.Fx>T (A > 0), mA : Fx* -* Fx+i, and cx : Fx > Fx+i. Assume
that fx has been previously defined. Consider the following diagram:
(M)+1(*?) U FXA
Given fx and ¡iA, the square is obtained by (epi, extremal mono)-factorization of the com
posite ([iA)(fxA). All maps in the triangle are extremal monomorphisms, obtained by
comparing subobjects of TA. Define
Fx+1A = (fxA)+\F¡A)UFxA
and
mx : FXA > Fx+iA
to be the map shown on the left side of the diagram. It follows from Constructions 4.1.7 and
4.1.6 that /A+i : FA+1 T is a subfunctor. Note that
{fx+i)(mxA) = (nA)(flA)
and mx is natural. For bookkeeping purposes, let us call
c\: Fx > Fx+1
the subfunctor which exhibits Fx C Fx+i.
If At is a limit ordinal, define
FK = \J{Fx : A < /t};


53
fK'-FK>T is a subfunctor by 4.1.6.
Defining the sequence of n* s: The following definition of nx is not recursive; nx
can be defined once we know fx, but the definition of f\ does not involve nx at all. Consider
the following commutative diagram of functors and natural transformations.
The natural transformations z and e are defined by condition that rj = ez is the (epi, extremal
mono)-factorization of 77, as described in 4.1.8. By assumption and the construction of the
sequence (Fx),
(r)idA)+1 CF0C Fx.
Let i\ denote the natural transformation such that e = fxiX) ix exists because e C i. Define
nx := i\z. Evidently,
V = fXnx
and
^a+i = cxnx.
What happens when the sequence terminates: Since A is extremally wellpow-
ered, for each A, the sequence (F\A)\ of subobjects of TA eventually terminates, say when
A = 1. Again using extremal wellpoweredness there is an ordinal, say k2, such that the
sequence (F\FK(A))x of subobjects of TFKlA terminates at k2. Define = nK2, f fK2
and m = mK2. These assignments give natural transformations; to check this, one considers
a map / : A B, and chooses k large enough that the subobject sequences (described
above) terminate for both A and B. It should be clear that each subobject sequence {FXA)x
terminates at FA.


54
Verifying that (F, n, m) is a monad. To prove the unit laws, it suffices that for
all A,
('mxA){FxnxA) = (mxA)(nxFxA) = cxA
(because, once the sequence terminates, the caAs become identity maps on F\A). Fix any
ordinal A. Naturality of fx and nx implies
(TnxA)(fxA) = (fxFxA)(FxnxA)
and
CnxTA){fxA) = (FxfxA)(nxFxA);
Note that 77 = fxnx and, by definition of horizontal composition,
flA = (TfxA)(fxFxA) = (fxTA)(FxfxA).
Calculating, one finds
(.flA){FxnxA) = (fxTA)(FxfxA)(FxnxA) def. noted above
= (fxTA)(Fxr]A) noted above
= (TrjA)(fxA). /a is natural
Thus,
UlA){FxnxA) (TVA)(fxA)
and (by a similar computation)
(fxA)(nxFxA) = (r,TA)(fxA).
To show (mxA)(FxnxA) = cxA, consider the following diagram:
F,A-I1^U FlA


55
the upper square commutes by an identity proved above; the lower square is the definition
of my, the upper triangle is trivial; the lower triangle is the unit law for (T, r¡, /i). Reading
the perimeter of the diagram, one finds
fx A = (fx+1A)(mxA)(FxnxA).
By uniqueness of maps manifesting inequalities between subobjects,
cxA = (mxA){FxnxA).
The diagram needed to prove
(mxA)(nxFxA) = cxA
is similar and omitted. This establishes the unit laws.
To prove the associative law for (F,,m), one may choose sufficently large A, then
draw a diagram comparing the associative squares for T and F, using appropriate powers of
fx to compare the corners. The comparison squares commute because of the definitions of
mx and horizontal composition.
Definition 4.2.3. Assume that A is cocomplete. In A, unions of chains are colimits, if
whenever D is a diagram, where Obj(D) consists of a chain of extremal subobjects of A and
Map(D) consists of all inclusions (in Sub(A)) that exist among elements of Obj(D), the map
c : colimD > A induced by colimit properties is an extremal monomorphism.
In this case, c : colimD > A is an extremal subobject which contains each element of
Obj(D); thus there is an extremal monomorphism colimD > (J Obj(D). Since (J Obj(D) is
the supremum of Obj(D) in the Sub(A), this map must be an isomorphism.
Remark 4.2.4. The condition that unions of chains are colimits is satisfied in many nat
urally occuring categories. For example, in relational categories like Top, P, and Set,
extremal subobjects are just subsets with the induced structure, so all unions are colimits.
Assume A is a category of finitary algebras. The (set theoretic)-union of a chain
(Ai) of subobjects is a subobject. The (set theoretic)-union A' of (A,) lies inside a smallest


56
extremal subobject, which by our notation is (J A. Evidently, the coproduct of the chain
(A) of subobjects is A'. So the issue about whether unions of chains are coproducts
amounts to checking whether A! = 1J A,. This depends on the delicate issue of whether
epimorphisms are surjective. If epis are not necessarily surjective, then 1J A, will be the
largest subobject in which A' is epi, which will generally be larger than A'.
Definition 4.2.5. As above, assume A is an SF-category, is a monad, and F is
a subfunctor of T which contains r?+1 id^. A partial algebra (A, a) is an object A equipped
with a map a : FA > A such that a(noA) = id^.
Let pAlg(Fo,no) denote the category of all partial algebras, with maps / : (A, a) >
(B, b) such that / : A > B is an A-map, and fa b(Fof).
Lemma 4.2.6. Continue with situation and hypotheses from f.2.2. Suppose A is cocomplete
and unions of chains are colimits. Suppose (A, a) is a partial algebra. The partial algebra
structure a extends to an F-algebra structure map a : FA > A if and only if for each ordinal
A there is a unique ax+\ making the diagram below commute.
Fl(A)
m\
*FX (A)
FX(A)^ -A
Proof. To prove a map a : FA > A exists, use transfinite induction. The hypothesis gives ax
at sucessor ordinals. To construct ax, when A is a limit ordinal use the coproduct property
of FXA Ua Fq(A).
Inductively one shows that
(ext) for each A, ax = aA+i(cAA) (each ax+i extends the preceding ax),
(unit) for each A, ax(nxA) = kU,


57
If (unit) holds for sucessor ordinals, then it holds for all ordinals. Suppose
k, is a limit ordinal. The map aK is epi, because it is a limit of epimorphisms. Note that aK
and aKnKaK are both compatible maps FKA > A; the definition of colimits implies that
because aK is epi, id^ = aKnK.
The algebra associative law holds for (^4, a). If a aA, then aA = aA+1, so by
the hypothesis regarding the existence of ax such that a\+i(rri\A) = a\(F\a\) proves the
associative law.
If (unit) holds for A, then (ext) holds for A. Note that the equation a\+\(rn\A)
a\{F\a\) defines a\+1. By induction hypothesis (unit),
{F\a\)(F\n\A) id/r^^;
therefore precomposing both sides of the defining equation for aA+i with (F\ri\A) shows
aA = ax+i(m\A)(FxnxA) = ax+i{cxA).
The second equality follows from (mxA){FxnxA) = (cxA), which was proved in verifying the
unit laws for F.
If (unit) holds at A, then it holds for A + 1. Consider the following diagram.
The left triangle commutes because (unit) holds for A; the bottom trapezoid commutes by
definition of ax+1; the top trapezoid commutes because
(nxA)(FxnxA)(mxA) = (nxA)(cxA) = nx+iA.


58
The first equality holds by proof of the unit law for F; the second equality holds because of
the compatibility between nAs. Thus,
aA+i(nA+i) = ax+i(mxA)(FxnxA)(nxA)
= ax(Fxax)(FxnxA)(nxA)
= ax(nxA)
= id a;
this establishes the unit law for (A, a).
Use the notation Alg(F0, no) to denote the full subcategory of pAlg(Fo, no) containing
objects satisfying the hypotheses of Lemma 4.2.6.
Lemma 4.2.7. Suppose the preceding lemmas apply. Also suppose each mx is epi. The map
(A, a) i> (A, a) induces an equivalence of categories; the functors involved are
E : Alg(F0, n0) -* AT,
defined by extension of structure and
F:AF> Alg(F0,n0)
given by restriction.
Proof. E is legitmately defined. Lemma 4.2.6 defines E on objects. Maps in AF are
A-maps compatible with the structure. So it suffices to show that for any Alg(Fo,n0) map
f): A B,
FXA FXB
bx
B


59
commutes for each ordinal A. By definition of Alg(F0, n0), the square commutes for A = 0.
Suppose it commutes for A; note that
a\(Fxa,\)
def. a^+i
= bx(Fx)(Fxax)
ind. hyp.
= bx(Fxbx)(F&)
ind. hyp.
= bx+,{mxB){Fl4>)
def. bx+i
= bx+i{Fx+i4>){mxA)
naturality mx.
Because mA is epi, one concludes ). At limit ordinals, properties of
colimits insure that the diagram commutes. By transfinite induction, and the defintion of
the extended structure, any map which preserves (Fo, n0)-structure preserves F-structure.
One also notes that restriction respects maps, because Fo is a subfunctor of F.
E and R form an equivalence. Obviously,
V(A, a0) G Obj(Alg(Fo, n0)), RE(A,ao) (A,a0).
The construction of the extended structure shows the restriction of any F-structure to Fo
uniquely determines the F-structure; hence
ER(A, a) = (A, a).

4.3 A Partial Algebra Which Does Not Extend
The category Set is an SF-category and unions are colimits, so Meseguers Lemma
applies to Set. We show the necessity of Lemma 4.2.6s hypothesis that a map a\+j such that
ax+\m\ = a\(F\a\) can be defined. The section discusses a subfunctor of the free-magma
monad, which has an algebra that cannot extend to a monad algebra. A magma is a set
with a binary operation, subject to no equations.
The free magma monad (T, rj, ¡i) has the following parts:
1. Given a set X, TX consists of all words with variables in A. A magma word is any
expression formed by finitely many applications of the rules:


60
(i) If x G X, then (^is a magma word.
(ii) If s, t are magma words, then
is a magma word the product of s and t. For ease in reading, we use binary tree
notation for products.
2. T defines a functor: given / : X > Y, to compute Tf we apply / to all members of
X in a given word, leaving the tree and circle structure unchanged.
3. The insertion of variables map r]X : X > TX. (rjX)(x) = (^.
4. The semantic composition map [iX : T2X > TX sends a word s G T2X of words
to a word in TX, by removing the circles around each element of TX used in making
s. n is also a natural transformation.
The notation takes a little while to soak in; to expedite the process, we consider a
calculation with TN. Suppose s G TN is the word
and ti = ss G T2(N) is the word


61
then (/N)(tx) is
Now we define a subfunctor F of T. Define the depth dep(n) of a node n in a binary
tree inductively by: the depth of the dep(root) = 0; if a is immediately below b, then the
dep(a) = dep(b) + 1. Define a leaf to be a node that has nothing below it. Let F consist of
all rooted, labeled, binary trees (i.e., magma words) such that the depth of each leaf is the
same. For example s and (/rN)(ti) are in F(N), but
is not, because dep{57) = 1 and dep(l) = dep(4) = 2. Evidently, F is a subfunctor, and for
any X, (r¡X)+1(X) C F(X). But, for any non-empty X, (pX)(F2(X)) is not contained in
F(X). One readily verifies that F = T.
Now we define an partial algebra structure on N that does not extend to a T-algebra
structure. As in Lemma 4.2.6, a structure map for a pair (F,rj : id^ > F) is a map
a : FA > A satisfying id = a(rjA). Define n : F( N) -t Non a tree r as follows: if the
depth of each leaf of r is odd then n(r) is the leftmost label, if the depth of each leaf of r
is even (or zero), then n(r) is the rightmost label. For example, n(s) 1 (s defined above)
and n(/iN)(ti) = 4.


62
In order for n : F(N) > N to extend to n : Fi(N)
making
F2( N)
Fn
F{ N)
/xN
F( N)
n i
*N
commute. No such a: can exist, for (/zN)(ti) is
N there must be a function rii
which equals (/iN)(t2), where t2 is
Finally, note that (Fn)(ti) is
so n(Fn)(t1) = 1. Note that (Fn)(t2) = (/rN)(t2), so n(Fn)(t2) = 4. Thus we have elements
11,12 G F2N, that /xN identifies and n(Fn) does not identify, so there is no function n\ such
that n\(nN) = n(Fn).


CHAPTER 5
FREE ALGEBRAS
This chapter explores a generalization of complete distributivity for P^-objects. In
several steps, the free complete distributive lattice monad is constructed, then Meseguers
Lemmas are used to construct an appropriate submonad, whose algebras generalize com
pletely distributive lattices.
Section 5.1 describes monads for complete semilattices; D is the monad for complete
join semilattices, 11 is the monad for complete meet semilattices.
The gist of section 5.2 is that there is a lifting of 'll over V (and a lifting of D over 11).
Therefore, UV and VU are monads over P. (See results of Beck [5] summarized in Section
3.4.) In Raney [27], it was shown that complete meet distributivity is the same as complete
join distributivity. Hence, the composite monads UV and Dll have the same category of
algebras. The objects in either category are complete lattices where meets distribute over
joins, and joins distribute over meets.
Completely distributive complete lattices have been thoroughly studied. The basic
structure is described in Raney [27], [28], and [29]. Free objects over Set were initially
described in Markowsky [24]. Tunnicliff [32] discusses properties of the free completely
distributive lattice over a poset. Free objects over P and the relationship between completely
distributive lattices and continuous lattices are described in Hoffman and Mislove [14]. The
approach here is apparently new, but yields obviously equivalent free objects.
Each pair (j,m) of subset systems gives rise to a subfunctor of It'D. Meseguers
Lemmas are applied to this subfunctor to produce a monad F. Any P^,-objects which
63


64
is P^-embeddable in a completely distributive complete lattice is an F-algebra. Any F-
algebra has a natural structure. Because of computational difficulties, no exact algebraic
characterization of F-algebras is given here.
A word about notation: the (functor parts) of the monads described below are given
by families of sets. Thus, checking the unit and associative laws requires working with many
levels of the power set tower. Roman letters S,T, denote sets. A subscipt designates the
power set complexity: Si G CP(A), S2 G S2A S2 is a family of sets, S3 G 73A S3 is a
family of families of sets, etc.
5.1 Complete semilattices
In this section, we describe the free complete join (resp. meet) semilattice on a poset
A, using the monad (V, d, fi) (resp. ('U,i,/x)). The description of D is well known; for
example, Meseguer [25] uses it. The reader will have noticed that q is used as a name for
two different natural transformations; this would normally be horrible notation, but in this
case, the formula for y is the same. Thus, our notational econony should cause no confusion.
The functors 'll and T> act on a poset A by
11(A) = {SCA:x>y6S =£> x G 5}
- the set of increasing subsets of A ordered by reverse inclusion- and
T>(A) = {SCA:x x G 5}
-the set of decreasing subsets of A ordered by inclusion.
Given monotone f : A > B, we define
Vf : 'D(A) -* V{B) : S {b £ B : 3s S, b < f(s)}
and
Uf : U(A) U(B) : S ^ {be B :3s e S,b> f{s)}
The following facts will be of later use.


65
1. t and D are functors: trivially they respect identity arrows. Observe that for any
S 6 A and montone functions / : A > B, g : B > C,
T> (gf)(S) = {cC:3seS,c = {ceC:3f(s)e'Df(S),c - (£ and, similarly, U(gf)(S) = (Ug)(Uf)(S).
2. For each poset A, D(A) is a complete lattice, with supremum operation given by set
theoretic union, and infimum given by intersection. tt(A) is a complete lattice, with
infimum given by union and supremum given by intersection.
3. For each S2 C D(A),
In particular, this holds if 6 is empty. So Vf preserves all suprema. Similarly, if
S2 C 'U(A), then
Uf(\j6)=\j{(Uf)(S):See}
so that Uf preserves all nfima.
Define
dA: A > T>(A) : a 1> {x G A : x < a},
iA : A 'U(A) :aH{i£i:i>a}, and
fiA : T>2{A) V(A) : S2 K2(A) ^ (J S2.
It may be puzzling that T>(A) is ordered by subset inclusion and li(.4) is ordered by
reverse subset inclusion. The fact that
x < y dA(x) C dA(y) <=> iA(y) C iA(x)


66
motivates the choice, because one wants iA to be order preserving. Moreover, one wants 1if
to preserve all intima, which only happens if the infimum operation in UA is union.
The reader may check that i, d and p are natural transformations. The monad
assocative law
Vy
D3
D2
yV
D2
V
holds for both D and 'll, because if S3 e T3(^4), then
UUs3 = U{l>:S^ss}.
The monad unit laws
D D2 D
hold for V, because if S 6 T)(A), then
5 = (J {(dA)(x) : x 5] = (J jr T>(A) : T C 5}.
Similar computation shows the unit laws hold for U.
Now to describe algebras over these monads:
Lemma 5.1.1. Let T C 1>(A), such that for all x E A, (dA)(x) T. Then the following
are equivalent:
1. there is an order preserving map a : T > A such that id^ = a(dA);
2. each S G T has a supremum;
3.dA : A T>A has a right adjoint.


67
The analogous conditions involving T C 1LA are also equivalent.
Proof. We give the proof for T C 1>A, leaving the upside-down argument for T C UA to the
reader.
(2 ==> 1) Define a(S) = \J S and compute.
(1 => 2) a(S) is an upper bound for S, because if x G S, then (dA)(x) C 5,
whence x = a(dA)(x) < a(S). Suppose for all x G S, x < u. Then (dA){x) C (dA)(u), so
S = U{(gL4)(x) : x G 5} C (dA)(u), therefore q(5) < u.
(2 3) By definition of V S, j x C S <*=> x <\J S.
Proposition 5.1.2. P is equivalent to the category of T)-algebras. Poo is equivalent to the
category ofU-algebras.
Proof. The first assertion is proved, leaving the second to the reader. By Lemma 5.1.1, any
D-algebra has a supremum for each decreasing set; the supremum of an arbitrary set S is
equal to the supremum of the decreasing set ]. S. By the definition D-algebra homomor-
phisms coincide with order preserving functions which preserve all suprema of decreasing
sets. It is easy to see that a map preserves suprema of all decreasing sets if and only if it
preserves suprema of all sets.
Conversely, if A G P it has structure map
a : T>(A) A : S >-*\J S.
In Lemma 5.1.1 it was noted that a satisfies the unit law for structure maps, a satisfies the
associative law by the order theoretic fact proved in Lemma 5.1.3 below.
Lemma 5.1.3. For any S2 G D2(A),
V {x A : 3S1 S2, X < V-Si} = VU^2'
Proof. To verify this equality, one notes that for all x G G S2 such that x 5!,
so x < V S\. Thus the left hand side dominates the right. If u is an upper bound for (J S2,
and 35i G S2, x < V Si, then x < u. So the left hand side is the least upper bound of [J S;
this establishes the lemma.


68
5.2 Completely Distributive Complete Lattices
In the following arguments, the forgetful functors are supressed from the notation.
The lemmas that follow establish that there is a lifting of 'll over V (and similar arguments
show that there is a lifting of V over It). Explicitly this means:
1. For any U-algebra A, there is a lt-structure on V(A).
2. It maps D-algebra maps to D-algebra maps, so we may view 'll as a functor P > P.
3. Both natural transformations i and p : U2 > U preserve all joins; similarly, d and
/x : D2 > D preserve all meets.
Lemma 5.2.1. For any poset A, both 11(A) and V{A) are complete lattices. Thus, 11(A)
and D(A) are both It and T> algebras.
Lemma 5.2.2. For any poset A,
1. dA : A > D(A) prserves all existing nfima.
2. iA : A 11(A) prserves all existing suprema.
Proof. Suppose S C A and u = f\S exists. Since dA is order preserving ,
(dA)(u) C P| jcL4(:r) : x G
For the reverse inequality, suppose i G f]{dA(x) : x E S}, i.e., i is a lower bound of S. Then
l < u, so £ E dA(u). The proof for uA is upside-down, but otherwise identical.
Lemma 5.2.3. If f : A B preserves all suprema, then It/ : 11(^4) > U(B) also pre
serves all suprema. Similarly, if f preserves nfima, so does T>(/). Therefore, for any order
preserving map f, both UT>f and VIIf preserve all nfima and suprema.
Proof. A proof of the first fact is given, the second is very similar, but notationally easier. Use
the fact that lt(A) is a complete sublattice of T(A), and therefore completely distributive.


69
Let (S\)xeL be an indexed subset of li(A); for each S\, choose a family (xx,K
that
S\ =T (x\,K k K).
Recall that:
joins in IL4 are intersections;
iA preserves suprema, which reads
T \Jyx = yx) = P| T y\,
A
for any indexed family (yx) C A.
One computes as follows:
- Mnib^.")
A K
= ^(LKn : c:L-+k}')
= (j{u(\Jxx,cW)-- C-.L-+K]
'a
= U { t V /(**-<*)) =
= U { Pi T /(^a,^a) : c-.L^k}
1 A
= n uf^
Corollary 5.2.4. For any poset A,
1. p, : T>2(A) > T)(A) prserves all nfima.
2. p : li2(A) > U(A) prserves all suprema.
: k K) such



70
Proof. Suppose S2,a £ X>2(^), for A L : claim 1 amounts to
unsw=nusw
A A
Clearly, the left hand side is contained in the right. Suppose that x is a member of the right
hand side, that is, for each A there exists Si,a 6 2,a with x Si,a- Then
Si = Pi Si,a
A
is in each S2,a because each S2,a is downward closed relative to the inclusion order in 2(.A).
Since a: Si, this proves that x is in the right hand quantity, and the desired equality of
sets holds.
The proof for the It works similarly.
The preceding lemmas show that there are liftings of 'll over (and vice versa), so
we have:
Corollary 5.2.5. It = (ItCD, 77, where
7] : idp It : a i-> {S VA :|aCS}
and
v : UDlt -> It : S4 ^ un*.
is a monad over IP whose algebras are completely distributive complete lattices.
Proof. The only thing left to be proved is the formulas for the natural transformations. By
Becks reasoning as summarized in Section 3.4, g is the horizontal composition i d. To
compute v : ltlt -4 It one uses the fact that vA is the structure map for the free algebra
UVA. Consider the situation in light of the discussion following Theorem 3.4.5. (Here It
plays the role of 5, while plays the role of T.) The relation a = ar{Tas) applied to the
object UVA implies
(vA)(s,) = u(n)(*)
= U{reBA:3S2ef|S,,rcf|S2}
= Ui>-


71
The assertion about algebras follows from Theorem 3.4.5.
Remark 5.2.6. UV is a monad because there is a lifting of U over P. One could use
the correspondence between liftings and distributive laws outlined in 3.4 to find a natural
transformation VU > UT>. This distributive law is not needed for the calculations which
follow, and is somewhat cumbersome, so its explicit description is omitted.
5.3 Some categories of algebras
In this section, Meseguers Lemmas are applied to the monads described above. Note
that P the category of posets and order preserving maps is a cocomplete SF-category in
which unions of chains are colimits. Note that extremal monomorphisms in P are inclusions
of subsets with the induced order: each order preserving map / : A > B factors as A >
f(A) y B, where f(A) is the set-theoretic image of A, with the order induced from B.
Thus, the results of Section 4.2 do apply to P.
The reader is advised to review the definition of subset system, given in 1.2.1, if
necessary.
Assume all subset systems Z are non-trivial in the sense that for each A, and a A,
{a} Z(A); this does not reduce the generality of the argument, because given any subset
system, one may adjoin all singletons to it without changing which optimum bounds are
preserved.
Theorem 5.3.1. [25] P-7 and Pm are monadic.
Proof. Given a subset system j
J0(A) = {iS:Sej(A)}
defines a subfunctor of V. This extends to a subfunctor j of T> such that (j, d, Jl) is a monad.
The natural transformations d and JL are obtained from d and /i by suitably modifying the
domains and codomains. The precise definition of these maps is contained in the proof of
Lemma 4.2.6.


72
If (A, a0) has a J0-structure, Lemma 5.1.1 shows that
ao(S) = \/5
for all 5 e Jo(A). Thus, to define ax one must show some map JXA A makes the following
diagram commute.
JSA J(V) *J0A
U
JXA-
Lemma 5.1.3 shows that \J S is defined for all S J\A and, moreover, that for S2 JqA,
VU S2 = V(JV )(>)
Identical arguments show show that for each A, if
a\(S) = \J S
that
aA+1(S) = V5-
It follows that any poset, A, in which each member of JqA has a supremum, each member of
J(A) has a supremum. Moreover, each map preserving J0-suprema also preserves J-suprema.
Thus, one obtains an equivalence of categories between P and PJ.
An upside-down version of this argument shows that Pm is a category of algebras.
Given subset systems m and j, F := M0J0 forms a subfunctor of
T ILD.
By Corollary 5.2.5, (T,u,rj) is a monad on P. By 4.2.1, there exists a smallest F above F
that is closed under v. This monad is used to discuss the categories defined below.


73
Definition and Remarks 5.3.2. Let DP^ denote the category of completely distributive
complete lattices with maps, which preserve order and all optimum bounds. DP^ denotes
the full subcategory of containing objects such that there is a PJm map <¡>: A > B, with
4> P-extremal mono, and B Obj(DP^).
denotes the category of F-algebras, where F is the monad described above.
It will be shown that
SpFm C mi c Ml.
To see the first inclusion, one notes that any family T of subsets of a set X which is also a
P^ object is in DP^; the inclusion T C 7(X) is a P^-embedding and T(X) is a completely
distributive complete lattice. Proposition 5.3.7 shows the second inclusion.
Let A 6 BP^, and <¡> : A + B be given as in the definition of OP^; let b : TB > B
be the T-structure for B. The verification of the hypothesis of 4.2.6 proceeds by comparing
the diagram to be completed with the lt!D-algebra associativity diagram for B. Prom this
point on, the notation of Lemma 4.2.2 is adopted with slight modification; v : T2 T is
the monad multiplication; v\ : F% F\+i is defined inductively (and plays the role that
was played by m\ in the proof of 4.2.2); r¡ : idp T is the unit; r¡\ : idp > F\ are defined
inductively, and play the role that was played by n\ in 4.2.2. The natural transformations
fxF\*T and c\ : F\ > F\+1 play the same roles as in Lemma 4.2.2.
Remark 5.3.3. By Lemma 3.4.5, the structure map b : TB > B is given by:
= A {V5i: 5i M-
Since is a P^-map, the following diagram commutes.
Fd> f\B
FA *-FB >TB
ao
B
B


74
The left square expresses that <¡> commutes with the F-structure; the right square expresses
that the F-structure on B is the restriction of the T-structure.
Lemma 5.3.4. Define ao : FA > A by a0(S2) = AC^ VX^O this map satisfies a0T]o = id^.
Proof. Let us begin by more explicitly calculating ao. Using the definition of li on maps we
find that
a0(S2) = A(UV)(*)
= f\{xeA:3S1eS2, \/Si = /\{\/S1:S1eS2}.
This definition makes sense; each Si G Jq(A) so each V Si exists because A G P^. (U V) is
an order preserving map, and S2 G mojo(A), which implies that
(U\J)(S2) e M0(A).
Thus, the meet defining ao(S2) exist.
Recall rj(x) = {S G DA x C 5}. Now one calculates
ao(?(*)) = /\{\/S:SeDA,lxQS}
= x

For later calculations, it is crucial to know that some of the maps are epi or mono.
Lemma 5.3.5. For each ordinal X, u\A is surjective. Therefore v\A is P-epi.
Proof. Because of the structure of P, unions of subposets actually are set theoretic unions.
Let 5 G F\+1 = F\ U (i/\A)+1(F%A). If S G (v\A)+1(FfA), then there is an S2 G F%A such
that (v\A)(S2) = S. Since c\A = (i'xA)(r]xFxA) so for any Si G FxA, T2 := (r]xFxA)(S) is a
member of FfA such that (uxA){T2) Si.
This lemma holds in any category where unions of subobjects actually are set-theoretic
unions.


75
After noting that extremal monomorphisms in P are injections / : A > B, where
f(A) has the order induced as a subset of B, one easily verifies:
Lemma 5.3.6. The functors U, T>, T, and F\ all preserve extremal monomorphisms.
Now one can verify the hypothesis of 4.2.6 is satisfied.
Proposition 5.3.7. Let A Obj(DP^), B e Obj(BP^), : A > B as in the definition of
BP,. Then
(ex) for each A, there is a unique map
A+i F\+iA A,
such that
a\(F\a\) a+iI'a;
(com) for each A, fiax = b(f\B)(F\).
Thus, OP^ C
Proof. Remark 5.3.3 established (com) for A = 0. If (com) holds for sucessor ordinals,
properties of unions insure that (com) holds at all ordinals.


76
Suppose (com) holds at stage A; we will establish (com) and (ex) hold for
stage A + 1. Consider the following diagram.
T2B >TB
The diagram commutes: the outer square commutes because of the T-algebra associative law
for B\ the left outer trapezoid commutes because of the definition of v\B\ commutativity of
the right outer trapezoid is obvious; the left inner trapezoid expresses naturality of v\\ the
top and right inner trapezoids commute by (com).
Establishing commutativity of the top outer trapezoid requires a bit more. Since
fx : F\ T is natural, (Tb)(fxTB) (fxB)(Fxb). By definition of horizontal composition
flB = (fxTB)(FxfxB).
Thus,
CTb){flB) = (Tb)(fxTB)(FxfxB)
= (fxB)(Fxb)(FxfxB)
This proves that the top outer square commutes.
Consider the (epi,extremal mono)-factorization of
b(fx+1B)(Fx+iy,


77
say it factors as
Fx+lA Ai'CB.
Since is epi, the (epi, extremal mono)-factorization of
q := b(fx+1B)(Fx+1)(uxA)
is
FXA A! C B.
The proof of Lemma 4.2.6 shows that each existing ax is split epi, which implies both ax and
(Fax) are epi. Since
q = 4>ax(Fxax),
the (epi, extremal mono)- factorization of q is also:
FXA a(4a) ACB.
Uniqueness of factorization implies there is an isomorphism i : A > A! compatible with the
factorizations. So a^+i = ia is a map making (com) hold for A + 1.
Reading off the diagram one finds that:
ax(Fax) b(Tb)U2xB)(F2x)
= b{vB){fxB)(Fx)
= b(fx+1B)(Fx+1)(vxA)
= The last step used that (com) holds for A + 1. Since 4> is a monomorphism, one concludes
that (ex) is satisfied for A + 1.
ax+1 is the only map which makes the inner square commute because vxA is epi, by
Lemma 5.3.5.


78
Corollary 5.3.8. DP^ is a full subcategory ofMPm.
Proof. By Lemma 5.3.3, any P^-map preserves the F-structure of a OP^ object. Lemma
4.2.7 implies that any P^-map extends to a map.
Corollary 5.3.9. The forgetful functor DP^ > P has a left adjoint.
Proof. First note that FA G Obj(DP^) for any A, because fA : FA > TA is an embedding
of FA in a completely distributive lattice. Because FA is the free F-algebra on A, if
/ : A B is an order preserving map from A G Obj(P) to Be Obj(DP^) C Obj(PF), then
there is a DP^-unique map f* : FA > B. This proves that rj has the universal property
described in 2.5.2.3; thus, F is the left adjoint to the forgetful functor DP^ P.
Corollary 5.3.10. The forgetful functors DP^ > OP^ and DP^ > have left adjoints.
Proof. Apply Proposition 3.3.6 and Corollary 6.2.8. (Note: the proof of 6.2.8 does not
depend on the arguments in this section, so the result is listed here. There is no circular
argument.)
There is some, rather limited, information about members of M^.
Proposition 5.3.11. If A G Obj(M^), then A G Obj(P^).
Proof. Define JA, MA, a, /?, 7, and 5, by the requirement that the squares below are
pullbacks.
JA ^ FA
7 a
T>A UHA
MA
FA
fA
UA ^UDA


By assumption, j(A) and rn(A) contain singletons, therefore
(iT>A)+1{J0A) C Mq Jo A C FA,
79
and
('UJA)+1(M0A) C Mq Jo A C FA.
It follows that
J0A C JA = (iDA)_1FA
and
Mo A C MA = (ltdA)_1FA
Again using the fact that j and m contain singletons,
(JA)+1(A) C JA
and
(A)+1(A) C MA,
so corestrictions dA\ : A > JA and iA| : A > MA satisfying a(dA)\ = dA and j(iA)| = fA
exist.
If a : FA > A is an F-algebra, then a(3(dA\) id^ and a5(iA\) = id^. Lemma 5.1.1
shows that for any subset 3 C DA containing each principal down-segment [ x, an order
preserving map y : 7 A satisfies y(dA\) id.4 if, and only if, y{S) = VS for each 5 J
(and analogously for subsets of 11 A). Thus, we conclude a(3 = \J and a6 = /\.
Remark 5.3.12. If j = m = u> the subset system which selects all finite subsets, then
SpF^ = DP^. For general j and m, the author does not know if equality holds, but he
suspects that the equality does not always hold.
Remark 5.3.13. A fundamental difficulty working with M^n is that one does not imme
diately know any order theoretic formula for the F-structure maps. One might hope that
each F-structure map is S2 /\(U \f)S2, but the author cannot presently substantiate such
hopes.


80
Remark 5.3.14. The author knows no examples of objects in Obj(M^) \ Obj(DP^). In
light of Corollary 5.3.10, any BP^ object is freely embedded in a completely distributive
lattice. In fact, A Obj(M^) is in Obj(BPJm) if, and only if, the unit of the adjunction
mentioned in 5.3.10 is P-extremal mono.
Li[18] explicitly constructs a map u : P > ISF(P) (not necessarily mono), with the
following properties:
ISF(P) is a complete, completely distributive lattice.
u preserves designated meets and joins.
If / : P > A is a map preserving designated meets and joins, and A is complete,
completely distributive, then there is a unique map /* : ISF(P) A such that
/ = f*u.
To explain the relationship between Lis results and the results here, note:
Li uses families SP C SP and IP C SP, which are only required to contain singletons.
Joins of SP-sets and meets o IP sets are required to be preserved. In this document,
the choice of distinguished optimum bounds is made for all posets at once, via a subset
system.
Comparing universal properties, one sees that Lis construction applied to P Obj(M^)
with SP = j(P) and IP m(P) yields the left adjoint mentioned in 5.3.10.
Unfortunately, it is very difficult to see when u is an embedding. Let [5] denote the
smallest downward closed family of A containing S which is closed under SP-joins. Let SP
denote the class of increasing sets, which are closed under IP-meets. Li defines IS(P) to
be the familly of all decreasing sets in SP, ordered by S < T if, and only if, there is an
indexed chain (5')6/ with I = Q fl [0,1] such that S0 = S, S¡ = T and whenever i < j,
Si fl [P \ Sj] = 0. The map u : P > ISP is defined by u(x) = {U 6 UA : U -< x}.


81
Nonetheless, BP^, it is a fairly nice category. In fact, it is complete.
Proposition 5.3.15. BP^ is complete.
Proof. Note that a poset A is a P^ object if and only if there is an order preserving map
a : J0A x M0A > A such that a(d0A x i0A) kU.
Suppose (Ai)ii is a family of P^ objects and for each i I,
(j)i ^ Pi
is an embedding of Ai into a completely distributive lattice The proof of Lemma 3.1.3
implies that the poset product At is a P^-object. Moreover, the product-induced map
$: ne/ A* -* n/ Bi is an embedding which preserves (j, m)-optiinum bounds. (Essentially
what is going on is that meets and joins are computed coordinate-by-coordinate.)
If /, g : A > B axe P, maps, then the proof of Lemma 3.1.3 implies eq(f,g) is
a (j, m)-complete subset of A. If A Obj(BP^), then A can be (j,m)-embedded into a
completely distributive lattice, so eq(/, g) may also be so embedded.
Thus, any set of objects in BP^ has a product, and any pair of BP^ maps has an
equalizer. By Borceux [6, Volume 1, 2.8.1], this implies BPfa is complete.


CHAPTER 6
COEQUALIZERS
6.1 Epis and Equalizers in P
The results of this section characterize epis and regular epis in the category P of posets.
The presentation and proofs (except Construction 6.1.2 which is discussed in Meseguer [25])
are the work of the author, but the author believes it likely that they are not new. In all
statements, A and B are arbitrary posets.
Lemma 6.1.1. Let f : A > B be a monotone map. f is epi if and only if f is onto.
Proof. Since the forgetful functor P > Set is faithful, if / is onto then / is epi. For the
converse, suppose b G B \ f(A). Define
Si =i (f(A)n | b)
and
S2 =1 b.
Evidently, both S\ and S2 have the same intersection with f(A), but b G S2 \ Si. Therefore
characteristic functions of B \ Si and B\ S2 are distinct maps (say
ci,c2 : B 2
are respectively the characteristic functions of B \ Si and B \ S2) such that Cif = c2f. Both
Q are order preserving, because Si and S2 are decreasing sets.
Definition and Remarks 6.1.2. The following construction is paraphrased, following Mese
guer [25] pp 73-74.
If / : A > B is any order preserving map, we may define a preorder X on 4 by
ai 4 a2 <=> /(ax) < /(a2).
82


83
This relation is obviously reflexive and transitive, but -because / may not be injective
may not be anti-symmetric.
Suppose (A, <) is a poset and ^ is a preorder strengthening <, i.e.,
i < a2 => ai < 2-
Then we have an order preserving map a : A > A/ <. (Recall that A/ < is the set of
equivalence classes
{x A : x -< a and a < x}
partially ordered by ;<.)
Define maps /1; h : A Bt to be equivalent, if there is an isomorphism, i : Bx > B2
such that
B2
commutes.
One verifies that:
1. The maps (f : A -* B) and (<)>-+ a \ A A/ < are mutually inverse
correspondences between
the set (modulo equivalence) of surjective maps / with domain A.
the set of preorders on A which strengthen <.
2. For any / : A* B, f = ca, where a : A > A/ ^ and c(a(a)) = /(a).
3. Given surjections fi,h'.A* B, there is a c : B\ > B2 if and only if
/i h
al ^ a2 ==> al ^ a2-


84
Remark 6.1.3. Note that any intersection of preorders is a preorder. Thus, the class of
preorders strengthening < is a complete lattice. We say a set S of ordered pairs generates
< if < is the smallest preordered containing S. Note that if 5 generates <, we can describe
-< explicitly: a0 < an if
clq dm or
there is a finite sequence ai,a2,--- a_j, such that for all i with 0 < i < n 1,
() t+i) £ S.
(The first bulleted condition insures that -< is reflexive. The second requirement insures
that < is transitive; generally, if S is a relation, the relation obtained by applying the second
bulleted item only is called the transitive closure of S.)
One may verify the following construction using 6.1.2.
Construction 6.1.4. Let f,g : A > B be order preserving maps. The coequalizer of
f and g is the quotient of (B, <) by the smallest preorder containing < and C(g, h)
{(gx,hx),(hx,gx) : x G A).
Lemma 6.1.5. Let f : A > B be a surjective order preserving map. The following are
equivalent.
1. f is the coequalizer of some pair g, h : Ao -h A.
2. There exist a poset Ao, and maps g, h : Ao > A such that the preorder < on (A, <) is
generated by < and C(g,h).
3. The preorder ^ on (A, <) is generated by < and K(f) {(x,y) E BxB : f(x) = f(y)}
f. The preorder < on (A, <) is generated by < and some equivalence relation.
Proof. (1 <=> 2) is evident from 6.1.4. (3 => 4) is trivial, because K{f ) is an equivalence
relation. We have
C(g,h)CK(f) c4


85
because fg = fh and
(x,y) eK(f) 4=^ f(x) < f{y) and f{y) < f(x).
f f
Since C(g,h) and For (4 => 1), let X be generated by < and the equivalence relation E C A x A
with the trivial order. Then the projection maps tt\, 1T2 : E > A, given by
TVi(xi,X2) = Xi
are order preserving. The coequalizer of 7Ti and 7r2 is /.
Example 6.1.6. Some readers may be surprised by the fact that surjections are not always
P-regular epi. An example of this situation is any map / : 2/a( 2 from the (trivially
ordered) two point set onto a two-point chain. Any such map / fails condition 3 from
Lemma 6.1.5.
6.2 Factorization of Maps Using Preorders
This section modifies the preorder factorization, to give a first approximation to
coequalizers in FJm.
Definition 6.2.1. Let Z be a subset system. Z is said to admit congruences if for all posets
A,
Z(A x A) = {S A x A : Z(A),i= 1,2}.
Lemma 6.2.2. Suppose both j and m are subset systems which admit congruences. Let
(A, <) be a poset and X be a preorder strengthening <. Then a : A A/ X preserves
(j, m)-optimum bounds if and only if X is a FJm-subobject of Ax A.
Proof. We show that q(Vx) = Va(r) for any (x : i G 1} j{A). Clearly a(Vx,) is an
upper bound of (a(x) : i I}. Since X is a P^-subobject of A x A, {x¡ : i 1} G j(A) and
for all i, Xi < a,
V{(xj, a) : i E 1} = (Vxi(a)


86
is a member of d', Vx < a. Thus, any upper bound of {a(x) : i £ 1} dominates a(Vx).
(The proof for meets is identical, so omitted.)
Conversely, if a preserves j-joins and m-meets, and (for example)
{xi'.i /}, {yt : i £ 1} £ j{A)
and for all i £ /, x d then Vx, d So ^ is a P^-subobject of ,4 x A.
Definition and Remarks 6.2.3. Suppose j and m admit congruences. Let (A, <) £
Obj(P^). If ^ is a preorder strengthening < and simultaneously a P^-subobject of A x A,
then we say < is a (j, m)-pocongruence.
Pocongruences of P^-objects behave very much like preorders of posets.
1. The maps (f : A B) >->4 and (^) a : A > A/ < are mutually inverse
correspondences between
the set (modulo equivalence) of surjective maps /, which preserve (j, m)-optimum
bounds and have domain A.
the set of (j, m)-pocongruences on A.
2. For any / : A > B, f = ca, where a : A > A/ < and c(a(a)) = /(a). If / preserves
(j, m)-optimum bounds, so do c and a.
3. Given (j,m)-optimum bound preserving surjections fi,f2'-A>Bl, there is c : B\
B2 if and only if
A Si
ai di a2 i di a2-
Consideration of Lemma 6.2.2 shows that if / : A > B is surjective and S £ j(A)
(resp. S £ rn(A)) then f(S) has a supremum (resp. infimum) in B. Moreover, sets of the
form f(S) for S £ j(A) (resp. S £ m(A)) are the only sets whose optimum bounds are guar
anteed to exist. Thus, we are motivated to offer the following conditions on subset systems
to guarantee surjective (resp. P-regular epi) images of P, objects are (j, m)-complete.


87
Definition and Remarks 6.2.4. Let Z be a subset system. We say Z preserves surjections
if whenever / : A B is a poset surjection,
Z(B) = {f(S):SeZ(A)}.
Similarly, we say Z preserves regular epis if whenever / : A B is a poset regular
epimorphism,
Z(B) = {f(S) : 5 Z(A)}.
Since poset regular epimorphims are surjective, any Z that preserves surjections also
preserves regular epimorphisms.
Any cardinal k, the associated subset system k preserves surjections. For if f : A > B
is surjective, then / is split epi (in Set). Hence, there is s : B > A (not necessarily order
preserving) such that idg = fs. Therefore, any set S C B with cardinality less than n is the
image of some set s(S) C A. The following remark provides an example of a subset system
that does not preserve surjections or regular epis. The author currently does not know of
any subset system that preserves regular epis without preserving surjections, but it seems
likely such a subset system exists.
A more interesting question, which the author also cannot currently answer is: do
there exist subset systems other than cardinals which preserve regular epimorphisms?
Remark 6.2.5. Let j = dir be the subset system which selects all upward directed subsets
of a poset. Let N denote a countable disjoint union of two-point chains. (See drawing below
for help visualizing, and to fix notation.)
to t\
h
bo
Evidently N is j-join complete.


88
Consider the function / : N N, where N = {0,1,2, } with the usual order,
defined by /(fc) = i (for i > 0) and /(_i) = i (for i > 1). Speaking roughly, f stacks
the two point chains. / is regular epi as a map of posets. Moreover, ^ is a j-subobject of
N x N, because / preserves all existing joins. But N is not j-join complete, because N is
upward directed but has no join!
Because of the pocongruence factorization for (j, m)-bound preserving maps (outlined
in 6.2.3) we have the following first approximation to the coequalizer in P^. The only thing
stopping map a : B > B/ < (described below) from actually being a coequalizer in P^ is
that B/ < is generally not {j, m)-complete.
Construction 6.2.6. Suppose f,g:A> (B, <) are ¥Jm maps, and C(f,g) is defined as in
6.I.4 Let X be the smallest pocongruence containing < and C(/, g). Then a : B > B/ X
has the following universal property: if h : B > C preserves (j, m)-optimum bounds and
hf = hg, then there is a unique map i : B/ < > C such that h ia.
Since the quotient map a : B > B/ ^ is a surjection we have -
Corollary 6.2.7. Continue with the notation from Construction 6.2.6. If j and m preserve
surjections, then B/ < is (j,m)-complete. Thus, a : B B/ -< is the coequalizer of f and
g in FJm. In particular, if j = k and m = A are cardinality subset systems, then a is the
coequalizer.
Corollary 6.2.8. DP has coequalizers.
Proof. The preceding shows that any pair f,g:A>Be P^ has a coequalizer. Suppose
B is completely distributive. Since a : B > coeqPoo (/, g) is a surjection which preserves all
meets and joins, coeqp* (/, g) is completely distributive. In particular if both A and B are
completely distributive, then coeqp(f,g) is also the coequalizer in DP.
6.3 Factorization of Meetsemilattice maps
Several simplifications occur describing coequalizers in P, if the objects have a meet
semilattice structure, i.e., m> uj.


89
Lemma 6.3.1. Let f : A > B be a meetsemilattice map. f is F-regular epi if and only if f
is surjective.
Proof. Since P-regular epimorphisms are always surjective, one implication is trivial. For
the other, suppose / : A B is a surjection preserving binary meets. Then
x k V <=> f(xAy) = f(x).
Thus we have the sequence x,x A y, y, where f(x) = f(x A y) and x Ay < y. By Lemma
6.1.5, this proves / is regular epi.
As noted in the proof, the relation < is completely described by K(f). If objects
have a meetsemilattice structure, one may use congruences (equivalence relations that are
simultaneously (j, m)-subalgebras) rather than the more complex pocongruences.
Finally we have:
Theorem 6.3.2. Suppose m > to and j and m preserve regular epis. Let f : A > B be a
P]m map. The following are equivalent.
1. f is surjective,
2. f is a quotient by some congruence K,
3. f is P regular epi,
4. f P^-regular epi,
5. f is Ff,-extremal epi.
Proof. The basic facts about pocongruences show (1 2). The preceding lemma shows
(1 <=> 3). (4 => 5) holds in any category.
To show (2 => 4), assume / : A > A/K. Since K C A x A, we have projection
maps ir i, 7t2 : K > A. f is plainly the coequalizer of 7ri,7r2.


90
(5 => 2) holds because we may factor any map / through its associated pocogruence.
If / is extremal epi, then / must coincide with the quotient by its associated pocongruence.

Corollary 6.3.3. Let j and m be subset systems which preserve regular epis. Let m > w.
The forgetful functors Up : FJm > P and USet : P, ~1' Set preserve and reflect regular epis.
Lemma 6.3.4. The forgetful functor Um^ : DP^ > Set reflects kernel pairs.
Proof. One verifies that the kernel pair of a map / : A B (calculated in DPJ consists of
the projections (from Ax A) restricted to the set
{(x,y) A x A : f(x) = f(y)}.

Applying Theorem 3.3.5, one obtains the following.
Corollary 6.3.5. Let j and m be subset systems which preserve regular epis. Let m > uj.
The forgetful functor Uset MP, ~1 Set is monadic.
6.4 Coequalizers in DP,
Theorem 6.4.1. For any pair g,h : A > B IMP^, the coequalizer coeq( Proof. The construction of the coequalizer of maps g and h is illustrated by the following
diagram. The maps on the diagram will be defined below.
~D{C)


91
Let a : B > B/ -< be the map given by Construction 6.2.6. Let / : B > C be any
BP^ map such that fg = fh. The existence and uniqueness of / is given by 6.2.6.
To construct the leftmost square, we apply the free completely distributive complete
lattice functor D = lD; r¡ is the natural transformation which injects a poset H into DH.
To construct the right-outside square, we note that (oo, oo)-pocongruences are closed
under intersection. So there is a smallest (in the sense that it makes the fewest possible
identifications) (oo, oo)-quotient
k : D(B/ <) > E(B/ X)
such that k(r](B/ X))a preserves all (j, m)-optimum bounds. Similarly, define
£ : D(C) -> E(C)
to be the smallest quotient such that £(r¡C)f preserves all (j, m)-bounds. Since
£D(J)(V(B/ l))a = £(VC)fa = £(rjC)f,
preserves all (j, m)-bounds we have the induced map E(f).
Define Eo to be the smallest BP^-subobject of E(B/ <) through which k(rj(B/ ^))
factors. (By Proposition 5.3.15, the intersection defining Eo exists.) The maps j : B/ < >
Eq and E0 > E(B/ X) are obtained by factoring (r¡(B/ -<))a) through E0.
By construction, the map £(r¡C) : C > E(C) has the universal property that any map
(j, m)-optimum bound preserving : C B, with B Obj(BP^) factors as (f) = t£(r¡C) for
a uniquely determined map t. Since C Obj(BP^r[), there is such a (j) which is P-extremal
mono. Thus, £(r]C) is P-extremal mono. Therefore, the P-(epi, extremal mono) factorization
of £(r)C) produces the factorization £(jgC) = qm. By construction, m is both P-epi and
P-extremal mono; so m is a P-isomorphism. Moreover, q : C > E(C) is necessarily the
smallest BP^-subobject of E(C') through which £{gC) factors.
We claim that coeq(g,h) = Eo via j : B > Eo. It suffices to show the existence
of a unique compatible map : Bo C. The commutativity of the largest rectangle


92
in the diagram above implies that k(rj(B/ ;<)) factors uniquely through the KMP^-object
E(f)~1(C) C E(B). Because E0 is the smallest BP^-object through which k(r](B/ :<))
factors, Eq C E(f)~l(C). This insures the existence and uniqueness of *.


CHAPTER 7
(j,m)-SPACES
This chapter studies spaces obeying a To-style separation axiom. Section 7.1
defines spaces and describes a functorial Galois connection, which specializes to Galois con
nections between DP^ and SJm. Section 7.2 develops a convenient description of epimorphisms
in SJm, which generalizes a known characterization of epimorphisms of To-spaces. Section 7.3
gives constructions of limits, similar to those for topological spaces. Section 7.4 describes
quotient maps, and characterizes extremal and regular epis as quotient maps.
The last two sections are related to the problem of finding reflections in SJm] Section 7.5
gives the flat spectrum (co)reflection on spatial objects which is equivalent to a reflection
on Sj. This reflection on spaces is a generalization of the Ti-reflection of topological spaces.
Last, but not least, Section 7.6 partially describes the epicomplete SJrn- objects; the
description is complete for T0-spaces. Epicomplete To-spaces are chains with the specializa
tion order. Products of epicomplete T0-spaces are not epicomplete, so there is no functorial
epicompletion in the category of To-spaces.
As mentioned in the introductory chapter, the research leading to this dissertation
began in an attempt to find reflections and coreflections in categories of generalized topo
logical spaces. After proving the results of this chapter, and reading Meseguer [25], the
author realized that additional assumptions where required on subset systems to insure that
subspaces could be reasonably defined. This realization prompted much of the thought sum
marized in Chapter 6 in particular Section 6.3; the author wanted to find when the theory
of this chapter was valid. These considerations, and construction of free DP^-objects, be
came the main focus of the dissertation. However, this state of affairs leaves many questions
concerning reflections and coreflections in SJm untouched.
93


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SUBSET SYSTEMS AND GENERALIZED DISTRIBUTIVE LATTICES
By
ERIC R. ZENK
A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL
OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT
OF THE REQUIREMENTS FOR THE DEGREE OF
DOCTOR OF PHILOSOPHY
UNIVERSITY OF FLORIDA
2004

ACKNOWLEDGMENTS
I would like to thank -
• Jorge Martinez. Sometimes words are inadaquate. He has been kind, generous, patient,
and interesting to work with. I have grown under his guidance.
• The people who inhabited the University of Florida math department from 1999 to
2004. They make the department the wonderful place it is.
• My friend Cielo: earth is wonderful, when one can see into the sky.
• My M. L.
• My mathematical siblings - other graduate students who worked with Jorge, especially
Ricardo Carrera.
• My family - Wayne, Phyllis, Margie, Jeff, and Rob.
• Those teachers who, by challenging me, caused me to improve.
• Participants in the “f-Rings and Ordered Algebraic Structures” conferences in Gaines¬
ville and Nashville during my time in graduate school. Because of these conferences, I
feel like I am joining a family of researchers, rather than just “getting a degree”. It is
an honor to know them.
• Those who helped proofread this document: constructive comments were made by
Jorge Martinez, Scott McCullough, and Pham Tiep.
• Anyone taking the time to read these words: a dissertation, like any book, is meant to
be read.
u

TABLE OF CONTENTS
ACKNOWLEDGEMENTS ü
ABSTRACT v
CHAPTERS
1 INTRODUCTION 1
1.1 Distributive Lattices 1
1.2 Subset Systems
1.3 Methods and Results 5
2 PRIMER ON CATEGORIES AND POSETS 8
2.1 Distinguished Maps
2.2 Bounds 12
2.3 Natural Transformations 17
2.4 (Co)Limits 19
2.5 Adjoint Functors 23
3 ALGEBRAS OF A MONAD 28
3.1 Categories of Algebras 28
3.2 Adjoint Connections induce Monads 32
3.3 Detecting Categories of Algebras 34
3.4 Distributive Laws 37
4 GENERATING SUBMONADS 42
4.1 Subfunctors 42
4.2 Meseguer’s Lemmas 50
4.3 A Partial Algebra Which Does Not Extend 59
5 FREE ALGEBRAS 63
5.1 Complete semilattices 64
5.2 Completely Distributive Complete Lattices 68
5.3 Some categories of algebras 71
6 COEQUALIZERS 82
6.1 Epis and Equalizers in P 82
iii

6.2 Factorization of Maps Using Preorders 85
6.3 Factorization of Meetsemilattice maps 88
6.4 Coequalizers in DU”, 90
7 (j,m)-SPACES 93
7.1 Spatial/Sober Functorial Galois Connection 94
7.2 The Skula Topology and Extremal Monos 97
7.3 Computing Limits 100
7.4 Quotients, Extremal and Regular Epis 101
7.5 Flat Spectra 104
7.6 Epicomplete objects in 105
REFERENCES 108
BIOGRAPHICAL SKETCH HO

Abstract of Dissertation Presented to the Graduate School
of the University of Florida in Partial Fulfillment of the
Requirements for the Degree of Doctor of Philosophy
SUBSET SYSTEMS AND GENERALIZED DISTRIBUTIVE LATTICES
By
Eric R. Zenk
August 2004
Chairman: Jorge Martinez
Major Department: Mathematics
Distributive lattices - alone, or with enriched structure - are mathematical objects
of fundamental importance. This text studies generalized distributive lattices: the general¬
ization is that certain infinite meets and joins are required to exist. Subset systems (natural
rules which select a family of subsets of each poset) j and m label which sets have joins and
meets, respectively.
A calculus of subfunctors is developed: using this calculus, it is shown that any
subfunctor F of a monad (containing the image of the unit) generates a submonad F. Under
suitable conditions, any partial F-algebra extends to an F-algebra. The monad F for the free
distributive (j. m)-complete lattice is the submonad of the completely distributive complete
lattice monad generated by a subfunctor obtained from j and m.
The category DP-^ of (j. m)-complete lattices which can be embedded in a completely
distributive complete lattice is a full subcategory of F-algebras. DP(n is complete and has
coequalizers.
(j, m)-complete families of subsets of a set (generalized topological spaces) are inves¬
tigated in analogy to classical point-set topology. Assuming suitable restrictions on j and

m, subspaces can be defined. Assuming these restrictions, there are well-behaved categories
corresponding to To- and sober- spaces.

CHAPTER 1
INTRODUCTION
1.1 Distributive Lattices
A semilattice is a set with an operation A satisfying the following universally quantified
equations
• a A a = a,
• a A b — b A a, and
• (a A b) A c = a A (b A c).
A and V, which are related by the condition
a V b — b.
< b to mean a A b = a; with this order a A b
is the largest thing smaller than both a and b and a V b is the smallest thing larger than both
a and b.
A lattice is distributive if either of the following, equivalent, universally quantified
equations hold
a A (b V c) — (a A b) V (a A c),
a V (6 A c) = (a V b) A (a V c).
Let us consider the concept and its relevance.
A distributive lattice bears some resemblance to ordinary arithemetic where A and
V correspond to addition and multiplication; the principal difficulty with this view is that
a A a = a V a = a, which does not hold in arithmetic. There is more symmetry in the
A lattice is a set with two semilattice operations
that
a A b = a 4=>-
A lattice may be partially ordered by defining a
1

2
equations defining distributive lattices than in ordinary arithmetic; formally “A” and “V” are
interchangable, and switching them reverses the order. Distributive lattices are interesting
algebraic structures in the same right as rings - structures with + and â–  sensibly defined.
Another perspective is that distributive lattices are models of logic, with “A” repre¬
senting “and” and “V” representing “or.” Obviously, the connectives “and” and “or” both
satisfy the semilattice rules. In this interpretation, the distributive laws are tautologies and
a < b means the proposition a implies b. Conventional logics are often described by dis¬
tributive lattices obeying extra equations, which correspond to additional tautologies to be
modeled.
A perspective particularly revelant to the author is that distributive lattices (with
some additional structure) describe topological situations. Intuitively, a topological space
is an amorphous blob, from which certain pieces can be cleanly removed. The removable
pieces are called closed parts and the complements (i.e, things left over after a closed part
has been removed) are called open parts. The usual definition of a topological space is a set
X, together with a designated family of open subsets, such that
• X and the empty subset are open,
• if U and V are open, so is the intersection U ft V, and
• if (Ui) is a family of open subsets, then the union |Jt is also open.
The lattice of open sets encodes how a space is woven together. Real analysis provides some
justification for the usual definition of a topological space. However, the author wondered
how the concept of a topology changes if one varies the definition by requiring either fewer
unions of open sets be open, or more intersections of open sets be open.
1.2 Subset Systems
A category is an abstract class of objects with structure, and maps (or, homomor-
phisms) which preserve the relevant structure. The maps in a category allow comparisons

3
between objects. Category theory allows formal comparisons between various theories of
algebraic objects: e.g., one can compare the category of all rings with the category of all
distributive lattices.
Many categories of partially ordered sets (with additional structure) fit into a simple,
general pattern. The objects are partially ordered sets in which certain intentionally distin¬
guished subsets have suprema or infima while the maps are order preserving functions which
preserve said infima and suprema. P denotes the category of all posets and order preserving
maps.
The challenge is how one selects subsets which have infima and suprema. We follow
Thatcher, Wright and Wagner [31], who introduced the useful (but blandly named) concept
of a subset system.
Definition and Remarks 1.2.1. A subset system Z, is a rule which assigns a family Z(A)
of subsets to any poset A, such that for any order preserving map / : A —> B,
{,f(S) : 5 G Z(A)} C Z(B).
Z-complete posets are posets A in which each set S G Z(A) has a supremum. (See Erne [7]
for more extensive bibliography regarding subset systems.)
The following examples of subset systems may convey the generality and usefulness
of the concept.
1. If k is a cardinal, we use k to denote the subset system which selects all subsets
with cardinality less than k. We use oo for the functor which places no restriction on
cardinality. This is a subset system because for any function / : A —> B, S C A, with
|A| < k implies |/(A)| < k.
2. A subset S C A is (upward) directed if for each x,y G S, there exists u G S such that
x < u and y < it. The rule dir which selects all directed subsets of a poset is a subset
system: if f{x),f(y) G /(S') and S is directed, then there is u G S such that x < u
and y < u. Thus f(u) is a common upper bound for f{x),f(y).

4
3. A subset S C A is (upward) compatible if for any x,y € S there is u € A such that
x < u and y < u. “Compatible” differs from “directed” because for the former u 6 S,
while for the later we only require u G A. Similar arguments show that compat, which
selects compatible subsets is a subset system.
4. A subset of S C A is a chain if x, y € S implies x < y or y < x. The rule ch which
selects all chains in a poset is a subset system.
5.We say a subset 5 C A is (upward) self-bounded if there is s 6 S such that for all
x € S, x < s; a self-bounded set contains a maximum element. The rule sb, which
selects all self-bounded sets, is a subset system. Note that any order preserving map
preserves joins of upward self-bounded sets.
6.A non-example: An anti-chain is a set of pairwise incomparable elements. The rule
ac which selects anti-chains is not a subset system, because there is an order preserving
surjection / : D —> 2, where D is the two-point anti-chain and 2 is the two point chain.
7.Generating examples: Let Q be any class of posets closed under order preserving
surjections. One may define a subset system Zq by
Za(A) = {S C A : S e Q}.
This construction shows there is a great multitude of subset systems. The subset sys¬
tem, compat, described above, is not generated this way, because one cannot determine
if S C A is compatible merely by looking at the poset S with its induced order.
We use subset systems, which we generically call j and m, to select which subsets
have joins and meets, respectively. Now we enumerate some categories of interest in this
discussion.
- the category of all (j, m)-complete posets: that is, posets in which j-suprema and
TO-infima exist and are preserved by all maps.

5
DP^ - the category of all completely distributive complete lattices.
DP^ - the full subcategory of P^ containing objects which can be P^-embedded in a
completely distributive complete lattice. DP^ is discussed in Section 5.3.
M¿, - the full subcategory of P^ containing posets with an F structure, where F is
the monad defined in Section 5.3.
SpFJm - the full subcategory of P£, containing spatial posets. See Chapter 7.
SJm - the category of generalized spaces with (j, m)-complete families of distinguished
subsets: see Chapter 7.
F£, - the full subcategory of SpTPJm containing spatial posets with flat spectrum: A is
defined to have flat spectrum if the maps A —+ 2 are trivially ordered: see Section 7.5.
1.3 Methods and Results
The discussion of DP^ and Mjn, which offer generalizations of distributive lattices,
uses the language of category theory. A quick introduction occurs in Chapter 2. The crucial
notion of a free object is formalized by monads, which are introduced in Chapter 3.
Chapter 4 describes a theory of subfunctors. The class of subfunctors of a functor
F bears a strong similarity to the power set lattice of a set X. Given a monad, with
functor part T, Meseguer’s Lemmas 4.2.1, 4.2.2, 4.2.6, and 4.2.7, show that any subfunctor
F of T containing all constants has a “monadic closure,” i.e., a smallest submonad F of T
which exceeds F. (Meseguer’s Lemmas were formulated and proved by the author, but the
technique is similar to one in Meseguer [25].) Intuitively, TX is the full set of polynomials (in
the sense of universal algebra) with variables in X, FX is a natural subset of polynomials,
and FX is the smallest natural subset of polynomials which is closed under composition
and contains FX. An algebra structure for T is a way of evaluating “all polynomials”; a
partial algebra structure for F is a way of “evaluating polynomials in F." Under suitable
conditions, partial algebras extend to F-algebras.

6
In Chapter 5, Meseguer’s Lemmas are brought to bear upon monads for free complete
semilattices and free completely distributive complete lattices. Given any subset systems j
and m, there is a submonad F of the free completely distributive complete lattice monad.
The category of F-algebras offers a (somewhat mysterious) generalization of the category
of distributive lattices. The subcategory BP^, containing all (j, m)-complete posets which
may be embedded in a completely distributive lattice is somewhat easier to understand and
still well behaved.
The existence of free objects in OP^ contrasts with the nonexistence of free objects
in P££ [9] and the category of complete Boolean algebras [8, 9], A fundamental difference
between these categories and DP¿, is the requirement that joins and meets obey a distributive
law.
The power of category theory comes as much from what it ignores as what it examines.
Significant conclusions are often obtained without examining the “grubby details” of what
is going on. But this innocence of “grubby details” limits the scope of investigation. In
the case of this document, several nicely behaved categories - M?m, BP^,, and SpPJrn - are
introduced. For general subset systems j and m, the author does not even know if these
categories differ! The end of Section 5.3 - from Corollary 5.3.10 onwards - describes most
of the author’s knowledge on the relationship between these categories.
Chapter 6 explores congruences, quotients and coequalizers in P, P3m and OP¿,. Much
classical algebra (ring theory, lattice theory, group theory, etc.) is simplified by the fact that
any surjection is a regular epimorphism. For the categories introduced here, the situation is
not so simple. Example 6.2.5 shows that a j-join preserving surjective image of a j-complete
poset need not be j-complete.
The results of Chapter 7 pre-date the other results presented here. Herrlich [12]
contains a detailed examination of reflections (and coreflections) in categories of topological
spaces. This dissertation aimed to generalize results summarized in Herrlich [12], for (j,m)-
spaces. A (j, m)-space consists of an underlying set and a family of “open” subsets which are

7
closed under j-unions and m-intersections. Continuous maps of (j, m)-spaces are functions
such that preimages of open sets are open. The initial aim was to find reflections and
coreflections of the category of (j, m)-spaces (obeying a To-style separation axiom), and
study how the existence and properties of reflections and coreflections varied depending
upon the subset systems j and m. An obvious prerequisite to such a project is knowledge
of factorizations of continuous maps.
The chapter contains a description of (j, m)-subspaces and (j, m)-quotients. In addi¬
tion, Section 7.5 describes a reflection of (j, m)-spaces that corresponds to the 7) reflection
of topological spaces. Lastly, Section 7.6 describes epicomplete (j, m)-spaces.

CHAPTER 2
PRIMER ON CATEGORIES AND POSETS
The text assumes a familiarity with the theory of sets typically used in mathematical
arguments. So familiar constructions - unions, intersections, cartesian products, quotients
by equivalence classes, functions, Zorn’s Lemma, and transfinite induction - are used without
further comment. (See Halmos [10] if this background is needed.) A basic familarity with
general topology is helpful.
Also some comfort with category theory is assumed. Namely, the reader can fill in
the blanks in the following informal definitions.
• A category A consists of a class of objects Obj(A) and maps Map(A), such that each
object has an identity map, and there is an associative notion of composition of maps.
• The set of A-maps from A\ to A2 is denoted A(Ai, A2).
• A functor F : A —> 23 assigns each A G Obj(A) an object F(A) € Obj(23) and
each A-map / : A\ —» A2 a 23-map Ff : F(A1) —> F(A2). The assignment respects
composition and identity arrows.
• If A is a category, Aop is the category with the same objects as A, but all arrows
reversed. For a category theoretic concept C, the dual is obtained by applying C to
A°p.
• A contravariant functor A —+ 23 is a functor A —► Bop.
• Diagrams are used to display the behavior of a collection of maps; a diagram commutes
if any composites with the same domain and codomain are equal. For example, the
8

9
diagram
W
f
h
9
Y
*Z
commutes if and only if gf — ih.
Recall the following properties of functors:
Definition 2.0.1. Let F : A —> B be a functor. For each Ai, A2 E A, F gives a function
from the hom-set A(Ai, A2) into B(FAi, FA2) by
(/ : Ax -» A2) ~ (Ff : Aj. - A2).
If, for each Ai and A2 this map is onto, then F is said to be full. If, for each A\ and A2 this
map is one-to-one, then F is said to be faithful.
A full subcategory of a category A is a category ‘B such that Obj(S) C Obj(A) and
all / : A —> B with A, B G Obj(B) are B-maps. B C A is full if, and only if, the inclusion
functor is full.
Good general references for category theory are MacLane [21], Borceux [6], and Her-
rlich and Strecker [13]. MacLane [21] gives a concise, high level summary of most category
theory and includes a chapter on monads. Herrlich and Strecker [13] is quite user friendly and
concretely describes many examples of adjoint functors. Borceux [6] covers a large amount
of material; the exposition is clear and very detailed.
2.1 Distinguished Maps
Definition and Remarks 2.1.1. Begin by defining a dual pair of concepts which coincide
with the notions “injective” and “surjective” in the category Set.
1. A map / : A\ —> A2 is epi, a.k.a epic (in noun form, an epimorphism) if
whenever g and h are maps A2 —> A3 such that gf = hf, then g = h.

10
2. A map / : A2 —> A3 is mono, a.k.a. monic (in noun form, a monomorphism)
if whenever g and h are maps A\ —> A2 and such that fg = fh, then g — h.
One may verify that a composition of epimorphisms (resp. monomorphisms) is epi (resp.
mono). Moreover, if f = ab is epi (resp. mono), then a is also epi (resp. b is also mono).
Definition 2.1.2. A map / : A —> B is an isomorphism, if there is g \ B —* A such that
id.4 = gf and idB = fg.
In most categories of “sets with structure”: a map is mono if and only if it is injective,
surjective maps are epi, but epimorphisms may not be surjective.
Example 2.1.3. Consider tfAb - the category of torsion-free abelian groups, i.e., abelian
groups such that
na = 0 => a = 0
for any natural number n and group element a, together with group homomorphisms. The
inclusion i : Z —» Q of the integers in the rational numbers is epi, but not onto.
In categories of “sets with relational structure,” bijective maps are not necessarily
isomorphisms.
Example 2.1.4. Consider Top - the category of topological spaces and continuous maps.
The identity function i : —> R from the reals (with discrete topology) to the reals (with
the usual topology), is a continuous bijection. However, the inverse function i_1 is not
continuous.
Example 2.1.5. Consider P - the category of partially ordered sets and order preserving
maps. Either bijection two elements is order preserving. But the inverse function 4>~l is not order preserving.
For further discussion and more examples of epimorphisms and monomorphisms see
Herrlich and Strecker [13, Section 6] and Borceux [6, Volume 1, Sections 1.7 and 1.8].

11
Definition and Remarks 2.1.6. Consider a pair of maps f,g:A\—* A2. A map i : Aq —■*
A\ right-identifies f and g if fi = gi. A map i : Aq —> A\ is called the equalizer of / and g
if:
(eql) i right-identifies / and g, and
(eq2) i has the feature that whenever j : B0 —> A\ right-identifies / and g, there
is a unique map e : Bo —> Aq such that j — ie.
The definite article is used for equalizers, because (eq2) implies that if i : Ao —* A and
i' : A'0 —> A are equalizers for / and g, then there is an isomorphism j \ Aq—* Aq such that
i' — ij. Notation: i = eq(/, If there are / and g such that i : A0 —* A\ is the equalizer of / and g, then i is regular
mono. Regular monomorphisms are monomorphisms. If / is epi and regular mono, then / is
an isomorphism. For proofs of the assertions in this paragraph and a discussion of examples,
see Borceux [6, Volume 1, Section 2] or Herrlich and Strecker [13, Section 16].
The definitions of “coequalizer” and “regular epi” are dual to “equalizer” and “regular
mono,” but are repeated for emphasis, e : A2 —* A3 left-identifies f and g if ef = eg. A
map e : A2 —* A3 is the coequalizer of / and g if
(coeql) e left-identifies / and g, and
(coeq2) e has the feature that whenever d : Ai —»• B3 left-identifies / and g, there is
a unique map c : A3 —» B3 such that e = cd.
Notation: e = coeq(/, g). If e is the coequalizer of some pair of maps, then e is called
regular epi The duals of all basic properties of regular monomorphisms hold for regular
epimorphisms.
Definition 2.1.7. An epimorphism / is extremal if whenever f = gh and g is mono, then g
is an isomorphism. Dually, a monomorphism / is extremal if whenever / = gh and h is epi,
then h is an isomorphism. For more detailed discussions, see Borceux [6, Volume 1, Section
4.3] and Herrlich and Strecker [13, Section 17].

12
Definition 2.1.8. The map / : A —► B is split mono if there exists g : B —> A such that
id¿ = gf. The map g : B —* A is split epi if there exists / : A —> B such that id^ = gf. For
more information, see Herrlich and Strecker [13, Section 5]. Note that our terminology differs
slightly from the reference; “section” and “split mono” are synonyms, as are “retraction”
and “split epi.”
Lemma 2.1.9. For A either “mono” or “epi,” consider the following statements.
1. f is split-A.
2. f is regular-A.
3. f is extremal-A.
4. f is A.
The implications 1 => 2 =4* 3 => 4 always hold. None of the converses generally hold.
For proof see: (1 => 2) Herrlich and Strecker [13, 16.15], (2 => 3) Herrlich and Strecker
[13, 17.11] or Borceux [6, Volume 1, 4.3.3(1)], (3 =» 4) holds by definition.
2.2 Bounds
Recall that a partial order on a set A is a relation < satisfying:
(pol) For all a 6 A, a < a.
(po2) Whenever a < b and b < c, a < c.
(po3) Whenever a < b and b < a, a = b.
A preorder is a relation that satisfies (pol) and (po2). If ^ is a preorder on A, define an
equivalence relation ~ on A by
a ~ b <=> a ^ b and b â– < a.
The relation is a partial order on A/ ~. A set A with a partial order (resp. preorder) is
called a partially ordered set (resp. preordered set).

13
Definition and Remarks 2.2.1. Let A be a preordered set. Define the up- and down-
closures of x 6 A by
| x = {a € A : a ^ x}
and
^ x = {a €. A : x ■< a}.
More generally, if 5 C A define
I 5 — U{| i:iGS} = {a€yl:3sGS,aXs}
and
| 5 = U{f i :i £ 5} = {a G j4 : 3s 6 S', s ^ a}.
If 5 C A an upper (resp. lower) bound for 5 is a £ A such that for all s G 5, s ■< a (resp.
a â– < s). Use the following notation for the set of upper bounds of 5,
1(5) = n{f x : x € 5} = {a G A : Vs 6 5, s ■< a}
and similar notation for the set of lower bounds
1(5) = n{j x : x € 5} = (a G A : Vs G 5, a ■< s}.
If a G A, S C A and f a = 1(5), then a is a least upper bound, a.k.a. join, a.k.a.
supremum of 5. An equivalent way to say this is
a < x <=> V s € 5, s < x.
If both a and a' are suprema of 5, then a,a' € 1(5). Thus a ~ a'. In a partially ordered
set, the (unique) supremum of 5 is written \/ 5. If a € A, S C A and j a — 1(5), then a is
a greatest lower bound, a.k.a. meet, a.k.a. infimum of 5. An equivalent way to say a is a
least upper bound is
x < a V s € 5, x < s.

14
In a partially ordered set, the (unique) infimum of 5 is denoted /\ 5. (In using this notion
with a preordered set (A, ^), one refers to the associated poset A/ For example, \J S
then denotes the equivalence class containing all suprema of 5.) If a = V 5 or a = f\ 5, a is
an optimum bound for 5.
The following are elementary properties of the bound-operators and optimum bounds.
Lemma 2.2.2. For any preordered set A (in particular any poset), the following properties
hold. Use IB to denote either operator B or B.
1. For x,y € A, x ^ y [ x C j y <==> ] y C] x.
2. For S,T C A, S CT => B(T) C B(5).
3. For SC A, 1(5) = I(| 5), 1(5) - 1(T 5).
4- For (5j)j6/ a family of subsets of A, B(u5¿) = nB(5¿).
5- If f : A —y B is order preserving, then ffE^S)) C Bs(/(5)).
Definition and Remarks 2.2.3. Let A and B be preordered sets. A Galois connection
between A and B is a pair of functions / : A —> B and g : B —> A such that
(gel) / and g are order reversing.
(gc2) For all a € A and b G B, a ^ g(f{a)) and b X f(g(b)).
Below, basic properties of Galois connections are listed. Symmetry in the definition
allows symmetry in proofs. For any true statement about Galois connections, then the
statement obtained by switching the roles of / and g, along with A and B is also a true
statement about Galois connections.
Note that / and g respect ~. The notation / ~ g means “for all a € A, f(a) ~ g(b)."
1 ■ 9 - gfg■ Suppose a £ A. By (gc2), a r< f(g{a)) and g(a) ■< g(f(g{a))). Using
a ^ f(g{a)) and (gel), g(f(g(a))) < g(a).

15
2. f(A/ ~) is dually order isomorphic to {6 G B : b ~ fg{b)}: This follows from the
preceding statement. Since / ~ fgf, any member f(a) G f(A/ ~) is equivalent to
f{g{f(a))). Moreover, if b G B and b ~ f(g(b)), then b G f(A/ ~) because b ~ f(g(b)).
3. a < g{b) b â– < /(a): if a ^ g(b), then b â– < f(g(b)) < f{a). The converse is proved
similarly.
4. For any S C A, Bf(S) = ^_1B(5). Calculate
x G B(/(5)) Vsg5,iS/(s)
<=^- Vs G S, s ^ 5(1)
<=* #)eI(S)
•<=>• a; G p-1®(5)
5. If A and B are posets, the previous item implies /\/(5) = /(V 5).
It is helpful to rephrase condition 3 for posets: ug(b) is the largest a with b < /(a).”
In symbols,
g(b) = \/{a E A:b -< /(a)}.
Basic information about Galois connections has been well-known sincethe 1940s;
Raney [28] contains a bibliography of this early literature. The basic properties and def¬
inition are listed in Herrlich and Strecker [13, Exercise 27Q]. The particular summary here
is by the author.
The concept of a Galois connection is symmetric, and allows one to transfer a great
deal of information between preordered sets. However, the fact that the functions involved
are order reversing is sometimes inconvenient. The concept of an adjoint connection between
posets is obtained by formally reversing one of the posets involved.

16
Definition and Remarks 2.2.4. Suppose A and B are preordered sets. A pair / : A —> B,
g : B —> A of order preserving functions is an adjoint connection between A and B if
(adl) For all a 6 A, a â– < g(f(a)).
(ad2) For all b e B, f(g(b)) < b.
Each basic property for Galois connections corresponds to a basic property of adjoint
connections. The basic properties of adjoint connections are listed below; proofs are omitted.
1 • / - fgf and g ~ gfg
2. f(A/ ~) = {b G B : b ~ g(f{b))} and g{B/ ~) = {a 6 A : a ~ f(g(a))}.
3. /(a) ^ b ■<=>• a ^ g(b)
4. For any SC A, Bg(S) — /_1B(5) and B/(S) = g_1B(5)
5. If A and B are posets, /(/\ S) = f\ f(S) and g(\J S) = \/ g(S).
Because of the asymmetry between / and g, and property 3, / is called the left adjoint
and g the right adjoint. Again there is an interpretation of 3 in words, “/(a) is the smallest
b such that a ■< g{b)\ g(b) is the largest a such that /(a) ^ 6.”
The definition and basic properties of adjoint connections are “folklore.” For another
discussion of them see Johnstone [16, Chapter I, Paragraph 3].
Definition 2.2.5. A poset A is complete if each subset SC A (including the empty set)
has a supremum. Since the supremum of set of lower bounds for S is a lower bound for 5,
A is complete if and only if each subset S C A (including the empty set) has an infimum.
There is a criterion for testing when a given order preserving (resp. reversing) map
between posets is part of an adjoint (resp. Galois) connection.
Theorem 2.2.6. Adjoint Existence Suppose A and B are posets.

17
1. Suppose f : A —> B is order preserving and A is complete. Then f is a left adjoint if
and only if f(\J S) = \J f(S) for all S C A.
2. Suppose g : A —» B is order preserving and A is complete. Then g is a right adjoint if
and only if g(f\ S) = /\ g(S) for all S C A.
3. Suppose f : A —> B is order reversing and A is complete. Then f is part of a Galois
connection if and only if f(\J S) = f\ f(S) for all SC A.
This theorem is the poset version of the adjoint functor theorem. Folklore: see Johnstone
[16, Chapter I, Section 4, Paragraph 2] or Joyal and Tierney [17, Chapter 1, Section 1].
Proof. A proof for 1 follows; the other items are similar. Define g : B —> A by
9ÍP) = \J{aeA: f(a) < b}.
Since / preserves suprema, f(g(b)) < b. Thus, g(b) is the largest a such that /(a) In adjoint (resp. Galois) connections, /(a) and g(b) can be defined as suprema or
intima. There is a sort of converse to this; one can view the supremum as an adjoint to a
particular map. See Lemma 5.1.1.
2.3 Natural Transformations
Maps compare objects in a category, functors compare categories, and natural trans¬
formations compare functors. This section contains no new results; results and some expo¬
sition are paraphrased from Herrlich and Strecker [13, Section 13].
Definition 2.3.1. Let F, G : A —> 3 be functors. A natural transformation a : F —> G is
a rule which assigns a map olA : FA —> GA to each A E A such that if / : A —> B is an
A-map the following diagram commutes.
Ff
FA —>FB
r\A
V B
GA
Gf
GB

18
Construction 2.3.2. Let F, G, H : X —» A be functors, a : F —> G and f3 : G —> H natural
transformations. (The situation is drawn in the diagram below.)
F
G
Q
H
' J
0
Then the assignment (/3a)A = (/3A)(aA) is a natural transformation.
Proof. Let / : A —> B be a map. Since a and (3 are natural, each square below commutes.
FA
Ff
â– *FB
GA-
Gf
GB
HA
as
HB
Therefore, the outside rectangle commutes; hence, (3a is natural. â–¡
Call /3a the vertical composition of a and (3.
Construction 2.3.3. If F : A —> B and G,H : 3 —> C are functors and a : G —> H is a
natural transformation, then (aF)A := a(FA) is a natural transformation aF : GF —> HF.
Construction 2.3.4. If F : “B —> C and G,H : A —► B are functors and a : G —> H is a
natural transformation, then (Fa)A := F(aA) is a natural transformation Fa : FG —> FH.
Construction 2.3.5. Suppose F, G : A —> Ü3 and H,J:T>—*G are functors, and a : F —>
G, (3 : H —> J are natural transformations. (The situation is drawn below.)
F H
0 e

19
Then for each object A € Obj(.A) the following square commutes.
FHA F0A > FJA
aHA
aJA
GHA—G0A >GJA
The assignment (/3 â–  a)A := (aJA)(F/3A) = (G(3A)(aHA) is a natural transformation.
Proof. The square commutes because a is a natural transformation; to see this, one applies
the natural property at the map (3A : HA —+ JA. To prove the natural property of Pa,
we use the squares which define Pa A and 0aB. Compare the corners of the squares using
maps obtained from / by application of the functors FH, GH, FJ, and GJ. The resulting
commutative cube shows Pa is a natural transformation. â–¡
Call P • a the horizontal composition of a and p.
Definition 2.3.6. A natural equivalence a : F —* G is a natural transformation such that
each component aA is an isomorphism. Given functors F and G are naturally equivalent if
there is a natural equivalence a : F —> G.
Categories A and B are equivalent if there are functors F : A —> B and G : B —> A
such that FG is naturally equivalent to ids and GF is naturally equivalent to id^.
A and B are dual if there exist contravariant functors F : A —> B and G : B —> A
such that FG is naturally equivalent to ids and GF is naturally equivalent to id^.
2.4 (Co)Limits
For intuition, it is useful to view the objects of a category as a preordered class, with
the preorder A < B if and only if there is a map f : A —* B. Categories are more complex
than preordered classes, because there may be many maps / which manifest A < B.
Definition and Remarks 2.4.1. Let A be a category. A diagram D in A (a.k.a. a small
subcategory of A) is a set of objects Obj(£>) C Obj(A) and maps Map(D) C Map(A)

20
between them; for technical reasons one requires that if f,g G Map(D) and fg is defined
then fg G Map(D), and that for all A G Obj(D), idA € Map(D). This definition of a diagram
is equivalent to, but differs from the one in the majority of the literature; see MacLane [21],
Borceux [6], or Herrlich and Strecker [13] for the standard definition.
Suppose D is a diagram. A source (S, {s^ : S —» A : A G Obj(£))}) for D consists of
S G Obj(A) and maps sA : S -> A such that if / : A -> A' is a map in D, fsA = sA>. A source
for D is a “lower bound” compatible with all maps in D. A source (L, {tA : A G Obj(Z))})
for D is the limit of D if whenever (5, : A G Obj(D)}) is a source, there is a unique map
c : S —► L such that for each A G Obj(D), sA — iAc.
Dually, define a sink, or cosource for D to be an object S together with maps iA :
A —> S (for A G Obj(D)) such that for each f : A—> A' in Map(D), iA = iA>f.
A colimit (C, {¡jla : A G Obj(D)}) for D is a sink for D, such that if (5, {iA : A G
Obj(D)}) is any sink there is a unique map c : C —> S such that for each A G Obj(D),
Ha = ciA.
Note that limits and colimits of D are unique up to compatible isomorphism. To
prove this (for limits), suppose (L,£A) and (L',£'A) are both limits for D. The limit property
guarantees that there are unique compatible maps c : L —> L' and d : L' —> L. But
cd : L' —> L' and dc : L -> L are both compatible maps. By uniqueness, cd = id'£ and
dc = idL. From now on, we shall use the definite article when writing about limits and
colimits.
Example 2.4.2. Let us consider Set. If X is a set of sets, form a diagram containing
all members of X and no functions. The limit for this diagram is the Cartesian product
(nx,7rx).
Recall that ¡Q X contains all functions / : X —* (J X,with the feature that f(X) G X
for all X G X. Such / are called choice functions, because they choose one member of each
X G X. The functions nx 'WX X are defined by nx(f) = f(X).

21
If there is a source (5, {s* : S -> X : X € X}), there is a unique function c : S -» f] ^
making
commute for each X. It is defined by c(a)(X) — sx(a).
Motivated by this example, one defines the category-theoretic product of a set X of
objects (in any category) as the limit of the diagram containing all members of X and no
maps. In most categories of “sets with structure” products (exist and) look like products in
Set, with suitable structure added.
In Set, the colimit of the diagram containing all members of X and no functions,
is the disjoint union ]J X, with inclusion maps px '■ X —i► JJ X. The colimit property is
satisfied by (JJ X, px) because if there is a sink (C, ix), the function c : JJ X —► C - defined
by c(x) = ix{x) for the unique X € X containing x - is the unique compatible map.
In categories other than Set, coproducts are defined identically. Usually the coprod¬
uct of a set X of ^.-objects is “the ^1-object freely generated by JJ X.”
Example 2.4.3. The [co]equalizer of /, g : A —> B is the [co]limit of the diagram containing
objects A and B along with maps / and g. The notation for equalizers is customarily
simplified by omitting the source map to B. Previously, i : E —> A was defined to be the
equalizer if fi = gi and i factors through any other map which right identifies / and g. i
is the source map to A. The source map to B is redundant: it must be fi = gi : E —* B.
Similar notational economy is applied to coequalizers.

22
Example 2.4.4. Let us consider the diagram
A
f
B - *C.
(The diagram also contains identities for all objects, but for brevity these are omitted.) The
limit (B Xc A, nA, ttb) of this diagram is called the pullback of / along g, or the pullback of g
along /. To be explicit, fnA = 7rBg and if (Q,qA : Q —* A,qs ■ Q —► B) satisfies fqA = qsg,
then there is a unique map i : Q —> B Xp A.
In Set,
B xc A = {(b, c) e B x A: g(b) = /(c)},
and the projection maps n are the restrictions of the projections from the cartesian product.
Pullbacks in any concrete category A - equipped with a limit preserving faithful functor
U : A —* Set - are computed identically. Two particular instances of pullbacks deserve
special attention.
First, let / : A —> C be any map and g : S —> C be a subset inclusion of S C C.
Note that (s, a) € S xc A if and only if g(s) = /(a); suppressing mention of g, this reads
S xc A — {(s,6) : f(b) = s}. Thus, this pullback is canonically isomorphic to the preimage
of S under /. This example partially motivates the name “pullback.”
Second, let / : A —> B be any map and consider the pullback of the diagram,
A
f
A *B
i.e., the pullback of / along itself. Using the computation for pullbacks in Set, given above,
A xB A = {(a, a') e A x A : f(a) = /(a')}.

23
This relation on A is often called the kernel of /. Thus, one calls (Axb A, nie/t, irnght) the
kernel pair of /. More in-depth discussion of pullbacks is given in Borceux [6, Volume 1,
Section 2.5] and Herrlich and Strecker [13, Section 21].
Definition and Remarks 2.4.5. A category is said to be complete if each diagram has a
limit; it is said to be cocomplete if each diagram has a colimit.
Unlike the situation for posets, a category may be complete without being cocomplete
and vice versa. (See Herrlich and Strecher [13, Section 23, pl61ff] for a detailed discussion
of this and related issues.)
Consider a functor F : A —> B. If D is a diagram in A, there is a diagram FD with
Obj (FD) = {FA : A e Obj(D)} and Map (FD) = {Ff : f 6 Map(D)}.
Let D be a diagram in A. A functor F : A —> B preserves the limit of D, if whenever
the limit (L,£a) exists in A, (DL, D(¿a)) is the limit of FD. F preserves limits if for any
diagram D, F preserves the limit of D.
If G : A —» B is contravariant, D is a diagram in A, and (L, {£a ■ A € Obj(D)}) is
the limit of D, then (GL, {GiA : GA —> GL}) is a sink for GD. If (GL, {G¿a '■ GA —► GL})
is the colimit for GD, then G takes limits to colimits. Analogously, may G take colimits to
limits.
2.5 Adjoint Functors
There are several useful concepts of adjoint connections between categories. There is
a strong analogy between preordered sets and categories; any category may be preordered
by
A^B <=> 3/ : A -> B.
Categories are more complex, because many maps could manifest A A B. We begin with
the concept analogous to 2.2.3.
Definition and Remarks 2.5.1. A (functorial) Galois connection between categories A
and 'B consists of contravariant functors F : A —> “B, G : B A, together with natural

24
transformations
77 : id* —» GF
and
e:W3 —> FG
such that (Fr]A)(eFA) = id^ and (GeB)(r]GB) = ides for each A 6 Obj(A) and B €
Obj(33); these equations are the so-called triangle identities. (Alternate terminology: if
(F,G,rj,e) is a functorial Galois connection, then F and G are adjoint on the right.)
1. The categories fix(r¡) and fix(e) - containing all objects such that r]A (resp eB) is an
isomorphism - are dual. [3, Section 4, Lemma 1]
2. There is a natural bijection A{A, GB) —* ®(5, FA) given by
(f : A —> GB) H-> F(f)(eB) : B —> FA.
The inverse map is
(g : B —> FA) >-> G(f)(vA) : A -> GB.
There are two (identical) calculations required to check that the functions are mutually
inverse. One is summarized by the diagram below.
GFA —(GFf) > GFGB —— * GB
The square commutes because 77 is natural. The triangle commutes because of the
identity ides = (GeB)(r¡GB). The reader may formulate and check what is meant by
“naturality” of the bijection.
3.F and G both take colimits to limits.

25
4. Each rjA has the following universal property: if / : A —► GB, there is a unique map
/ = F(f)(eB) : B —► EM which makes the following diagram commute:
A ——*■ GFA
Each eB has the analogous universal property.
Much of the literature just deals with adjoint connections, where both functors are covariant.
By duality, any such result can be translated in terms of Galois connections. Banaschewski
and Bruns [3] includes a reasonably thorough expository section on functors which are adjoint
on the right.
The “(functorial) Galois connection” concept is symmetric, but the contravariance of
the functors involved is sometimes awkward. An analogous asymmetric concept, with the
functors both covariant is described below.
Definition and Remarks 2.5.2. A (functorial) adjoint connection between catego-ries A
and 3 consists of functors F : A —> 3 and G : 3 —► A and natural transformations
77 : idyi —> GF and e : FG —> id® such that (GeB)(r¡GB) = id^s and (eFA)(Fr¡A) = id^;
these equations are the so-called triangle identities. In this situation, one also says “F and
G are adjoint functors,” “F is the left adjoint” and “G is the right adjoint.” The basic
properties of functorial adjoint connections closely correspond to the basic properties of
functorial Galois connections.
1. The categories fix(r/) and fix(e) - containing all objects such that 7/.4 (resp eB) is an
isomorphism - are equivalent.
2. There is a natural bijection A(A, GB) —> 3(FA, B) given by
(f : A—* GB) t-> (eB)F(f) : FA —> B.

26
The inverse map is
(g:B-+FA)~ G(f)(r¡A) : A —> GB.
3. F preserves colimits; G preserves limits. Borceux [6, Volume 1, 3.2.2]
4. Each rjA has the following universal property: if / : A —> GB, there is a unique map
/ = (eB)F(f) : FA —> B which makes the following diagram commute:
A ——>GFA
Each eB has the analogous universal property: if / : FA —► B, there is a unique
/ = G(f)(r)A) : A —* GB such that
commutes. Any functor for which there is a natural transformation q with the above
universal property is part of an adjoint connection.
For discussions of adjoint functors, see MacLane [21, Chapter IV], Herrlich and Strecker [13,
Sections 26, 27, 28], or Borceux [6, Volume 1, Chapter 3].
There is a criterion for determining when functors are adjoints, which corresponds to
Theorem 2.2.6.
Theorem 2.5.3. (Adjoint Functor Theorem, [21, V.6.2],) Let A be a complete category.
A functor G : A —> B has a left adjoint if and only if
1. G preserves limits, and

27
2. (solution set condition) for each B € 3 there is a set I and an I-indexed family of
maps fi'.B—> GAi such that any map f : B —> GA can be written as h = (Gt)fi for
some i 6 / and t : A¡ —> A.

CHAPTER 3
ALGEBRAS OF A MONAD
Monads (a.k.a. triples, a.k.a. standard constructions) and their (Eilenberg-Moore)
algebras provide a concise formulation of many important categorical aspects of universal
algebra. The category of algebras for a monad has special properties, which are summarized
in Section 3.1; a category of algebras is often complete and cocomplete, and always has
“free objects.” Each (functorial) adjoint connection induces a monad; the correspondence
between adjunctions and monads is discussed in Section 3.2. In Section 3.3, the question of
when an adjoint connection connects A to a category of algebras is addressed. In Section
3.4, the question “when does the composite of two monadic adjunctions yield a monadic
adjunction?” is discussed.
The results in this chapter are reasonably well known. MacLane [21, Chapter VI],
Borceux [6, Volume 2, Chapter 4], Barr and Wells [4, Chapters 3 and 9], Manes [23] and
the introduction to the seminar notes [1] contain good expositions of monads from various
perspectives.
3.1 Categories of Algebras
Definition 3.1.1. A monad T = (T, rj, /i) on A consists of a functor
T:A^A,
a natural transformation
V : idA -> T,
and a natural transformation
/i: T2 —> T,
28

29
such that the following identities (expressed by commutative diagrams) hold: the unit laws
(In these diagrams, Tn denotes the n-fold composite of T with itself.)
Intuitively, TA is the free object on A; r]A is the “insertion of variables” map; fjA
is the “semantic composition,” i.e., a map which allows one to view a polynomial with
polynomial variables as a polynomial. See Example 3.2.2 for a concrete illustration of the
roles of T, 77 and ¡1.
Definition and Remarks 3.1.2. Let T be a monad on A. A T-algebra (A, a) consists of
A € Obj(A) and a : TA —> A (the so-called structure map) such that a(rjA) — kU (unit
law) and
T2A >TA
M a
TA

30
(associative law) commutes. A homomorphism / : (A, a) —> (B, b) of T-algebras is an A-map
/ such that
Tf
TA —>TB
a b
commutes. The category of all T-algebras and T-algebra homomorphisms is denoted A1.
There is a forgetful functor UT : Ar —♦ A; it is defined by UT(A, a) = A and
UT(f) = f. UT has a left adjoint FT : A —► AT defined by
Ft{A) = (TA,iiA)
and
FT(f) = Tf.
The associated natural transformations are
7?r = 77: kU - UtFt = T
and
eT : FtUt —> id^r : eT(A, a) a.
[21, VI.2.Theorem 1]
Limits in Ar are “computed in A” in the following sense.
Proposition 3.1.3. [4, 3.3.4], [6, Volume 2, 4.3.1] Suppose (T,rj,p) is a monad on A. If D
is a diagram in AT such that UT(D) has a limit (L,pu(A)), then there is a unique structure
map I :TL —> L such that each pu(A) a T-algebra homomorphism.
Proof. Let D be a diagram in AT, such that UTD has a limit (L, IA). For each A G Obj(D),
name the structure map s¿ : TA A. The goal is to produce a structure map s : TL —> L
such that each : L —> A is a T-algebra map.

31
The requirement that each is T-algebra map amounts to: for each A, the diagram
below commutes.
FL
SA
Since (L,£a) is a source for UD, (FL, sa(FIa)) is a source for UD. Thus, there is a unique
A map s : FL —> L making each diagram above commute.
Checking the unit law, s(iL) = id¿: the diagram
commutes using the definition of s, and that i is a natural transformation. Thus, for each
A, £a — £as(íL). Prom the uniqueness of the map from the limit of D to any other source
for D, it follows that id/, = s(iL).
A similar “comparision of squares diagram” can be used to verify that the algebra
associative law holds. â–¡
Corollary 3.1.4. If A is complete and T is a monad on A, then AT is complete.
If A is cocomplete, AT is often also cocomplete. The following theorem was originally
proved with fewer hypotheses in Linton [20], Other expositions are given in Borceux [6,
Volume 2, 4.3.4] and Barr and Wells [4, Section 9.3].
Theorem 3.1.5. Let A be cocomplete and T be a monad on T. AT is cocomplete if and
only if AJ has coequalizers.

32
3.2 Adjoint Connections induce Monads
The following proposition gives a correspondence between monads and adjoint con¬
nections. It should be emphasized that the correspondence is not bijective. Each monad
gives rise to a unique adjoint connection, but in general many adjunctions induce the same
monad.
Proposition 3.2.1. Correspondence between monads and adjoint connections
1. Let F : A —> 3, G : "B —> A, r¡ : kU —> GF, e : FG —> id® be an adjoint connection.
Then (GF,r¡,GeF) is a monad on A.
2. If T = (T, T], /i) is a monad, then
GtFt = T,
T
V =v,
and
GTeTFT = fi.
The proof of the preceding Proposition consists of verifying identities: the triangle
identities for adjoint connections imply the unit laws; the associative law holds because
the square defining horizontal compositions commutes. Details are given in MacLane [21,
VI. 2. Theorem 1],
This correspondence allows construction of many examples of monads. Often, but not
always, “a naturally occurring” adjoint connection corresponds to the category of algebras
over the induced monads.
Example 3.2.2. Consider the category Grp of groups. The forgetful functor Ucrp ■ Grp —»
Set and the free group functor FgtP : Grp —» Set form an adjoint connection between Grp
and Set. Recall that ForpiX) is the set of all reduced words

33
where x¿ 6 X, s¿ 6 Z; a word is reduced if for all i with 2 < i < n, x,_i / x¿. The operation
on Fcrp(X) is concatenation of words, followed by reduction. T = Ucrp^Grp is the functor
part of the induced monad; 77 : idset —> T is the natural transformation whose component at
X sends x € X to word x1; if w“l • • • iu“n 6 T2(X) and for each i, = x^1 • • • then
•••<■) = (*5y-^a{r-tóí-*í3!3r-.
6l.m(U'l
Each group G is an T-algebra; the group multiplication and inversion give a map from the free
group on the underlying set of G to G. Conversely, each T-algebra structure gives a group
multiplication and inversion. Grp is equivalent to the category of Setr. (This example is
typical in the sense that any category of finitary algebras - in the sense of universal algebra
- is also a category of monad algebras. The parts T, 77, and /r of a monad generally have the
same roles as in this example.)
Example 3.2.3. Some (non-trivial) adjunctions involving Set induce the trivial monad.
For example, consider the category Top of topological spaces and continuous maps. The
forgetful functor
U'Xop ■ Top —* Set
has a left adjoint
FtbP : Set ->• Top,
which sends a set X to the discrete topological space with underlying set X. One easily
checks that T = GTopFTop is the identity functor on Set, and that all associated natural
transformations are identities.
Definition 3.2.4. Suppose F : A —G:B—>Fl, 77: id^ -7 GF, and e : FG —> id®
forms an adjoint connection. By Proposition 3.2.1, the adjoint connection induces a monad
(T = GF, 77, GfiF). If B is equivalent to AT, then we say the adjoint connection (sometimes
just the right adjoint G) is monadic. Often a concrete category B has a “canonical” forgetful
functor G : B —> A] in this case, one may even say B is monadic over A - omitting mention
of G.

34
These examples illustrate the qualitatively different behaviors of monadic and non-
monadic adjoint connections. Monadic categories are determined by the combinatorial struc¬
ture of the free algebra functor FT. Most “relational” categories - like Top, P, the category
of graphs, etc. - have free functors, but these free functors do not add any structure to the
underlying set; they merely attach the “most discrete” possible relation to the given set.
3.3 Detecting Categories of Algebras
Because of the special properties of AT, the question of when an adjoint connection
is monadic has great practical importance.
Remark 3.3.1. Let (A, a) be a T-algebra. Because of the associative and unit laws, and
because 77 is natural, the following equations hold:
(nat) (Ta){r[TA) — (rp4)a,
(unitl) (fiA)(rjTA) = idTA,
(unit2) a(r)A) = id¿, and
(assoc) a(Ta) = a(fiA).
These equations imply that a — coeq(¡iA,Ta)\ for if b : TA —► B right-identifies ¡1A and Ta,
then b(r]A) : A —> B and
b(r¡A)a = b(Ta){T]TA) by (nat)
= b(fiA)(r]TA) since b right identifies
= b by (unitl).
One verifies that / = b{qA) the unique map / : A —> B with fa = for if fa = b, then
/ = fa(rjA) = b{r¡A).
These equations give a great deal of information about a. Thus, the key hypothesis
in Beck’s theorem - the criterion for when an adjunction is monadic - is the preservation
and reflection coequalizers obeying the equations described above.
Using the above described equations for motivation we offer the following definitions.

35
Definition and Remarks 3.3.2. Consider maps f,g:A—>B.
1. If e : B —» C has the feature that for any functor F, Fe = coeq(Ff, Fg) then we say e
is an absolute coequalizer.
2. If there are maps e : B —> C, sc : C —> B and sb : B —> A such that
fsB
id/i = gsB,
idc = esc,
ef = eg,
then we say e is a split coequalizer.
Note that for every T-algebra {A, a), a is a split coequalizer of Ta and pA. It is also easy to
see that every split coequalizer is absolute.
The following theorem is due to Beck (unpublished). Linton [19] contains a detailed
discussion of variations on the theorem. The theorem was originally phrased in terms of
split coequalizers only; Pare [26] refined the theorem to include the “absolute coequalizer”
condition. Many variations on the hypotheses exist; the version stated here is found in
MacLane [21, VI.7.1], Other expositions of the theorem may be found in Barr and Wells [4,
Section 3.3] and Borceux [6, Volume 2, Section 4.4].
Theorem 3.3.3. Let (F : A —> 25, G : 3 —> A, r?, e) be an adjoint connection, and T = GF,
rj — q, and p = GeF be the associated monad. The following conditions are equivalent:
1. The adjunction (F,G,r],e) is monadic.
2. If f,g : A —► B 6 Map(!B) and the pair (Gf,Gg) has an absolute coequalizer e!, then
e = coeq(f,g) exists and Ge — e'.

36
3. If /, g : A —* B € Map(!B) and the pair (Gf,Gg) has a split coequalizer e', then
e = coeq(f,g) exists and Ge = e'.
Recognizing monadic adjunctions is complex because compositions of monadic func¬
tors are not generally monadic.
Example 3.3.4. The category Ab - of abelian groups - is monadic over Set, and tfAb (see
Example 2.1.3) is monadic over Ab, because any reflection is monadic. Each free abelian
group is torsion free, so the monad on Set induced by the free abelian group functor is the
same as the monad induced by the “free torsion free abelian group functor.” Thus, if tfAb
were monadic over set, then tfAb and Ab would be equivalent categories - both equivalent
to a suitable category of monad algebras. The categories tfAb and Ab are obviously not
equivalent. (The preceding example is summarized from Borceux [6, Volume 2, Example
4.6.4].)
If G : A —* Set, the hypotheses of Beck’s theorem can be reformulated to make it
easier to check whether G is monadic.
Theorem 3.3.5. [11, Theorem 4.2] Let G : A —> Set be a functor with a left adjoint.
Suppose that A is complete and has coequalizers. The following are equivalent:
1. G is monadic.
2. G satisfies the following conditions
• G preserves and reflects regular epimorphisms;
• if f : GA —> X is an isomorphism, then there is a unique map g : A —> B such
that Gg — f;
• G reflects kernel pairs.
The following lemma provides useful information concerning when a functor between
two categories of algebras is monadic.

37
Lemma 3.3.6. [6, Volume 2, Corollary 4.5.7] Suppose U : 23 —> A, V : G —> A, and
Q : 23 —> C are functors. IfU — VQ, U and V are monadic, and 23 has coequalizers, then Q
is monadic. In particular, Q has a left adjoint.
3.4 Distributive Laws
Compositions of monadic adjunctions are not generally monadic; when a composition
of monadic adjunctions is monadic, it indicates a distributive law between the two structures.
The following section summarizes the results later needed from Beck’s [5]; this exposition
follows Beck’s notation, except that composition of functions here will read right-to-left.
For the duration of the section, assume there are two monads T = (T, t]t, pT) and S =
(S, rjs, ps) over some base category A. For any monad, we use F and U, with superscripts for
the name of the monad, to denote the free algebra functor and forgetful functor associated
with a monad, respectively. See 3.1.2 for the definitions of F and U.
This material is rather abstract, so it helps to have an example in mind: after each
definition and theorem, we will illustrate what it means using S - the free monoid monad
over Set - and T - the free abelian group monad over Set; the composite monoid TS gives
the free ring. For a set X, SX consists of all “strings” from X, with concatenation as the
operation: 5 acts on functions by
Sf(xxx2 • • • xn) = f(x1)f(x2) ■ • • f(xn).
TX consists of all formal (finite) linear combinations of elements of X, with integer coeffi¬
cients. T acts on functions in the expected way.
Despite the concrete illustrations in terms of these monads, all theorems and defintions
will apply to any monads S and T.
Definition and Remarks 3.4.1. A distributive law of S over T is a natural transformation
£ : ST->TS

38
satisfying the compatibility conditions:
T S
If there is a distributive law £ : ST —► TS of S over T, then the composite monad is TS =
(TS,r]TS := r]TT]s,pTS := pT ps(T£S)). In the definitions of r¡TS and pTS, juxtaposition
denotes horizontal composition of natural transformations. The verification that TS actually
defines a monad is omitted; for more information see Beck [5],
In the case when “S=free monoid” and “T=free abelian group,” the natural trans¬
formation i : ST —> TS expresses a product-of-sums as a sum-of-products in the usual
way:
n 5^ aitk»-+ n ac(k),k
keK iei c keK
where c ranges over all choice functions c : K —> I. It is a somewhat enlightening exercise
to check that Í - defined this way - is a natural transformation ST —► TS.
A distributive law may of T over S also be viewed as a “way of lifting T to a monad
over S.

39
Definition 3.4.2. T has a lifting into ./Is, if there is a monad T such that TUS = UST,
UsrjT — r]TUs and Usp^ = pTUs. To sketch the situation, T is a lifting of T onto if
commutes.
Theorem 3.4.3. There is a bijective correspondence between liftings of T to /Is and dis¬
tributive laws of S over T.
The proof is outlined in Beck [5]. The correspondence is defined as follows: if l is a
distributive law, and (A, a) is an 5-algebra, then T is defined as follows:
T(A,a) = (TA,(Ta)(£A)),
rjT(A, a) = t]TA : (A, a) -> T(A, a),
and
fiT(A, a) = pTA : TT(A, a) -» T(A, a).
Consider the following diagram:
the square commutes because r¡T is a natural transformation; the triangle commutes because
l is compatible with rf. Since the perimeter of the diagram commutes, rjTA is an 5-algebra

40
map, A similar diagram, which uses the compatibility between l and pT, shows pF is an
5-algebra map. Since the underlying maps are defined in A and obey the monad laws, tjt
and pT also obey the monad laws. If T is a lifting of T over As, i is defined to be the
following composition:
ST STS = STUsFs = USFSUSTFS usflfs USTFS = ST.
(In the above equation, es is the counit of the adjunction (Fs, Us, r/5, es); es(A. a) = a.)
After some detailed computation, one verfies that this is a distributive law, and that the
correspondences described are mutually inverse.
Corollary 3.4.4. If T has a lifting to As, then there is a composite monad TS.
Theorem 3.4.5. Suppose I: ST —> TS is a distributive law. The categories ATS and (AS)T
are equivalent.
Let us consider this in more detail. An object in (AS)T consists of an A-object,
A, along with an 5-structure as : SA —> A, T-structure ar : TA —+ A, and 5-structure
t: ST A —> A for TA such that the following diagram commutes.
SciT
ST A 1—- SA
t aS
TA — >A
Given a T5-algbra {A, a), A has a T-structure aT map defined by
aT = a(Tr]SA) :TA->A
and an 5-structure map as defined by
as = a(r]TSA) : SA —> A.

41
The following diagram - which expresses the distributivity of the structure maps - commutes.
Thus we have a map ATS —> (AS)T. given by,
(A, a) e ATS i * (A, ar, as,Tas(£A))
which is functorial, because both rj1' and rf are natural.
The inverse functor (AS)T —> ATS, maps (A,ar,as,t) to asS(ar) — art. The verifi¬
cations for these assertions is given in Beck [5],

CHAPTER 4
GENERATING SUBMONADS
In this chapter, a technique for creating monads is laid out; before proceeding formally,
let us consider a rough outline of the technique. Suppose (T, r], /r) is a monad on a reasonable
category, and that / : F —* T is a subfunctor of T. The goal is to extend F to a submonad
of T.
Intuitively speaking, F is a natural collection of polynomials. One should therefore
require that for all A, (r/A)(A) C F(A) - i.e., that for each A and a G A, FA contains the
“constant polynomial with value a”. F is not generally “closed under composition”; i.e., it
is not generally true that (nA)(F2A) C A. To correct this problem, we start with F and
iteratively add polynomials obtained by composing members of FA.
The purpose of the chapter is to formalize the preceding vague outline. The author
abstracted and clarified [25, Proposition 3.4], which may be viewed as a special case of this
result. Thus, the main results of this chapter are called Meseguer’s Lemmas - 4.2.1, 4.2.2,
4.2.6, and 4.2.7 - to acknowledge the analogy which prompted the technique. The author
believes the general formulation of the technique is new.
Section 4.1 details requirements on a “reasonable category”, explains precisely what
is meant by “subfunctor”, and gives methods for constructing subfunctors. Section 4.2 gives
a proof of Meseguer’s Lemmas. Section 4.3 describes an example showing the necessity of a
technical hypothesis of the lemmas.
4.1 Subfunctors
This section describes a general theory of subfunctors. A subfunctor of F : X —> Set
is a natural transformation r] : E —+ F such that each component is a subset inclusion. The
42

43
concept “subfunctor” is quite useful, but in some categories such as P, Top, and Locale
(explained in Example 4.1.2), there is either
• no obvious meaning for subset inclusion (in Locale), or
• more than one possible structure on each subset (in P and Top).
The following section expounds a theory in which “subfunctor” means “natural transforma¬
tion whose components are all extremal mono”.
Axioms 4.1.1. Throughout, the base category A is assumed to have the following features:
1. A is complete.
2. A is an (epi, extremal mono) - category. Recall that an (epi, extremal mono) - category
is a category with the features that
• For each map /, it is possible to factor / as / = me where e is epi and m is
extremal mono. This factorization is unique, in the sense that if / = m'e' is
another way of writing / as an epi followed by an extremal mono, there is an
isomorphism i such that
commutes.
• if a and b are extremal mono and ab is defined, then ab is extremal mono.
3.A is extremally-wellpowered. This means: for each A, there is a set
{fx 'â–  S\-+ A}
of extremal monomorphisms, such that if m : T —► A is any extremal monomorphism,
there is an index A and isomorphism i such that f\ — mi.

44
If A has these features, then it is an SF-category. (“SF” stands for “subfunctor”.)
Example 4.1.1. By Herrlich and Strecker [13, 34.5], any wellpowered category with in¬
tersections and equalizers is an (epi, extremal mono) - category. So if A is complete and
wellpowered then A is an SF-category.
It follows that P, Set, Top and practically any reasonable category of topological
spaces, is an SF-category.
Example 4.1.2. The category of frames - i.e., complete lattices in which
a a\J S = ^ {a A s \ s E S}
holds for all elements a and subsets S - is complete and cocomplete. This category also has
(regular epi, mono) factorizations. Thus, Locale, the dual category to the category of frames,
is an SF-category. Locale is extremally-wellpowered, but not wellpowered. The author’s
interest in locales motivated him to use the given definition (which only requires extremal¬
wellpoweredness) for SF-category rather than defining SF-category to mean “complete and
wellpowered”, so that the theory of subfunctors would apply to localic subfunctors in addition
to the previous examples. (For background on frames and their relation to point-set topology,
see Isbell [15], Johnstone [16], Joyal and Tierney[17] and Madden [22].)
Lemma 4.1.3. Assume that A is an (epi, extremal mono) - category. Then
1. (diagonalization) [13, 33.3] If ge = mf, where e is epi and m is extremal mono, then
there exists k such that mk = g and ke = f.
2. [13, 34.2(2)] Any intersection of extremal subobjects is extremal.
3. [13, 34.2(3)] Pullbacks of extremal monomorphisms are extremal mono.
Definition and Remarks 4.1.4. Defining Sub(A) - the lattice of extremal subob¬
jects: In an SF-category A, the lattice of (equivalence classes) of extremal subobjects has
particularly nice structural features.

45
An extremal subobject of A is an extremal monomorphism s : S —> A. Define a
preorder on the class of extremal subobjects by
(s : S —► A) C (t : T —* A) -*==>■ 3c : S —* T, s — tc.
For brevity, one often only mentions the map or object part of an extremal subobject. To
avoid confusion, the same letter will be used to denote both parts, with the lower case letter
used for the function.
Since t is mono, there is at most one map c which manifests s C t. If s C t and t C s,
then there are cj and c2 such that s — tc\ and t — sc2. So t = icjc2 and s = sc2cj. Since
c2ci shows s C s, c2cj = ids. Similarly, cic2 = idy. One identifies extremal subobjects s and
t if s C t and f C s, or equivalently, when there is an isomorphism c such that s — tc. The
set of equivalance classes under this relation is denoted Sub(A).
Any map c, which exhibits s C t, is extremal mono. For if ca — cb, then sa =
tea — teb = sb, so (since s is mono) a — 6; so c is mono. If c = me, where e is epi, then
s = tc = tme. Since s is extremal mono, e must be an isomorphism. Thus c is extremal
mono.
Recall that the intersection of a set M of monomorphisms with a common codomain
is the limit of the diagram generated by M. Since A is complete and intersections of extremal
monomorphisms are extremal, Sub(A) is a complete lattice, with meet operation - f) - and
join operation - |J. In general, joins in Sub(A) are not disjoint unions; in general, (J At is
computed using
(jAi = p|{A':Vz, ACA'}.
(See Borceux [6, Volume 1, 4.2.2, 4.2.3, 4.2.4].)
A map f : A —* B induces an adjoint connection between Sub(A) and Sub(B). (See
Borceux [6, Volume 1, 4.4.6]; Borceux’s results are phrased in terms of “strong monomor¬
phisms”. Under our assumptions a map is strong mono if and only if it is extremal mono.)
If s : S —* A € Sub(A), use the (epi, extremal mono) - factorization to obtain a unique

46
s' 6 Sub(£?) such that
A
f
B
S
S -/+1(S)
commutes. The notation /+1(5) is used for the image of S, to remind the reader of the
analogy to ordinary set theoretic images of subsets under maps. The map (s : S —> A) i—►
{s' : S' = f+1{S) —> B) is the left adjoint. The right adjoint f~l : Sub(S) —> Sub(A) is
obtained by taking the pullback of s : S —> B along /. Thus,
f+1{S) SCf-\T),
/+,(US') = U^1(S<).
and
r1{f]Si) = f]r1{Si).
Definition and Remarks 4.1.5. Let F : X —♦ A be a functor. A subfunctor E of F is a
rule that selects an extremal subobject eX : EX —> FX for each X G Obj(X) such that if
f : X —>Y, then
(Ff)+1(EX) C EY.
Any such assignment E of F gives rise to a functor E : X —> A. Ef : EX —> EY is defined
to be the composition EX —► (Ff)+1{EX) C EY.
Since eY : EY —> FT is mono, F/ is uniquely determined by the condition that
Ff
FX —>-FY
eX
eY
EX ——EY
commutes. Thus, (Ff)+1{EX) C EY implies that Ef can be defined to make the above
square commute.

47
The converse also holds: if there is a map Ef such that the square commutes, then
(Ff)+1(EX) C EY. To prove this, suppose there is a map Ef which makes the square
commute. Use the unique factorization Ef = ab where a : A —► EY is extremal mono and b
is epi. Note that (Ff)(eX) = (eY)(Ef) — (eY)ab gives a factorization of (Ff)(eX) into an
epi b followed by an extremal mono (eY)a. Thus,
[(eY)a : A —> FT] — [(Ff)+1(EX) -* FT]
and (Ff)+1(EX) C EY. Thus, a subfunctor is exactly “a natural transformation whose
components are all extremal monomorphisms”.
Let Subfun(F) denote the class of subfuctors of F : X —> A. Define a preorder on
Subfun(F) by
E\ C F2 <=> W1 € Obj(.A), Ei(A) C E2(A).
If Ei C E2, then, for each A, there is a unique extremal monomorphism cA such that
e\A — (e2A)(cA). One easily verifies that cA are the components of a natural transformation
c : Ex —> E2; in fact, c : Ex —► E2 is a subfunctor.
The following constructions show that the class Subfun(F) behaves very much like
Sub(j4).
Construction 4.1.6. Subfun(F) is complete. If {F¿ : i € 1} is any class of sub functors,
there is a supremum (JF¿ and infimum P) Et. The supremum is given by (IJEi)(A) —
UFj(j4). The infimum is given by (PF¿)(A) = P| Ei(A).
Proof. Since A is extremally wellpowered, for each A, the class {EfA) : i 6 1} has a
representative set. Thus, the object-by-object definitions for |J F, and fj Ex make sense. It
is obvious that if U F¿ and p) F¿ are subfunctors that they are the optimum bounds in the
subfunctor lattice.
Let f : X —*Y be any map. A h-> P|¿ EX(A) is a subfunctor because for each i € I,
(F/)+1(P| Eí(A)) C (Ff)+1(Ei(A)) C Et(B),

48
so (F/^a £i(¿))c a £(£)•
^ Ei(A) is a subfunctor because,
(F/)+1(LJf¿(A)) = U(Ff)+1(Ei{A)) C U^(5).
i i
a
Construction 4.1.7. Suppose a : F —+ G is a natural transformation. Then there is an
(order theoretic) adjoint conection
a+1 : Subfun(F) «-> Subfun(G) : a-1.
The left adjoint a+1 is defined by (a+1E)(A) = (aA)+1(EA). The right adjoint a-1 is defined
by (a~1E)(A) = a~1(EA).
Proof The order on Subfun(F) is defined “object-by-object”, and whenever
f : A B,
f+1 and f~l form an adjoint connection between Sub(.4) and Sub(F). So it suffices to check
that a+1E and a~lE define subfunctors.

49
Let E € Subfun(F), / : X —> Y be a map, and consider the following diagram.
(â– aA)+1(EA) (aB)+1(EB)
The trapezoids (i) and (iii) are obtained by factoring (aB)(eB) and (aA)(eA), respectively;
in each case 2 is the epi part and e' is the extremal mono part. The square (iv) expresses
the naturality of a. The trapezoid (ii) expresses that e : E —* F is a subfunctor.
To show a+1E is a subfunctor of G, it suffices to show there is a map
k : (aA)+\EA) -* (aB)+1(EB)
that makes the top trapezoid commute. For this, use the diagonalization property from
Lemma 4.1.3. Define / = zB(Ef), g = (Gf)e'A, e = zA and m = e'B; note that ge - mf
with e epi and m extremal mono. Thus, the diagonalization property guarantees the desired
k exists.
Assume f : A —> B. A similar diagram is used to verify that a_1E is a subfunctor
of F whenever E e Subfun(G). The missing map (aA) 1{EA) —► (aB) X{EB) is obtained
from the pullback property of (aB)~1(EB). â–¡

50
Corollary 4.1.8. Let a : F —> G be a natural transformation, and e : E —> F be a subfunc¬
tor of F. Let e' : a+1E —> F be the subfunctor described in Construction 4.1.7. There is a
natural transformation z : a+1E —> F such that for each object A, (aA) — (e'A)(zA) is the
(epi, extremal mono)-factorization of aA.
Proof. One defines e' and 2 as in the preceding proof. Examining the comparison of squares
diagram used to produce a+1E/, shows that 2 and e! are natural. â–¡
Construction 4.1.9. Assume E,F,G,H are functors A —> A. Suppose that
1. a : E —* F and (3 : G —> H are sub functors,
2. Either E or F preserves extremal monomorphisms.
then the horizontal composition (3a : EG —* FH is a subfunctor.
Proof. The horizontal composition of natural transformations is always natural. Hypothesis
2 implies that, for any A, (3aA is an extremal monomorphism, because the class of extremal
monomorphisms is closed under composition and
((3a)A := (aHA){E(3A) = (F(3A)(aGA),
by definition of horizontal composition (see Construction 2.3.5). â–¡
4.2 Meseguer’s Lemmas
Lemma 4.2.1. Suppose A is an SF-category, (T, p, rj) is a monad on A, and F is a sub¬
functor ofT, such that (r]A)+1{A) C FA. There is a smallest subfunctor F of T such that
F C F C T and for all A 6 Obj(^l)
(pA)(F2A) C FA.
If the equation
(fj,A)(G2A) C GA
holds for G, we say G is closed under p.

51
Proof. Let T denote the class of all subfunctors of T, that are larger than F and closed
under p. S' is nonempty because it contains T. By Construction 4.1.6, F = exists.
Since S (pA)(S) is an order preserving map Sub(T2v4) —* Sub(2L4), so for each A
{pA)(F2 A) C FA. â–¡
The preceding gives an easy candidate for the functor part of the monad generated
by F. One needs a more complex argument if one wants detailed information about the
natural transformations - related to r¡ and /i - which make F into a monad.
The proofs of the following lemmas require detailed computation. For clarity, the
goal of each paragraph in the proof is written in bold-face.
Lemma 4.2.2. (Assume notation and hypotheses from Lemma If.2.1.) There are natural
transformations ñ : id^ —» F and m : F2 —> F which make (F, n, m) a monad.
Proof. One defines four sequences of natural transformations:
1. f\'.F\—>T- the subfunctor generated at stage A,
2. n\ : id^ —» F\ - a natural transformation obtained from rj by suitably modifying the
domain and codomain,
3. m\ : Ff —> F\+1 - a natural transformation obtained from p by suitably modifying
domain and codomain, and
4. c\ : F\ —> F\+i - the inclusion.
Define F0 = F; use the notation f0 : F0 —> T. m0 and Co are defined according to the same
pattern that defines later “m”s and “c”s, which is described below.

52
Defining f\ : F\ —> T (A > 0), mx : Fx* -* Fx+i, and cx : Fx —» Fx+1. Assume
that fx has been previously defined. Consider the following diagram:
(M)+1(F\)u F*A
Given fx and (aA, the square is obtained by (epi, extremal mono)-factorization of the com¬
posite (fj,A)(fxA). All maps in the triangle are extremal monomorphisms, obtained by
comparing subobjects of TA. Define
FX+1A = (nA)+\F¡A) U FXA
and
mx : FXA —> Fx+iA
to be the map shown on the left side of the diagram. It follows from Constructions 4.1.7 and
4.1.6 that fx+1 : Fx+i —► T is a subfunctor. Note that
{fx+i)(mxA) = (nA)(flA)
and mx is natural. For bookkeeping purposes, let us call
c\: Fx —> Fx+1
the subfunctor which exhibits Fx C Fx+i.
If k, is a limit ordinal, define
Fk = : A < k};

53
fK'-FK—>T is a subfunctor by 4.1.6.
Defining the sequence of “n*” s: The following definition of nx is not recursive; n\
can be defined once we know f\, but the definition of f\ does not involve nx at all. Consider
the following commutative diagram of functors and natural transformations.
The natural transformations z and e are defined by condition that rj = ez is the (epi, extremal
mono)-factorization of 77, as described in 4.1.8. By assumption and the construction of the
sequence (Fx),
(r)idA)+1 CF0C Fx.
Let i\ denote the natural transformation such that e = fxiX) ix exists because e C i. Define
nx := i\z. Evidently,
V = f\nx
and
^a+i = cxnx.
What happens when the sequence terminates: Since A is extremally wellpow-
ered, for each A, the sequence (F\A)\ of subobjects of TA eventually terminates, say when
A = «1. Again using extremal wellpoweredness there is an ordinal, say k2, such that the
sequence (F\FK(A))x of subobjects of TFKlA terminates at k2. Define ñ = nK2, f — fK2
and m = mK2. These assignments give natural transformations; to check this, one considers
a map / : A —* B, and chooses k large enough that the subobject sequences (described
above) terminate for both A and B. It should be clear that each subobject sequence {FXA)x
terminates at FA.

54
Verifying that (F, n, m) is a monad. To prove the unit laws, it suffices that for
all A,
(mxA){FxnxA) = (mxA)(nxFxA) = cxA
(because, once the sequence terminates, the “caA”s become identity maps on FXA). Fix any
ordinal A. Naturality of fx and nx implies
CTnxA)(fxA) = (fxFxA)(FxnxA)
and
(nxTA)(fxA) = (FxfxA)(nxFxA);
Note that rj = fxnx and, by definition of horizontal composition,
flA = (TfxA)(fxFxA) = (fxTA)(FxfxA).
Calculating, one finds
(flA){FxnxA) = (fxTA)(FxfxA)(FxnxA) def. noted above
= (fxTA)(Fxr]A) noted above
= (TrjA){fxA). /a is natural
Thus,
UlA){FxnxA) - (TVA)(fxA)
and (by a similar computation)
(.flA){nxFxA) = (rjTA)(fxA).
To show (mxA)(FxnxA) = cxA, consider the following diagram:
FXA

55
the upper square commutes by an identity proved above; the lower square is the definition
of mx; the upper triangle is trivial; the lower triangle is the unit law for (T, r¡, /i). Reading
the perimeter of the diagram, one finds
fx A = (fx+1A)(mxA){FxnxA).
By uniqueness of maps manifesting inequalities between subobjects,
cxA = (mxA)(FxnxA).
The diagram needed to prove
{m\A)(nxFxA) = cxA
is similar and omitted. This establishes the unit laws.
To prove the associative law for (F,ñ,m), one may choose sufficently large A, then
draw a diagram comparing the associative squares for T and F, using appropriate powers of
fx to compare the corners. The comparison squares commute because of the definitions of
mx and horizontal composition. â–¡
Definition 4.2.3. Assume that A is cocomplete. In A, unions of chains are colimits, if
whenever D is a diagram, where Obj(D) consists of a chain of extremal subobjects of A and
Map(D) consists of all inclusions (in Sub(A)) that exist among elements of Obj(D), the map
c : colimD —> A induced by colimit properties is an extremal monomorphism.
In this case, c : colimD —> A is an extremal subobject which contains each element of
Obj(D); thus there is an extremal monomorphism colimD —> (J Obj(D). Since (J Obj(D) is
the supremum of Obj(D) in the Sub(A), this map must be an isomorphism.
Remark 4.2.4. The condition that unions of chains are colimits is satisfied in many nat¬
urally occuring categories. For example, in “relational” categories like Top, P, and Set,
extremal subobjects are just subsets with the induced structure, so all unions are colimits.
Assume A is a category of finitary algebras. The (set theoretic)-union of a chain
(Ai) of subobjects is a subobject. The (set theoretic)-union A' of (A,) lies inside a smallest

56
extremal subobject, which by our notation is (JA¿. Evidently, the coproduct of the chain
(Ai) of subobjects is A'. So the issue about whether “unions of chains are coproducts”
amounts to checking whether A! = 1J At. This depends on the delicate issue of whether
epimorphisms are surjective. If epis are not necessarily surjective, then 1J A2 will be the
largest subobject in which A! is epi, which will generally be larger than A'.
Definition 4.2.5. As above, assume A is an SF-category, is a monad, and F is
a subfunctor of T which contains r?+1 id^. A partial algebra (A, a) is an object A equipped
with a map a : FA —> A such that o(noA) = id^.
Let pAlg(Fo,no) denote the category of all partial algebras, with maps / : (A, a) —>
(B, b) such that / : A —> B is an A-map, and fa — b(Fof).
Lemma 4.2.6. Continue with situation and hypotheses from f.2.2. Suppose A is cocomplete
and unions of chains are colimits. Suppose (A, a) is a partial algebra. The partial algebra
structure a extends to an F-algebra structure map a : FA —> A if and only if for each ordinal
A there is a unique a^+i making the diagram below commute.
H(A) ——
m\
-Fx(A)
FX(A)—^ >A
Proof. To prove a map a : FA —> A exists, use transfinite induction. The hypothesis gives ax
at sucessor ordinals. To construct ax, when A is a limit ordinal use the coproduct property
of FXA — U«a
Inductively one shows that
(ext) for each A, ax — aA+i(cAA) (each ax+i extends the preceding ax),
(unit) for each A, ax(nxA) = kU,

57
If (unit) holds for sucessor ordinals, then it holds for all ordinals. Suppose
k, is a limit ordinal. The map aK is epi, because it is a limit of epimorphisms. Note that aK
and aKnKaK are both compatible maps FKA —> A; the definition of colimits implies that
because aK is epi, id^ = aKnK.
The algebra associative law holds for (^4, a). If a — ax, then ax = aA+1, so by
the hypothesis regarding the existence of ax such that a\+i(rri\A) = ax(Fxax) proves the
associative law.
If (unit) holds for A, then (ext) holds for A. Note that the equation a\+\(rn\A) —
a\{F\a\) defines a>+1. By induction hypothesis (unit),
(â– Fxa\)(FxnxA) = idFj,4;
therefore precomposing both sides of the defining equation for aA+i with (F\ri\A) shows
aA = ax+i(mxA)(FxnxA) = ax+i{cxA).
The second equality follows from (mxA)(FxnxA) = (cxA), which was proved in verifying the
unit laws for F.
If (unit) holds at A, then it holds for A + 1. Consider the following diagram.
The left triangle commutes because (unit) holds for A; the bottom trapezoid commutes by
definition of ax+1; the top trapezoid commutes because
(nxA)(FxnxA)(mxA) = (nxA)(cxA) = nx+1A.

58
The first equality holds by proof of the unit law for F; the second equality holds because of
the compatibility between “nA”s. Thus,
aA+i(nA+i) = ax+1(mxA)(FxnxA)(nxA)
= ax(Fxax)(FxnxA)(nxA)
= ax(nxA)
= hU;
this establishes the unit law for (A, a). â–¡
Use the notation Alg(F0, no) to denote the full subcategory of pAlg(Fo, no) containing
objects satisfying the hypotheses of Lemma 4.2.6.
Lemma 4.2.7. Suppose the preceding lemmas apply. Also suppose each mx is epi. The map
(A, a) i—> (A, a) induces an equivalence of categories; the functors involved are
E : Alg(F0, n0) -*• AT,
defined by extension of structure and
> Alg(F0,n0)
given by restriction.
Proof. E is legitmately defined. Lemma 4.2.6 defines E on objects. Maps in AF are
A-maps compatible with the structure. So it suffices to show that for any Alg(Fo,n0) map
f): A B,
FXA FXB
a\
bx
B

59
commutes for each ordinal A. By definition of Alg(F0, n0), the square commutes for A = 0.
Suppose it commutes for A; note that
ax(Fxax)
def. a^+i
= bx(Fx)(Fxax)
ind. hyp.
= bx(Fxbx)(F^)
ind. hyp.
= bx+l{mxB){Fl4>)
def. bx+1
= bx+l(Fx+i4>)(mxA)
naturality mx.
Because mx is epi, one concludes ). At limit ordinals, properties of
colimits insure that the diagram commutes. By transfinite induction, and the defintion of
the extended structure, any map which preserves (Fo, n0)-structure preserves F-structure.
One also notes that restriction respects maps, because Fo is a subfunctor of F.
E and R form an equivalence. Obviously,
V(A, a0) G Obj(Alg(Fo, n0)), RE(A,ao) — (A,a0).
The construction of the extended structure shows the restriction of any F-structure to Fo
uniquely determines the F-structure; hence
ER(A, a) = (A, a).
â–¡
4.3 A Partial Algebra Which Does Not Extend
The category Set is an SF-category and unions are colimits, so Meseguer’s Lemma
applies to Set. We show the necessity of Lemma 4.2.6’s hypothesis that a map ax+j such that
ax+imx = ax(Fxax) can be defined. The section discusses a subfunctor of the free-magma
monad, which has an algebra that cannot extend to a monad algebra. A magma is a set
with a binary operation, subject to no equations.
The free magma monad (T, rj, fi) has the following parts:
1. Given a set X, TX consists of all words with variables in A. A magma word is any
expression formed by finitely many applications of the rules:

60
(i) If x G X, then (^)is a magma word.
(ii) If s, t are magma words, then
is a magma word - the product of s and t. For ease in reading, we use binary tree
notation for products.
2. T defines a functor: given / : X —> Y, to compute Tf we apply / to all members of
X in a given word, leaving the tree and circle structure unchanged.
3. The “insertion of variables” map r¡X : X —> TX. (rjX)(x) = (^.
4. The “semantic composition” map [iX : T2X —> TX sends a word s G T2X of words
to a word in TX, by removing the circles around each element of TX used in making
s. n is also a natural transformation.
The notation takes a little while to soak in; to expedite the process, we consider a
calculation with TN. Suppose s G TN is the word
and ti = ss G T2(N) is the word

61
then (/¿N)(tx) is
Now we define a subfunctor F of T. Define the depth dep(n) of a node n in a binary
tree inductively by: the depth of the dep(root) = 0; if a is immediately below b, then the
dep(a) = dep(b) + 1. Define a leaf to be a node that has nothing below it. Let F consist of
all rooted, labeled, binary trees (i.e., magma words) such that the depth of each leaf is the
same. For example s and (/rN)(ti) are in F(N), but
is not, because dep(57) = 1 and dep(l) = dep(4) = 2. Evidently, F is a subfunctor, and for
any X, (r¡X)+1(X) C F(X). But, for any non-empty X, (pX)(F2(X)) is not contained in
F(X). One readily verifies that F = T.
Now we define an partial algebra structure on N that does not extend to a T-algebra
structure. As in Lemma 4.2.6, a structure map for a pair (F,rj : id^ —» F) is a map
a : FA —> A satisfying id¿ = a(rjA). Define n : F( N) -t Non a tree r as follows: if the
depth of each leaf of r is odd then n(r) is the leftmost label, if the depth of each leaf of r
is even (or zero), then n(r) is the rightmost label. For example, n(s) — 1 (s defined above)
and n(/iN)(ti) = 4.

62
In order for n : F(N) —> N to extend to n : Fi(N)
making
F2( N)
Fn
â– F{ N)
/xN
F( N)
n i
â– *N
commute. No such a: can exist, for (/zN)(ti) is
N there must be a function rii
which equals (/iN)(t2), where t2 is
Finally, note that (Fn)(ti) is
so n(Fn)(ti) = 1. Note that (Fn)(t2) = (/rN)(t2), so n(Fn)(t2) = 4. Thus we have elements
11,12 G F2N, that /xN identifies and n(Fn) does not identify, so there is no function n\ such
that n\(nN) = n(Fn).

CHAPTER 5
FREE ALGEBRAS
This chapter explores a generalization of complete distributivity for P^-objects. In
several steps, the free complete distributive lattice monad is constructed, then Meseguer’s
Lemmas are used to construct an appropriate submonad, whose algebras generalize com¬
pletely distributive lattices.
Section 5.1 describes monads for complete semilattices; D is the monad for complete
join semilattices, 11 is the monad for complete meet semilattices.
The gist of section 5.2 is that there is a lifting of 'll over T> (and a lifting of D over 11).
Therefore, UT> and T>U are monads over P. (See results of Beck [5] summarized in Section
3.4.) In Raney [27], it was shown that complete meet distributivity is the same as complete
join distributivity. Hence, the composite monads UT> and Dll have the same category of
algebras. The objects in either category are complete lattices where meets distribute over
joins, and joins distribute over meets.
Completely distributive complete lattices have been thoroughly studied. The basic
structure is described in Raney [27], [28], and [29]. Free objects over Set were initially
described in Markowsky [24]. Tunnicliff [32] discusses properties of the free completely
distributive lattice over a poset. Free objects over P and the relationship between completely
distributive lattices and continuous lattices are described in Hoffman and Mislove [14]. The
approach here is apparently new, but yields obviously equivalent free objects.
Each pair (j,m) of subset systems gives rise to a subfunctor of UT>. Meseguer’s
Lemmas are applied to this subfunctor to produce a monad F. Any P^,-objects which
63

64
is P^-embeddable in a completely distributive complete lattice is an F-algebra. Any F-
algebra has a natural structure. Because of computational difficulties, no exact algebraic
characterization of F-algebras is given here.
A word about notation: the (functor parts) of the monads described below are given
by families of sets. Thus, checking the unit and associative laws requires working with many
levels of the power set tower. Roman letters S,T, - â–  â–  denote sets. A subscipt designates the
“power set complexity”: Si G 7(A), S2 G 72A - S2 is a family of sets, S3 G 73A - S3 is a
family of families of sets, etc.
5.1 Complete semilattices
In this section, we describe the free complete join (resp. meet) semilattice on a poset
A, using the monad (D, d, /u) (resp. ('U,i,/x)). The description of D is well known; for
example, Meseguer [25] uses it. The reader will have noticed that q is used as a name for
two different natural transformations; this would normally be horrible notation, but in this
case, the formula for y is the same. Thus, our notational econony should cause no confusion.
The functors 'll and T> act on a poset A by
11(A) = {SCA:x>y6S =£> x G 5}
- the set of increasing subsets of A ordered by reverse inclusion- and
T>(A) = {SCA:x x G 5}
-the set of decreasing subsets of A ordered by inclusion.
Given monotone f : A —> B, we define
Vf : 'D(A) -* D(B) : S ^ {b G B : 3s G S, b < f(s)}
and
Uf : U(A) -» U(B) : S ^ {be B :3s e S,b> f(s)}
The following facts will be of later use.

65
1. tí and D are functors: trivially they respect identity arrows. Observe that for any
S € A and montone functions / : A —> B, g : B —> C,
T>(gf)(S) = {c€C:3seS,c = {ceC:3f(s)e'Df(S),c = (%)('D/)
and, similarly, U(gf)(S) = (Ug)(Uf)(S).
2. For each poset A, D(A) is a complete lattice, with supremum operation given by set
theoretic union, and infimum given by intersection. tt(A) is a complete lattice, with
infimum given by union and supremum given by intersection.
3. For each S2 C D(A),
In particular, this holds if 6 is empty. So Vf preserves all suprema. Similarly, if
S2 C 'U(A), then
Uf{\j6)=\j{(Uf)(S):See}
so that Uf preserves all Ínfima.
Define
dA: A —> T)(A) : a {x € A : x < a},
iA : A —* 'U(A) :aH{i£i:i>a}, and
fiA : T>2{A) V(A) : 52 € D2(A) ^ (J S2.
It may be puzzling that D(A) is ordered by subset inclusion and li(.4) is ordered by
reverse subset inclusion. The fact that
x < y dA(x) C dA(y) <=> iA(y) C iA(x)

66
motivates the choice, because one wants iA to be order preserving. Moreover, one wants 1if
to preserve all ínfima, which only happens if the infimum operation in UA is union.
The reader may check that i, d and fi are natural transformations. The monad
assocative law
D3 ——*T>2
[iD
D2
â– V
holds for both D and 'll, because if S3 e IP3 (.A), then
Ul> = U{Us’:S*e-4
The monad unit laws
hold for V, because if S G D(A), then
S = \J {{dA)(x) : x e 5} = U [t € T>(A) : T C 5}.
Similar computation shows the unit laws hold for It.
Now to describe algebras over these monads:
Lemma 5.1.1. Let T C T>(A), such that for all x € A, (dA)(x) € T. Then the following
are equivalent:
1.there is an order preserving map a : T —> A such that id^ = a(dA);
2.each S e T has a supremum;
3.dA : A —* T>A has a right adjoint.

67
The analogous conditions involving T C 1LA are also equivalent.
Proof. We give the proof for T C T>A, leaving the upside-down argument for T C UA to the
reader.
(2 => 1) Define a(S) = \J S and compute.
(1 => 2) a(S) is an upper bound for S, because if x G S, then (dA)(x) C S,
whence x = a(dA)(x) < a(S). Suppose for all x G S, x < u. Then (dA)(x) C (dA)(u), so
S = |J{(eL4)(:r) : x G 5} C (dA)(u), therefore a(S) < u.
(2 3) By definition of V S, j x C S <*=>• x <\J S. □
Proposition 5.1.2. P°° is equivalent to the category of T)-algebras. Poo is equivalent to the
category ofU-algebras.
Proof. The first assertion is proved, leaving the second to the reader. By Lemma 5.1.1, any
D-algebra has a supremum for each decreasing set; the supremum of an arbitrary set S is
equal to the supremum of the decreasing set ]. S. By the definition D-algebra homomor-
phisms coincide with order preserving functions which preserve all suprema of decreasing
sets. It is easy to see that a map preserves suprema of all decreasing sets if and only if it
preserves suprema of all sets.
Conversely, if A G P°° it has structure map
a : V{A) —* A : S S.
In Lemma 5.1.1 it was noted that a satisfies the unit law for structure maps, a satisfies the
associative law by the order theoretic fact proved in Lemma 5.1.3 below. â–¡
Lemma 5.1.3. For any S2 G T>2(A),
V {x € A : 3S1 G S2, x<\JSr] = \J\JS2.
Proof. To verify this equality, one notes that for all x G G S2 such that x € Si,
so x < V Si. Thus the left hand side dominates the right. If u is an upper bound for (J S2,
and 3Si G S2, x < \J Si, then x < u. So the left hand side is the least upper bound of [J 6;
this establishes the lemma.
â–¡

68
5.2 Completely Distributive Complete Lattices
In the following arguments, the forgetful functors are supressed from the notation.
The lemmas that follow establish that there is a lifting of It over V (and similar arguments
show that there is a lifting of V over It). Explicitly this means:
1. For any U-algebra A, there is a lt-structure on V(A).
2. It maps D-algebra maps to D-algebra maps, so we may view U as a functor P® —> P®.
3. Both natural transformations i and p \ It2 —> It preserve all joins; similarly, d and
fi : D2 —*■ D preserve all meets.
Lemma 5.2.1. For any poset A, both lt(A) and V{A) are complete lattices. Thus, li(.A)
and V(A) are both 'll— and D— algebras.
Lemma 5.2.2. For any poset A,
1. dA : A —> V(A) prserves all existing Ínfima.
2. iA : A —» U(j4) prserves all existing suprema.
Proof. Suppose S C A and u = f\S exists. Since dA is order preserving ,
(dA)(u) C P| jcL4(:r) : x G
For the reverse inequality, suppose i G f]{dA(x) : x G S'}, i.e., i is a lower bound of S. Then
l < u, so £ G dA(u). The proof for uA is upside-down, but otherwise identical. □
Lemma 5.2.3. If f : A —* B preserves all suprema, then It/ : 11(^4) —> U(B) also pre¬
serves all suprema. Similarly, if f preserves Ínfima, so does T>(/). Therefore, for any order
preserving map f, both UT>f and VIIf preserve all ínfima and suprema.
Proof. A proof of the first fact is given, the second is very similar, but notationally easier. Use
the fact that lt(A) is a complete sublattice of tP(A), and therefore completely distributive.

69
Let (S\)\eL be an indexed subset of li(A); for each S\, choose a family (x\^K : k E K) such
that
S\ =T {x\,K : k € K).
Recall that:
• joins in UA are intersections;
• iA preserves suprema, which reads
t \] y\ = (iA)(\J y\) = 0 T y
A
for any indexed family (y\) C A.
One computes as follows:
= u/(niu^")
A K
= U/(U{ n t xa,c(A) : C : L -* /i})
= u{ t/(VxV(A)) : c:L->k}
'a
= c:L->k}
= c-.L-K}
1 A
= n uf^
â–¡
Corollary 5.2.4. For any poset A,
1. p : T>2(A) —> D(A) prserves all ínfima.
2. p : li2(A) —> U(A) prserves all suprema.

70
Proof. Suppose S2,a £ X>2(^), for A e L : claim 1 amounts to
unsw=nusw
A A
Clearly, the left hand side is contained in the right. Suppose that x is a member of the right
hand side, that is, for each A there exists S\t\ € ¿2,a with x € Si,a. Then
Si = Pi Si,a
A
is in each S2,a because each S2,a is downward closed - relative to the inclusion order in ©2(.A).
Since a: € Si, this proves that x is in the right hand quantity, and the desired equality of
sets holds.
The proof for the It works similarly. â–¡
The preceding lemmas show that there are liftings of 'll over © (and vice versa), so
we have:
Corollary 5.2.5. It© = (ItCD, 77, where
rj : idp —► It© : a i-> {S € VA :|aCS}
and
1/: UDlt© —* It© : S4 i—> un*.
is a monad over IP whose algebras are completely distributive complete lattices.
Proof. The only thing left to be proved is the formulas for the natural transformations. By
Beck’s reasoning - as summarized in Section 3.4, g is the horizontal composition i ■ d. To
compute v : lt©lt© -4 It© one uses the fact that vA is the structure map for the free algebra
'll©A Consider the situation in light of the discussion following Theorem 3.4.5. (Here It
plays the role of 5, while © plays the role of T.) The relation a — ar{Tas) applied to the
object UVA implies
Mxs.) = u(®n)(*)
= U{reBA:3S2ef|S,,rcf|S2}
= Ui>-

71
The assertion about algebras follows from Theorem 3.4.5. â–¡
Remark 5.2.6. IÍD is a monad because there is a lifting of 'll over P®. One could use
the correspondence between liftings and distributive laws outlined in 3.4 to find a natural
transformation DU —> UD. This distributive law is not needed for the calculations which
follow, and is somewhat cumbersome, so its explicit description is omitted.
5.3 Some categories of algebras
In this section, Meseguer’s Lemmas are applied to the monads described above. Note
that P - the category of posets and order preserving maps - is a cocomplete SF-category in
which unions of chains are colimits. Note that extremal monomorphisms in P are inclusions
of subsets with the induced order: each order preserving map / : A —> B factors as A —>
f(A) —y B, where f(A) is the set-theoretic image of A, with the order induced from B.
Thus, the results of Section 4.2 do apply to P.
The reader is advised to review the definition of subset system, given in 1.2.1, if
necessary.
Assume all subset systems Z are non-trivial in the sense that for each A, and a € A,
{a} € Z(A); this does not reduce the generality of the argument, because given any subset
system, one may adjoin all singletons to it without changing which optimum bounds are
preserved.
Theorem 5.3.1. [25] P-7 and Pm are monadic.
Proof. Given a subset system j
J0(A) = {iS:Sej(A)}
defines a subfunctor of D. This extends to a subfunctor j of D such that (j, d, Jl) is a monad.
The natural transformations d and JL are obtained from d and /i by suitably modifying the
domains and codomains. The precise definition of these maps is contained in the proof of
Lemma 4.2.6.

72
If (A, a0) has a J0-structure, Lemma 5.1.1 shows that
ao{S) = \J S
for all 5 e J0(A). Thus, to define ax one must show some map JXA A makes the following
diagram commute.
JSA J°(V) *J0A
U
JXA-
Lemma 5.1.3 shows that \J S is defined for all S € J\A and, moreover, that for S2 € JqA,
VU S2 = V(J«V )(«>)•
Identical arguments show show that for each A, if
ax(S) = \/S
that
ax+1(S) = \/S.
It follows that any poset, A, in which each member of JqA has a supremum, each member of
J(A) has a supremum. Moreover, each map preserving J0-suprema also preserves J-suprema.
Thus, one obtains an equivalence of categories between P and PJ.
An upside-down version of this argument shows that Pm is a category of algebras. â–¡
Given subset systems m and j, F := M0J0 forms a subfunctor of
T := UV.
By Corollary 5.2.5, (T,u,r]) is a monad on P. By 4.2.1, there exists a smallest F above F
that is closed under v. This monad is used to discuss the categories defined below.

73
Definition and Remarks 5.3.2. Let OP^ denote the category of completely distributive
complete lattices with maps, which preserve order and all optimum bounds. PP^ denotes
the full subcategory of containing objects such that there is a P3m map : A —> B, with
4> P-extremal mono, and B € Obj(DP^).
denotes the category of F-algebras, where F is the monad described above.
It will be shown that
SpWm c mi c Ml.
To see the first inclusion, one notes that any family T of subsets of a set X which is also a
P^ object is in DP^; the inclusion T C 7(X) is a P^-embedding and T(A') is a completely
distributive complete lattice. Proposition 5.3.7 shows the second inclusion.
Let A 6 BP^, and : A —+ B be given as in the definition of BP^; let b : TB —> B
be the T-structure for B. The verification of the hypothesis of 4.2.6 proceeds by comparing
the diagram to be completed with the lt!D-algebra associativity diagram for B. Prom this
point on, the notation of Lemma 4.2.2 is adopted with slight modification; u : T2 —> T is
the monad multiplication; v\ : F% —► F\+\ is defined inductively (and plays the role that
was played by m\ in the proof of 4.2.2); r¡ : idp —► T is the unit; r¡\ : idj» —> F\ are defined
inductively, and play the role that was played by n\ in 4.2.2. The natural transformations
fx : Fx —> T and c* : F\ —* F\+i play the same roles as in Lemma 4.2.2.
Remark 5.3.3. By Lemma 3.4.5, the structure map b : TB —> B is given by:
b^ = A{V5i:5l€52}-
Since is a P^-map, the following diagram commutes.
Fd> f\B
FA —>FF >TB
ao
B
B

74
The left square expresses that <¡> commutes with the F-structure; the right square expresses
that the F-structure on B is the restriction of the T-structure.
Lemma 5.3.4. Define ao : FA —> A by a0(S2) = f\(U \J)(S2); this map satisfies a0T]o = id^.
Proof. Let us begin by more explicitly calculating ao. Using the definition of li on maps we
find that
a0(S2) = A(UV)(*)
= /\{xeA:3S1eS2, \/Si = /\{\/S1:S1eS2}.
This definition makes sense; each Si G Jq(A) so each \J S\ exists because A G P^. (IX V) is
an order preserving map, and S2 G mojo(A), which implies that
(U\J)(S2) e M0(A).
Thus, the meet defining ao(S2) exist.
Recall rj(x) = {S € DA x C 5}. Now one calculates
ao(>?(*)) = /\{\/S:SeDA,lxCS}
= x
â–¡
For later calculations, it is crucial to know that some of the maps are epi or mono.
Lemma 5.3.5. For each ordinal X, u\A is surjective. Therefore v\A is P-epi.
Proof. Because of the structure of P, unions of subposets actually are set theoretic unions.
Let 5 G Fx+i = Fx U (i>\A)+1(FfA). If S € (v\A)+1(FfA), then there is an S2 G F%A such
that (v\A)(S2) = S. Since c\A = (nxA)(rixFxA) so for any Si G FxA, T2 := (rjxFxA)(S) is a
member of FfA such that (uxA){T2) — S\.
This lemma holds in any category where unions of subobjects actually are set-theoretic
unions. â–¡

75
After noting that extremal monomorphisms in P are injections / : A —> B, where
f(A) has the order induced as a subset of B, one easily verifies:
Lemma 5.3.6. The functors U, T>, T, and F\ all preserve extremal monomorphisms.
Now one can verify the hypothesis of 4.2.6 is satisfied.
Proposition 5.3.7. Let A € Obj(DP^), B e Obj(BP^), : A —> B as in the definition of
BP^,. Then
(ex) for each A, there is a unique map
“A+i • F\+iA A,
such that
a\(F\a\) — üa+iI'a;
(com) for each A, fiax = b(f\B)(F\).
Thus, OP^ C
Proof. Remark 5.3.3 established (com) for A = 0. If (com) holds for sucessor ordinals,
properties of unions insure that (com) holds at all ordinals.

76
Suppose (com) holds at stage A; we will establish (com) and (ex) hold for
stage A + 1. Consider the following diagram.
T2B >TB
The diagram commutes: the outer square commutes because of the T-algebra associative law
for B\ the left outer trapezoid commutes because of the definition of vxB\ commutativity of
the right outer trapezoid is obvious; the left inner trapezoid expresses naturality of vX] the
top and right inner trapezoids commute by (com).
Establishing commutativity of the top outer trapezoid requires a bit more. Since
fx:Fx->Tis natural, (Tb)(fxTB) — (fxB)(Fxb). By definition of horizontal composition
flB = (fxTB)(FxfxB).
Thus,
CTb){flB) = (Tb)(fxTB)(FxfxB)
= (fxB)(Fxb)(FxfxB)
This proves that the top outer square commutes.
Consider the (epi,extremal mono)-factorization of
b(fx+1B)(Fx+1y,

77
say it factors as
Fx+lA ^A'CB.
Since vxA is epi, the (epi, extremal mono)-factorization of
q := b(fx+1B)(Fx+1)(vxA)
is
FXA A! C B.
The proof of Lemma 4.2.6 shows that each existing ax is split epi, which implies both ax and
(Fax) are epi. Since
q = 4>ax(Fxax),
the (epi, extremal mono)- factorization of q is also:
FXA “a(4“a) ACB.
Uniqueness of factorization implies there is an isomorphism i : A —> A! compatible with the
factorizations. So a^+i = ia is a map making (com) hold for A + 1.
Reading off the diagram one finds that:
ax(Fax) - b(Tb)tf2xB)(F2x)
= b{vB){flB){Fl)
= b(fx+1B)(Fx+1)(vxA)
= The last step used that (com) holds for A + 1. Since is a monomorphism, one concludes
that (ex) is satisfied for A + 1.
ax+1 is the only map which makes the inner square commute because vxA is epi, by
Lemma 5.3.5. â–¡

78
Corollary 5.3.8. DP^ is a full subcategory ofMPm.
Proof. By Lemma 5.3.3, any P^-map preserves the F-structure of a B)Pjn object. Lemma
4.2.7 implies that any P^-map extends to a map. â–¡
Corollary 5.3.9. The forgetful functor DP^ —> P has a left adjoint.
Proof. First note that FA G Obj(DP^) for any A, because fA : FA —> TA is an embedding
of FA in a completely distributive lattice. Because FA is the free F-algebra on A, if
/ : A —► B is an order preserving map from A G Obj(P) to Be Obj(DP^) C Obj(PF), then
there is a DP^-unique map f* : FA —> B. This proves that rj has the universal property
described in 2.5.2.3; thus, F is the left adjoint to the forgetful functor DP^ —»P. □
Corollary 5.3.10. The forgetful functors DP^ —> OP^, and DP^ —> have left adjoints.
Proof. Apply Proposition 3.3.6 and Corollary 6.2.8. (Note: the proof of 6.2.8 does not
depend on the arguments in this section, so the result is listed here. There is no circular
argument.) â–¡
There is some, rather limited, information about members of M^.
Proposition 5.3.11. If A G Obj(M^), then A G Obj(P^).
Proof. Define JA, MA, a, ¡3, 7, and 5, by the requirement that the squares below are
pullbacks.
UA
*WD A

By assumption, j(A) and rn(A) contain singletons, therefore
(â– iT>A)+1{J0A) C MqJqA C FA,
79
and
(UJA)+1(M0A) C Mq Jo A C FA.
It follows that
J0A C JA = (iDA)_1FA
and
Mo A C MA = (ltdA)_1FA
Again using the fact that j and m contain singletons,
(JA)+1(A) C JA
and
(¿A)+1(A) C MA,
so corestrictions dA\ : A —> JA and iA| : A —> MA satisfying a(JA)| = JA and 7(1 A) | = fA
exist.
If a : FA —> A is an F-algebra, then a/3(JA|) = id^ and a5(iA\) = id^. Lemma 5.1.1
shows that for any subset 3 C DA containing each principal down-segment [ x, an order
preserving map y : 7 —* A satisfies y(JA|) = id.4 if, and only if, y(S) = V5 for each 5 € J
(and analogously for subsets of ICA). Thus, we conclude a(3 = \J and aS = /\. â–¡
Remark 5.3.12. If j = m = u> - the subset system which selects all finite subsets, then
SpF^ = DP£. For general j and m, the author does not know if equality holds, but he
suspects that the equality does not always hold.
Remark 5.3.13. A fundamental difficulty working with M^n is that one does not imme¬
diately know any order theoretic formula for the F-structure maps. One might hope that
each F-structure map is S2 /\(U \f)S2, but the author cannot presently substantiate such
hopes.

80
Remark 5.3.14. The author knows no examples of objects in Obj(M^) \ Obj(DP^). In
light of Corollary 5.3.10, any object is freely embedded in a completely distributive
lattice. In fact, A € Obj(M^) is in Obj(DP^) if, and only if, the unit of the adjunction
mentioned in 5.3.10 is P-extremal mono.
Li[18] explicitly constructs a map u : P —> ISF(P) (not necessarily mono), with the
following properties:
• ISF(P) is a complete, completely distributive lattice.
• u preserves designated meets and joins.
• If / : P —> A is a map preserving designated meets and joins, and A is complete,
completely distributive, then there is a unique map f* : ISF(P) —> A such that
/ = f*u.
To explain the relationship between Li’s results and the results here, note:
• Li uses families SP C TP and IP C TP, which are only required to contain singletons.
Joins of SP-sets and meets of IP sets are required to be preserved. In this document,
the choice of distinguished optimum bounds is made for all posets at once, via a subset
system.
• Comparing universal properties, one sees that Li’s construction applied to P G Obj(M^)
with SP — j(P) and IP = m(P) yields the left adjoint mentioned in 5.3.10.
Unfortunately, it is very difficult to see when u is an embedding. Let [5] denote the
smallest downward closed family of A containing S which is closed under SP-joins. Let TP
denote the class of increasing sets, which are closed under /P-meets. Li defines IS(P) to
be the familly of all decreasing sets in TP, ordered by S â– < T if, and only if, there is an
indexed chain (S¿)j€/ , with I = Q D [0,1] such that S0 = S, Si = T and whenever i < j,
Si fl [P \ Sj] — 0. The map u : P —> ISP is defined by u(x) = {U e 1L4 : U ^ x}.

81
Nonetheless, BP^, it is a fairly nice category. In fact, it is complete.
Proposition 5.3.15. BP^ is complete.
Proof. Note that a poset A is a P^ object if and only if there is an order preserving map
a : J0A x M0A —> A such that a(d0A x i0A) = kU.
Suppose (Ai)i€i is a family of P^ objects and for each i € I,
(j)i • ^ Pi
is an embedding of Ai into a completely distributive lattice The proof of Lemma 3.1.3
implies that the poset product At is a P^-object. Moreover, the product-induced map
: n¿€/ A* —► n¿€/ Bi is an embedding which preserves (j, m)-optiinum bounds. (Essentially
what is going on is that meets and joins are computed coordinate-by-coordinate.)
If /, g : A —> B axe P¿, maps, then the proof of Lemma 3.1.3 implies eq(f,g) is
a (j, m)-complete subset of A. If A € Obj(DP^), then A can be (j,m)-embedded into a
completely distributive lattice, so eq(/, g) may also be so embedded.
Thus, any set of objects in BP^ has a product, and any pair of BP^ maps has an
equalizer. By Borceux [6, Volume 1, 2.8.1], this implies BPfa is complete. â–¡

CHAPTER 6
COEQUALIZERS
6.1 Epis and Equalizers in P
The results of this section characterize epis and regular epis in the category P of posets.
The presentation and proofs (except Construction 6.1.2 which is discussed in Meseguer [25])
are the work of the author, but the author believes it likely that they are not new. In all
statements, A and B are arbitrary posets.
Lemma 6.1.1. Let f : A —> B be a monotone map. f is epi if and only if f is onto.
Proof. Since the forgetful functor P —> Set is faithful, if / is onto then / is epi. For the
converse, suppose b G B \ f(A). Define
Si =i (f(A)n | b)
and
S2 =1 b.
Evidently, both S\ and S2 have the same intersection with f(A), but b G S2 \ Si. Therefore
characteristic functions of B \ Si and B\ S2 are distinct maps (say
Cj, c2 : B —> 2
are respectively the characteristic functions of B \ Si and B \ S2) such that Cif = c2/. Both
Q are order preserving, because Si and S2 are decreasing sets. â–¡
Definition and Remarks 6.1.2. The following construction is paraphrased, following Mese¬
guer [25] pp 73-74.
If / : A —> B is any order preserving map, we may define a preorder ■< on A by
ai 4 a2 <=> /(ax) < /(a2).
82

83
This relation is obviously reflexive and transitive, but -because / may not be injective
may not be anti-symmetric.
Suppose (A, <) is a poset and ^ is a preorder strengthening <, i.e.,
Üi < Ü2 => a\ ■< 02.
Then we have an order preserving map a : A —► A/ ■<. (Recall that A/ •< is the set of
equivalence classes
{x € A : x -< a and a ■< x}
partially ordered by X.)
Define maps /x, /2 : A —* to be equivalent, if there is an isomorphism, i : Bx —> B2
such that
B2
commutes.
One verifies that:
1. The maps (f : A -* B) and (<)>-+ a \ A A/ < are mutually inverse
correspondences between
• the set (modulo equivalence) of surjective maps / with domain A.
• the set of preorders on A which strengthen <.
2. For any / : A—* B, f = ca, where a : A —> A/ ^ and c(a(a)) = /(a).
3. Given surjections fi,h'.A—* B¿, there is a c : B\ —> 52 if and only if
/i h
al ^ a2 => «1 ^ «2-

84
Remark 6.1.3. Note that any intersection of preorders is a preorder. Thus, the class of
preorders strengthening < is a complete lattice. We say a set S of ordered pairs generates
^ if ^ is the smallest preordered containing S. Note that if 5 generates â– <, we can describe
-< explicitly: a0 â– < an if
• clq dm or
• there is a finite sequence ai,a,2,--’ , a„_i, such that for all i with 0 < i < n — 1,
(®¿) ®t+i) £ S.
(The first bulleted condition insures that -< is reflexive. The second requirement insures
that â– < is transitive; generally, if S is a relation, the relation obtained by applying the second
bulleted item only is called the transitive closure of S.)
One may verify the following construction using 6.1.2.
Construction 6.1.4. Let f,g : A —> B be order preserving maps. The coequalizer of
f and g is the quotient of (B, <) by the smallest preorder containing < and C(g, h) —
{(gx,hx),(hx,gx) : x G A).
Lemma 6.1.5. Let f : A —> B be a surjective order preserving map. The following are
equivalent.
1. f is the coequalizer of some pair g, h : Ao -hâ–º A.
2. There exist a poset Ao, and maps g, h : Ao —> A such that the preorder ■< on (A, <) is
generated by < and C(g,h).
3. The preorder X on (A, <) is generated by < and K(f) — {(x,y) E BxB : f(x) = f(y)}
f. The preorder â– < on (A, <) is generated by < and some equivalence relation.
Proof. (1 <=> 2) is evident from 6.1.4. (3 => 4) is trivial, because K(f ) is an equivalence
relation. We have
C(g,h)CK(f) ¿

85
because fg = fh and
(x,y) €K(f) 4=^ f{x) < f(y) and f{y) < f(x).
f f
Since C(g,h) and generate X, K(f) and < generate â– <.
For (4 => 1), let 4 be generated by < and the equivalence relation E C A x A
with the trivial order. Then the projection maps tt\, tt2 : E —> A, given by
ni(xi,x2) = Xi
are order preserving. The coequalizer of 7Ti and 7r2 is /. â–¡
Example 6.1.6. Some readers may be surprised by the fact that surjections are not always
P-regular epi. An example of this situation is any map / : 2/¿a( —» 2 from the (trivially
ordered) two point set onto a two-point chain. Any such map / fails condition 3 from
Lemma 6.1.5.
6.2 Factorization of Maps Using Preorders
This section modifies the preorder factorization, to give a first approximation to
coequalizers in FJm.
Definition 6.2.1. Let Z be a subset system. Z is said to admit congruences if for all posets
A,
Z(A x A) = {S € A x A : tt<(5) € Z(A),i= 1,2}.
Lemma 6.2.2. Suppose both j and m are subset systems which admit congruences. Let
(A, <) be a poset and ■< be a preorder strengthening <. Then a : A —> A/ ■< preserves
(j, m)-optimum bounds if and only if < is a FJm-subobject of Ax A.
Proof. We show that q(Vx¿) = Vq(x¿) for any (x, : i G 1} € j{A). Clearly a(Vx,) is an
upper bound of (a(x¿) : i € I}. Since ^ is a P^-subobject of A x A, {x{ : i G 1} G j(A) and
for all i, Xi < a,
V{(x¿, a) : i e 1} = (VXj, a)

86
is a member of X; Vx¿ < a. Thus, any upper bound of {a(x¿) : i £ 1} dominates a(Vx¿).
(The proof for meets is identical, so omitted.)
Conversely, if a preserves j-joins and m-meets, and (for example)
{xi'.i € /}, {yt : i £ 1} £ j{A)
and for all i £ /, x¿ ■< yt, then V¿x, X V,yt. So ^ is a P^-subobject of ,4 x A. □
Definition and Remarks 6.2.3. Suppose j and m admit congruences. Let (A, <) €
Obj(P^). If ^ is a preorder strengthening < and simultaneously a P^-subobject of A x A,
then we say â– < is a (j, m)-pocongruence.
Pocongruences of P^-objects behave very much like preorders of posets.
1. The maps (/ : A —♦ B) and (^) a : A —> A/ ■< are mutually inverse
correspondences between
• the set (modulo equivalence) of surjective maps /, which preserve (j, m)-optimum
bounds and have domain A.
• the set of (j, m)-pocongruences on A.
2. For any / : A —> B, f = ca, where a : A —> A/ ■< and c(a(a)) = /(a). If / preserves
(j, m)-optimum bounds, so do c and a.
3. Given (j,m)-optimum bound preserving surjections fi,f2'-A—>Bl, there is c : B\ —►
Z?2 if and only if
A Si
ai di a2 a,i ^ ü2-
Consideration of Lemma 6.2.2 shows that if / : A —> B is surjective and S £ j(.4)
(resp. 5 £ rn(A)) then f(S) has a supremum (resp. infimum) in B. Moreover, sets of the
form f(S) for S £ j(A) (resp. S £ m(A)) are the only sets whose optimum bounds are guar¬
anteed to exist. Thus, we are motivated to offer the following conditions on subset systems
to guarantee surjective (resp. P-regular epi) images of P¿, objects are (j, m)-complete.

87
Definition and Remarks 6.2.4. Let Z be a subset system. We say Z preserves surjections
if whenever / : A —> B is a poset surjection,
Z(B) = {f(S):SeZ(A)}.
Similarly, we say Z preserves regular epis if whenever / : A —* B is a poset regular
epimorphism,
Z(B) = {f(S) : 5 € Z(A)}.
Since poset regular epimorphims are surjective, any Z that preserves surjections also
preserves regular epimorphisms.
Any cardinal k, the associated subset system k preserves surjections. For if f : A —> B
is surjective, then / is split epi (in Set). Hence, there is s : B —> A (not necessarily order
preserving) such that idg = fs. Therefore, any set S C B with cardinality less than n is the
image of some set s(S) C A. The following remark provides an example of a subset system
that does not preserve surjections or regular epis. The author currently does not know of
any subset system that preserves regular epis without preserving surjections, but it seems
likely such a subset system exists.
A more interesting question, which the author also cannot currently answer is: “do
there exist subset systems other than cardinals which preserve regular epimorphisms?”
Remark 6.2.5. Let j = dir be the subset system which selects all upward directed subsets
of a poset. Let N denote a countable disjoint union of two-point chains. (See drawing below
for help visualizing, and to fix notation.)
to t\
bo b\
Evidently N is j-join complete.

88
Consider the function / : N —» N, where N = {0,1,2, •••} with the usual order,
defined by /(fc¿) = i (for i > 0) and /(í¿_i) = i (for i > 1). Speaking roughly, “/ stacks
the two point chains.” / is regular epi as a map of posets. Moreover, ^ is a j-subobject of
N x N, because / preserves all existing joins. But N is not j-join complete, because N is
upward directed but has no join!
Because of the pocongruence factorization for (j, m)-bound preserving maps (outlined
in 6.2.3) , we have the following first approximation to the coequalizer in P^. The only thing
stopping map a : B —> B/ < (described below) from actually being a coequalizer in P^ is
that B/ â– < is generally not (j, m)-complete.
Construction 6.2.6. Suppose f,g:A—> (B, <) are ¥Jm maps, and C(f,g) is defined as in
6.I.4■ Let X be the smallest pocongruence containing < and C(/, g). Then a : B —> B/ X
has the following universal property: if h : B —> C preserves (j, m)-optimum bounds and
hf = hg, then there is a unique map i : B/ ■< —* C such that h — ia.
Since the quotient map a : B —> B/ ^ is a surjection we have -
Corollary 6.2.7. Continue with the notation from Construction 6.2.6. If j and m preserve
surjections, then B/ < is complete. Thus, a : B —► B/ -< is the coequalizer of f and
g in FJm. In particular, if j = k and m — A are cardinality subset systems, then a is the
coequalizer.
Corollary 6.2.8. DP“ has coequalizers.
Proof. The preceding shows that any pair f,g:A—>Be P^ has a coequalizer. Suppose
B is completely distributive. Since a : B —> coeqPoo (/, g) is a surjection which preserves all
meets and joins, coeqp* (/, g) is completely distributive. In particular if both A and B are
completely distributive, then coeqp™(f,g) is also the coequalizer in DP“. □
6.3 Factorization of Meetsemilattice maps
Several simplifications occur describing coequalizers in P¿, if the objects have a meet¬
semilattice structure, i.e., m> uj.

89
Lemma 6.3.1. Let f : A —> B be a meetsemilattice map. f is F-regular epi if and only if f
is surjective.
Proof. Since P-regular epimorphisms are always surjective, one implication is trivial. For
the other, suppose f : A —> B is a surjection preserving binary meets. Then
x k y <=> f(xAy) = f(x).
Thus we have the sequence x,x A y, y, where f(x) = f(x A y) and x Ay < y. By Lemma
6.1.5, this proves / is regular epi. â–¡
As noted in the proof, the relation â– < is completely described by K(f). If objects
have a meetsemilattice structure, one may use congruences (equivalence relations that are
simultaneously (j, m)-subalgebras) rather than the more complex pocongruences.
Finally we have:
Theorem 6.3.2. Suppose m > to and j and m preserve regular epis. Let f : A —> B be a
P]m map. The following are equivalent.
1. f is surjective,
2. f is a quotient by some congruence K,
3. f is P regular epi,
4. f P^-regular epi,
5. f is Ff,-extremal epi.
Proof. The basic facts about pocongruences show (1 2). The preceding lemma shows
(1 <=> 3). (4 => 5) holds in any category.
To show (2 => 4), assume / : A —> A/K. Since K C A x A, we have projection
maps 7Ti,n2 : K —> A. f is plainly the coequalizer of 7ri,7r2.

90
(5 => 2) holds because we may factor any map / through its associated pocogruence.
If / is extremal epi, then / must coincide with the quotient by its associated pocongruence.
â–¡
Corollary 6.3.3. Let j and m be subset systems which preserve regular epis. Let m > w.
The forgetful functors Up : FJm —> P and USet : P¿, ~1' Set preserve and reflect regular epis.
Lemma 6.3.4. The forgetful functor Um^ : DP^ —> Set reflects kernel pairs.
Proof. One verifies that the kernel pair of a map / : A —> B (calculated in DP¿J consists of
the projections (from Ax A) restricted to the set
{(x,y) € A x A : f(x) = f(y)}.
â–¡
Applying Theorem 3.3.5, one obtains the following.
Corollary 6.3.5. Let j and m be subset systems which preserve regular epis. Let m > uj.
The forgetful functor Uset '■ HME”, —> Set is monadic.
6.4 Coequalizers in DP¿,
Theorem 6.4.1. For any pair g,h : A —> B € IMP^, the coequalizer coeq( Proof. The construction of the coequalizer of maps g and h is illustrated by the following
diagram. The maps on the diagram will be defined below.
~D{C)

91
Let a : B —> B/ -< be the map given by Construction 6.2.6. Let / : B —> C be any
BP^ map such that fg = fh. The existence and uniqueness of / is given by 6.2.6.
To construct the leftmost square, we apply the free completely distributive complete
lattice functor D = líD; r¡ is the natural transformation which injects a poset H into DH.
To construct the right-outside square, we note that (oo, oo)-pocongruences are closed
under intersection. So there is a smallest (in the sense that it makes the fewest possible
identifications) (oo, oo)-quotient
k : D(B/ ■<) —> E(B/ X)
such that k(r](B/ X))a preserves all (j, m)-optimum bounds. Similarly, define
£ : D(C) -> E(C)
to be the smallest quotient such that £(r¡C)f preserves all (j, m)-bounds. Since
£D(J)(V(B/ l))a = £(VC)fa = £(rjC)f,
preserves all (j, m)-bounds we have the induced map E(f).
Define Eo to be the smallest BP^-subobject of E(B/ â– <) through which k(rj(B/ ^))
factors. (By Proposition 5.3.15, the intersection defining Eo exists.) The maps j : B/ < —>
Eq and E0 —> E(B/ X) are obtained by factoring (r¡(B/ -<))a) through E0.
By construction, the map £(r¡C) : C —> E(C) has the universal property that any map
(j, m)-optimum bound preserving : C —* B, with B € Obj(BP^) factors as (f) = t£(r¡C) for
a uniquely determined map t. Since C € Obj(BP^r[), there is such a (j) which is P-extremal
mono. Thus, £(r]C) is P-extremal mono. Therefore, the P-(epi, extremal mono) factorization
of £(r)C) produces the factorization £(jgC) = qm. By construction, m is both P-epi and
P-extremal mono; so m is a P-isomorphism. Moreover, q : C —> E(C) is necessarily the
smallest BP^-subobject of E(C') through which £{gC) factors.
We claim that coeq(g,h) = Eo via j : B —> Eo. It suffices to show the existence
of a unique compatible map * : Bo —» C. The commutativity of the largest rectangle

92
in the diagram above implies that k(r¡(B/ ;<)) factors uniquely through the BP^-object
£'(/)_1(C') C E(B). Because E0 is the smallest BP^-object through which k(r](B/ :<))
factors, Eq C E(f)~1(C). This insures the existence and uniqueness of *. â–¡

CHAPTER 7
(j,m)-SPACES
This chapter studies spaces obeying a To-style separation axiom. Section 7.1
defines spaces and describes a functorial Galois connection, which specializes to Galois con¬
nections between DP^ and SJm. Section 7.2 develops a convenient description of epimorphisms
in SJm, which generalizes a known characterization of epimorphisms of To-spaces. Section 7.3
gives constructions of limits, similar to those for topological spaces. Section 7.4 describes
quotient maps, and characterizes extremal and regular epis as quotient maps.
The last two sections are related to the problem of finding reflections in SJm] Section 7.5
gives the flat spectrum (co)reflection on spatial objects - which is equivalent to a reflection
on S¿j. This reflection on spaces is a generalization of the Ti-reflection of topological spaces.
Last, but not least, Section 7.6 partially describes the epicomplete SJrn- objects; the
description is complete for T0-spaces. Epicomplete To-spaces are chains with the specializa¬
tion order. Products of epicomplete T0-spaces are not epicomplete, so there is no functorial
epicompletion in the category of To-spaces.
As mentioned in the introductory chapter, the research leading to this dissertation
began in an attempt to find reflections and coreflections in categories of generalized topo¬
logical spaces. After proving the results of this chapter, and reading Meseguer [25], the
author realized that additional assumptions where required on subset systems to insure that
subspaces could be reasonably defined. This realization prompted much of the thought sum¬
marized in Chapter 6 - in particular Section 6.3; the author wanted to find when the theory
of this chapter was valid. These considerations, and construction of free DP^-objects, be¬
came the main focus of the dissertation. However, this state of affairs leaves many questions
concerning reflections and coreflections in SJm untouched.
93

94
This chapter assumes slightly more background then the rest of the text. Closure
operators are used without comment. If c is a closure operator,
fix(c) := {x : c(x) = x}.
7.1 Spatial/Sober Functorial Galois Connection
Definition and Remarks 7.1.1. Let APos denote the category of whose objects A consist
of
(obi) an underlying partially ordered set, denoted A,
(ob2) a family of designated subsets 3 A, such that \J 5 exists for all
Se3A,
(ob3) a family of designated subsets MA, such that f\ S exists for all
S € MA.
Items (ob2) and (ob3) of the data defining a member of Obj (APos) will be referred to as,
the signature of A. Note that a given poset A may have several possible signatures. A
APos-map 4> '■ A —> B is a function A —* B such that:
(mapO) cj) is monotone, i.e., a < b e A => 4>{a) < (f>(b).
(mapl) 4>[3A] := {(5) : 5 G 3A} C 0P, and VS € 3A, S) = \J (f>{S)-
(map2) (/)[MA\ C MB, and VS G MA, 0(A S) = A 4>{S).
The category AS has objects (X,D(X), E(X), A(X)) where A is a set, O(X) is a family of
subsets of A, and E(A), A(A) are families of subsets of D(A), such that (£>(A), E(A), A(A))
is a APos-object, in which the optimum bound operations - meet and join - are the set the¬
oretic operations of intersection and union. AS maps are functions f : X —* Y such that
/-1 : 0(Y) —> £>(A) is a APos-map.
Definition and Remarks 7.1.2. We describe the functorial Galois connection between AS
and APos.

95
1. D : AS —♦ APos is the contravariant functor which sends a space X to the APos-object
(O(X), £(JA), A(A)), and a map /, to the associated inverse image map, which sends
U G D(Y) to f~\U) G D(X).
2. A prime filter on A G Obj (APos) is a set P C A such that
(pFill) P is increasing-, that is, if a G P and a < b then b G P.
(pFil2) P is 0-inaccessible; that is, if S G 0A and \J S G P, then Sf]P is
non-empty.
(pFil3) P is M-closed; that is, if S G MA and S C P, then f\ S G P.
For A G Obj (APos), there is a natural bijection between prime filters, prime ideals (decreas¬
ing, 0-closed and M-inaccessible subsets of A) and characters A —* 2 G AS: if x : A —> 2 is a
map, then x-1(0) is a prime ideal and x_1(l) is a prime filter, x can be recovered either from
x-1(0) or x-1(l). For a € A, let coz(a) denote the collection of all prime filters containing a.
1. Let 'LA denote the collection of all prime filters of A. 'LA is a AS-object if we define
D(^(A)) = coz[A] := {coz(a) : a G A},
E(A) := coz[[0A]] := {{coz(a) : a G 5} : S G 0A}, and
A (A) := cozfpVtA]].
By definition of maps in AS, 'L extends to a contravariant functor.
2. In Banaschewski and Bruns [3] it is verified that >L and D are adjoint on the right,
where the unit natural transformations have components
VÁa) = coz(a)
and
ex(x) = {U G 0{X) :xeU}

96
In verifying the adjunction, one shows that
coz (V*) = U{ coz(a) : a G sj
for any S € 8(A), and
coz (A5)=n{ coz(a) : a G 5|,
for any S G JVt(A).
3. We also follow Banaschewski and Bruns [3] in defining SpAPos to be the full subcate¬
gory of APos containing all A such that r¡,\ is an isomorphism, and So AS to be the full
subcategory of AS containing X all X such that ex is an isomorphism. These categories
are said to consist of spatial posets and sober spaces (respectively). Banaschewski and
Bruns [3] show that SpAPos is onto-reflective in APos and that So AS is reflective in
AS.
In order to apply the results of this section, we identify with the full subcategory
of APos, containing all (j, m)-complete objects with appropriate signature, i.e.,
Obj(Fm) = {A G APos : 8A = j(A), JVC A = m(A)}.
Rather than AS, we discuss subcategories that roughly correspond to the Pjn categories.
We use the following notations: given X G Obj(AS) and x G X, ex(x) denotes the class of
all open sets (members of £>) that contain x. We refer to members of Obj(AS) as spaces.
We assume all spaces satisfy a separation axiom, which generalizes the usual T0 axiom.
Namely, we require:
(sep) Vx, y G X, e(x) = e(y) =â–º x = y
As in general topology, whenever the axiom (sep) holds for X, the soberification map ex is
injective.
Define SJm by
Si = {X G AS : D(X) G PJm, (sep) holds for X}.

97
Remark 7.1.3. To strengthen the analogy between topological spaces and §Jm-objects, we
assume that the subset systems j and m preserve regular epimorphisms.
The functorial Galois connection of Banaschewski and Bruns [3] restricts to a duality
between and Sp¥3m. Because of (sep),
e : id¡.j —> Dí'
£>m
is a monoreflection on S^.
7.2 The Skula Topology and Extremal Monos
Assume j and m preserve regular epimorphisms. If S C X € Obj(S^), we may put a
S^-structure on S by defining
D(S) = {U n S : U € D(X)}.
By analogy with general topology, we call S with this S3m a (j, m)-subspace. Because the
poset map dual to the inclusion S C X is onto, and j and m are subset systems that preserve
regular epis, D(S) is (j, m)-complete.
Prompted by the treatment of similar problems in To and sober topological spaces,
we define a topology on (underlying sets) of objects from S^,; see Skula [30], We define the
Skula topology on A € Obj(S^) to be the topology with base
D{X) U {X \ U : U € D(X)}.
It may seem somewhat confusing to put an actual topology on a generalized topological
space, but it is also useful. Corollaries 7.2.2 and 7.2.3 identify epimorphisms and extremal
monomorphisms using the Skula topology. Without it, the results become more cumbersome
to state and less intuitive.
Lemma 7.2.1. The Skula-closure of S C X, denoted b(S), is given by the formula
b(S) = {x € X : VGX, G2 € D(X), G1n5 = G2n5=>x^ GxXG2}

98
Proof. Let b'(S) denote the right hand side of the given formula. The operator b' is order
preserving, and increasing. If S C U, we have a trace map from 0(U) —> O(S), given
by tr(A) = A fl S. Note that b'(S) is the largest subspace of X so that the trace map is
injective. Thus, b' is also idempotent, and, in fact, a closure operator. It now suffices to
show fix(h') = fix(6).
We first show fix(b') C fix(6). Suppose b'(A) = A, and let U be the complement of A
and x E U. We will give a Skula-open neighboorhood of x contained in U. If x £ b'(A), then
we have Gi, G2 G O(X) with x E Gi\G2 and Gx fl A = G2 fl A. Evidently, Gi fl (X \ G2)
is a Skula neighboorhood of x contained in U.
For fix(6) C fix(6'), it suffices to show SD(X\T) is b' closed, whenever 5,T E D(X).
Let x $ b'(S n{X\ T)), with S,T € O(X). Then we have GUG2 E D(X) with
G1nsn(X\T) = G2nsn(X\T)
and x E Gi\ G2. It follows that x S n (X \ T). â–¡
Corollary 7.2.2. Let j and m be subset systems which preserve regular epis. A map f :
X —> Y E is epi if and only ifb(f(X)) = Y.
We say f : X —* Y is a sub space embedding, if / corestricted to f(X) -
considered as a (j, m)-subspace of Y - is an §Jm isomorphism.
Corollary 7.2.3. Let j and m be subset systems which preserve regular epis. A map f :
X —» Y is extremal mono if and only if f is a (j,m)-subspace embedding and b(f(X)) —
/(*)•
Proof. Suppose / is extremal mono. We may factor / = X —* b(f(X)) —► Y; the first
factor is epi, ergo an isomorphism. Conversely, consider the map b(f(X)) C Y. If this
factors as gs, where s is epi with codomain Z, we can factor g as Z —* b(g(Z)) C Y. The
composition b(f(X)) —> Z —> b(f(X)) is an epi containment of subspaces of Y. By 7.2.2,
b(f(X)) — b(g(Z)), so s is an isomorphism. □

99
Corollary 7.2.4. Let j and m be subset systems which preserve regular epis. Suppose X 6
Obj(SoS¿J is sober. Then S C X is sober if and only ifb(S) = S.
Proof. Since SoSJm is monoreflective in and Skula-closed subobjects are extremal, if
b(A) = A, then A is sober. For the converse, suppose A is sober. We have the epi con¬
tainment A C b(A) of sober spaces. Since the containment is epi, its dual is injective. Since
it is a containment, the dual is surjective. We now apply the fact that bijective maps in P^
are isomorphisms. â–¡
Lemma 7.2.5. Let f,g:A-+Be DP^ and
c = coeqmin(f,g) : B —*C.
The SpP3m-coequalizer of (f,g) is obtained by following c with the spatial reflection.
Theorem 7.2.6. Let j and m be subset systems which preserve regular epis. Suppose f :
X —» Y G SoSf is a map of sober objects, with dual 1. f is extremal mono.
2. (¡> is extremal epi.
3. f is a subspace inclusion.
j. f is a quotient map (in P3m) such that the codomain is spatial.
5. f is onto.
6. f is regular epi.
7. / is regular mono.
Proof. (1 o 2) and (6 «=> 7) hold because SoSJm and SpF3m are dual. (2 o 3) follows from
7.2.3 and 7.2.4. (3 =>• 4) holds because the SbS^-structure on a subspace is defined by a
P^-congruence. (6 4^ 4 «=> 5) holds by 6.3.2 and 7.2.5. (7=> 1) holds in any category; see
Lemma 2.1.9. â–¡

100
7.3 Computing Limits
We describe how to calculate limits in §Jm. To this end we prove a result analogous
to the fact in general topology that any family of subsets generates a smallest topology.
Lemma 7.3.1. Let X be a set. If $ is a collection of families of subsets of X, such that
each t G $ is closed under j-unions and m-intersections, then a := f'|5 is also closed under
j-unions and m-intersections.
Proof. For any r G 5, the inclusion map from a —> r preserves subset inclusion. Thus, for
any r 6 J, j(o) C j(r) and m(cr) C m(r), so if S G j{cr) (resp. S G rn(a)) then |JS € t
(respectively, p| 5 G r). Hence a = p| 5 is closed under unions of j-families and intersections
of m-families. â–¡
If a is a family of subsets of X, we consider 5, the collection of all (j, m)-complete
families of sets of X that contain a; f] j? is called the (j, m)-complete family generated by a.
Definition and Remarks 7.3.2. Defining S^-products. Consider a set {X¿} C Obj(S^)
The underlying set of X is the cartesian product nX¿, with canonical projections 7r, :
X —* Xi. O(X) is the (j, m)-complete family generated by sets of the form R~X{U) where
U C O(Xi).
1. The object X thus defined satisfies (sep): if x, y G X and x ^ y, there exists i such that
7ri(x) 7Ti(y). Since satisfies (sep), there is U G O(Xi) with |U fl {/xix,'Kiy}\ = 1.
Thus there is V G O(X) (namely V = n~l(U)) with |V fl {x,y}| = 1.
2. X is the categorical product of (XJ: since the underlying set of X is the cartesian
product of {Xj}, whenever we have an object Y G Obj(S^) and functions :Y —> X¿,
there is a unique function / : V —» X so that for all i, -Rif = fi. If each /¿ is an
S^j-map, then because of our definition of D(X), / is also an S^-rnap.
Definition and Remarks 7.3.3. S^-equalizers. Let j and m be subset systems which
preserve regular epis. Let f,g : X —> Y be a pair of maps in S^. The set E := {x G X :

101
f(x) = g(x)} has a natural S^-structure 0(E) = {U fl E : U € D(X)}. The relation R on
D(X) given by
urv u n E = v n E
is a P^-congruence because
f)uinE= (f|üi)n£7
i i
and
(J^n£=((Jt/¿)n£
i i
for any set E and family of subsets (t/j). Since O(E) is the quotient of O(X) by a P^-
congruence, and j and m preserve regular epis, O(E) is (j,m)-complete. E satisfies (sep),
because X does. If h : Z —> X equalizes /, g, then the image of h must be contained in E.
Since h is an S^-map, the inclusion of the image of h in E is also an S^-map.
7.4 Quotients, Extremal and Regular Epis
Definition and Remarks 7.4.1. Let X G Obj(S^), Eqv(X) denote the set of equivalence
relations on X, and denote the set of subfamilies of O(X). We consider equivalence
relations to be sets of ordered pairs. If R € Eqv(X) and x € X, we use the notation
R(x) = {y £ X : (x, y) G R} for the equivalence class of x.
1. If R e Eqv(X), we say U € O(X) is R-saturated if (x, y) € R and x G U imply that
y € U\ let R* denote the family of all R saturated members of O(X). If R\ C R2, then
r*2 c r*v
2. If a € ?P, we define an equivalence relation a* on X by (x, y) G a* if and only if for
each U G (J, we have x G U y G U. Note that cq C &2 ^ av
3. If a G *}3. then each member of a is cr*-saturated. In symbols, a C a**.
4. Suppose R G Eqv(X). U G D(X) is R-saturated if and only if for all (x,y) G R
x G U <=$â–  y G 17. Thus, R C R**.

102
5. To summarize, * gives a Galois connection between Eqv(X) and *}3.
Given R € Eqv(X), we define an S^-structure on the set X/R of equivalence classes, by
defining 0(X/R) to be the family {{-R(x) : x G U} : U G R*}. Since arbitrary unions and
intersections of saturated sets are saturated and since the inclusion of R* in D(X) is order
preserving, 0(X/R) is (j, m)-complete.
We omit the proofs of the following lemmas, which are straightforward verifications.
Lemma 7.4.2. If f : X —*Y G Map(S^), R G Eqv(X), and f is constant on each equiva¬
lence class R(x), then f factors as X —> X/R —> Y, where X —» X/R is the quotient map,
and X/R —> Y is defined by R(x) f{x). For all U G D(T), f~l(U) is R-saturated and in
0(X).
Lemma 7.4.3. Let R\,R2 € Eqv(X) with quotient maps : X —> X/Ri- There is a
quotient map q : X/R\ —* X/R2 such that q2 = qq\ if and only if R\ C R2.
Lemma 7.4.4. For R G Eqv(X) the following are equivalent:
1. R** = R,
2. there is a £ p such that R = a*,
3. X/R satisfies (sep).
Proof. The equivalence of the first two conditions is trivial. If R = a* and R(x) ^ R(y),
then there exists U G a such that |U fl {a;,y}| = 1. So (2 =>• 3).
(3 =$■ 1) Since R C R**, we have a quotient map q : X/R —► X/R** as described,
in the preceding lemma. It suffices to show that q is injective. If R(x) / R(y) and X/R
satisfies (sep), then there is U G R* with JC7 n {x,y}\ — 1. Thus, R**(x) ± R**{y)- □
Corollary 7.4.5. If X is a To topological space and R G Eqv(X), then X/R is To if and
only if R — R**.

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Lemma 7.4.6. If f : X —> Y sS^ and R G Eqv(X) is defined by
R = {(*>2/) : fix) = f{y)}
then R** = R
Proof. We use Lemma 7.4.2. Since Y satisfies (sep), and for all U G 0(Y), f~x{U) is R-
saturated, X/R satisfies (sep): if x, y G Y and x ^ y, then we have U G 0(F) containing
exactly one member of {x, y}. â–¡
Definition and Remarks 7.4.7. Coequalizers in §Jm. If /, g : X —> Y, let a — {U G
D{X):f-\U) = g-\U)}.
The coequalizer of /, g is the quotient map c : Y —► Y/o*. First, one shows that
cf — eg: if c(/(x)) ^ c(g(x)) there is U G a which contains exactly one of {f{x),g(x)}.
But f~l{U) = 5_1(f/), since U G cr; this contradicts the fact that x is in exactly one of
/-1(t/), <7-1(i7). This contradiction shows that cf = eg.
Suppose h : Y —> Z and hf = hg. We claim that h — we for a unique w. Note that
h factors as Y —* Y/R —* Z, where R = {(x,y) : h(x) — h(y)}. By Lemma 7.4.3, it suffices
to show that a* C R. If h(x) ± h(y), then we have U G 0{Z), which contains exactly one
member of {h(x),h(y)}. Since hf — hg, h~l{U) G a, so (x, y) & a*.
We say / : X —► Y is a quotient map, if there is an isomorphism i : Y —> X/R, and
an equivalence relation R = R** such that R(x) — if{x), for all x.
Proposition 7.4.8. For f : X —+ Y G §£,, the following are equivalent:
1. f is a quotient map,
2. f is regular epi,
3. f is extremal epi.
Proof. (1 => 2). Suppose that / is a quotient map, with associated equivalence relation
R = R** and family of R-saturated sets, R*. Define a S^-object Z to have underlying set

104
jJ{C x C : C 6 X/Rj - a disjoint union - and define O(Z) = 7(Z). D(Z) is complete
(hence (j, m)-complete) so Z G Obj(S^); moreover any set map from Z —> X is a S^-map.
We now define two maps g,h : Z —> X by g(x, y) = x and h(x, y) = y. By construction, / is
the coequalizer of g, h as computed in Set. Since / is also a S^-quotient map, / is also the
coequalizer of g, h in Sjn.
(2 => 3) holds in any category - [13, 17.11].
To show (3 =£■ 1) suppose / is extremal epi. We may factor / into a quotient map
q followed by an injection i, using Lemma 7.4.2. Since / is epi, then i is also epi; it now
follows, by the extremal property of / that i is an isomorphism. So / is a quotient map. â–¡
7.5 Flat Spectra
Let j and m be subset systems which preserve regular epis. We now examine a
monocoreflective category of Sp¥3m. To define the category, we need to put a partial order
on A, for A € ¥Jm. (Recall that 7A is the set of functions A —> 2 which preserve order and
(j, m)-optimum bounds.) We think of x, y G Í'A as characters x, y : A —* 2. We define x < y
to mean Va 6 A.x(a) < y(a). Clearly, x < y if and only if e(x) = {a 6 A : x(a) — 1} C e(y).
The relation < is anti-symmetric by virtue of the fact that our spaces satisfy (sep).
Now we define to be the full subcategory of 5 G containing all A with the
property that for all x, y G 4/(^4), x < y implies x — y. To show FPm monocoreflective, we
will use the category theoretic dual of the following theorem.
Theorem 7.5.1. [13, 37.1] Suppose A is a cowellpowered complete category, and 3 is a full,
isomorphism closed subcategory. B is epireflective if and only if it is closed under products
and extremal subobjects.
Proposition 7.5.2. is monocoreflective in Sp¥3m.
Proof. SpFJm is cowellpowered and cocomplete, because it is wellpowered, complete and has
a coseparator 2. [13, 23.14] We show that F^ is closed under coproducts. Suppose the

105
coproduct JJ Ai has characters x < y. Since x ^ y, there is an index i such that xpi ± ypi.
But x\ii,yiii are characters of Ai. Thus, some At has characters x < y.
We show that is closed under extremal epis. Suppose A has flat spectrum and
A —> A/K is a quotient. If A is non-trival, then it cannot have constant characters. Thus, if
A has flat spectrum, all characters are onto. Now we use 6.3.2 to conclude that characters
of A/K are precisely characters of A whose kernel contains K. It follows that A/K has flat
spectrum. â–¡
Remark 7.5.3. The theorem would remain true if we redefined to be a full subcategory
of containing flat spectrum objects. The rationale for not using this definition of F¿, is
that non-spatial flat spectrum objects behave rather differently than spatial objects. Without
assuming A spatial, the spectrum gives incomplete information. For instance, there are A
with no characters whatsoever! These objects trivially have flat spectrum. We wish to avoid
this problem.
7.6 Epicomplete objects in SJm
Assume throught this section that j and m preserve regular epimorphisms.
Definition 7.6.1. An object A in any category is epicomplete if any map f : A —> B that
is both epi and mono is an isomorphism.
Note: any monoreflection map is epi; see Herrlich and Strecker [13, Section 36],
Soberification is a monoreflection on SJm. Therefore, any epicomplete object in SJm must be
sober. We now examine which sober spaces are epicomplete.
We recall the partial order on characters A —* 2 that was used in the preceding
subsection. For a sober space, the points x € X are in bijective correspondence to S^-maps
D(X) —> 2. If A is a space, x < y means e(x) C e(y). Note that if U 6 O(X), x G U and
x < y, then y € U] in other words, each U 6 O(X) is a increasing subset of X. Also note
that any continuous map preserves this order.
Lemma 7.6.2. Suppose X € Obj(S^) with to C j fl m.

106
1. X is a chain if and only if D(X) is a chain.
2. If X is not a chain, then there is a (j,m)-structure r on X, such that r C D(X) and
(X,t) satisfies (sep).
Proof. 1. If X is a chain, then the family 3 of all increasing subsets of X is a chain (ordered
by inclusion). Since £>(X) C 3, £>(X) is a chain. Conversely, if £)(X) is a chain, then the
set of prime ideals of D(X) is a chain, so X is a chain.
2. Suppose D(X) is not a chain. Then we have incomparable U,V € D(X). Let
r = {W E D(X) : W C U or U C W}. One may easily verify that r is closed under all
unions and intersections which exist in O(X); since j(r) C j(Q(X)) and m(r) C m(D(X)),
this shows that r is a (j, m)-complete.
We now verify that (X, r) satifies (sep). Suppose x, y € X are distinct points; without
loss of generality, there is G E O(X) such that x € G and y & G. If x E U then G fl U E r
separates x and y. If x $ U and y $ U, then G U U separates x and y. Lastly, if x £ U, and
y G U, then U separates x and y. â–¡
Theorem 7.6.3. Let X G Obj(5oS^), ui C j n m. We have the implications (i) => (ii) =>
(in) =» (iv). If j — oo, then (iv) => (i).
(i) X is epicomplete
(ii) If f : X —>Y is SoPm-monic, then f(X), b f(X), and X are all homeomorphic.
(in) There is no (j,m)-structure, r on X, such that r C O(X) and (X,t) satisfies (sep).
(iv) X is a chain satisfying (Hi).
Proof. (i=> ii). Suppose / : X —> Y is monic. By corestriction of / we obtain a S^-map
X —> f(X). Composing with the inclusion map of f(X) in bf(X) we obtain a monic epic
SoPm map with domain X\ (i) forces this map to be an isomorphism.
(ii => iii). Suppose r is a (j,m)-structure on X and r C D(X). The identity map on
X is a continuous bijection (X, O(X)) —> (X, r). Following this map by e(x» : (X, r) —»

107
Í'T, we obtain a mono epi map (X,D(X)) —» i'r. (ii) implies that all maps involved are
isomorphisms.
(iii => iv). This follows from Lemma 7.6.2.
(iv => i). When j = oo, we have more information about which (j, restructures
correspond to a given partially ordered set. Suppose (X, <) is a poset and r is a topology
on X such that x < y if and only if each open set containing x also contains y. The union
of the family {U € t : x £ U} is open and equals {y 6 X : y x}. Thus, r must contain
the weak topology generated by the sets {y € X : y x} for x € X.
Suppose X is a chain, and / : X —> Y is an epic monomorphism. Since / is injective
and preserves order, f(X) is a chain order isomorphic to X. Ergo, the f(X) has a topology
finer than the weak topology induced by the partial ordering. Since the corestriction / : X —►
f(X) is continuous, the topology on X is finer than the topology on f(X). Thus, X and
f(X) are homeomorphic. Since X is sober, f(X) is sober, which implies b(f(X)) — f(X).
Since / is epi, b(f(X)) = Y, so X and Y are homeomorphic. â–¡

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BIOGRAPHICAL SKETCH
Eric Richard Zenk was born on February 2, 1977, in Fridley, MN. He lived in An¬
dover, MN, for the following eighteen years. Then, Eric attended St. Johns University in
Collegeville, MN, where he majored in mathematics and physics. Upon graduating in 1999,
Eric moved to Florida where he studied mathematics at the graduate level at the University
of Florida.
110

I certify
able standards
dissertation for
I certify
able standards
dissertation for
I certify
able standards
dissertation for
I certify
able standards
dissertation for
I certify
able standards
dissertation for
that I have read this study and that in my opinion it conforms to accept-
of scholarly presentation and is fully adequate, in scope and quality, as a
the degree of Doctor of Philosophy.
Jorge Martinez, Chairman
Professor of Mathematics
that I have read this study and that in my opinion it conforms to accept-
of scholarly presentation and is fully adequate, in scope and quality, as a
the degree of Doctor of Philosophy.
Pham Tiep
Associate Pr
essor of Mathematics
that I have read this study and that in my opinion it conforms to accept-
of scholarly presentation and is fully adequate, in scope and quality, as a
the degree of Doctor of Philosophy.
Alexander Dranymnikov
Professor of Mathematics
that I have read this study and that in my opinion it conforms to accept-
of scholarly presentation and is fully adequate, in scope and quality, as a
the degree of Doctor of Philosophy.
/L U'-A-'
Scott McCullough
Professor of Mathematics
that I have read this study and that in my opinion it conforms to accept-
of scholarly presentation and is fully adequate, in scope and quality, as a
the degree of Doctor of Philosophy.
Reyíialdó Jimenez
Associate Professor of Romance Languages and
Literatures
This dissertation was submitted to the Graduate Faculty of the Department of Math¬
ematics in the College of Liberal Arts and Sciences and to the Graduate School and was
accepted as partial fulfillment of the requirements for the degree of Doctor of Philosophy.
August 2004
Dean, Graduate School

LD
1780
20 OA