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# Subset systems and generalized distributive lattices

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Title:
Subset systems and generalized distributive lattices
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vi, 110 leaves : ; 29 cm.
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English
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Zenk, Eric R
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## Notes

Thesis:
Thesis (Ph. D.)--University of Florida, 2004.
Bibliography:
Includes bibliographical references.
Statement of Responsibility:
by Eric R. Zenk.
General Note:
Printout.
General Note:
Vita.

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Full Text

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 inadequate. 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.

Jorge Martinez, Scott McCullough. and Pham Tiep.

* Anyone taking the time to read these words: a dissertation, like any book, is meant to

ACKNOWLEDGEMENTS .................... ............ ii

ABSTRACT . . . . .. v

CHAPTERS

1 INTRODUCTION .................... ............. 1

1.1 Distributive Lattices ................... .......... 1
1.2 Subset Systems ................... ............. 2
1.3 Methods and Results ............................... 5

2 PRIMER ON CATEGORIES AND POSETS .................. 8

2.1 Distinguished Maps ............................. 9
2.2 Bounds .............. ...................... 12
2.3 Natural Transformations ........................... 17
2.4 (Co)Lim its . . . . 19

3 ALGEBRAS OF A MONAD ............................ 28

3.1 Categories of Algebras ............................ 28
3.3 Detecting Categories of Algebras ................ ...... 34
3.4 Distributive Laws ................... ........... 37

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

6.2 Factorization of Maps Using Preorders ....... . 85
6.3 Factorization of Meetsemilattice maps .... ..... 88
6.4 Coequalizers in DP3 ............................. 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 ............. ................... 110

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 DPI of (j, m)-complete lattices which can be embedded in a completely

distributive complete lattice is a full subcategory of F-algebras. DP' 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

aAa = a,

aAb=bAa, and

(aAb)Ac=aA(bAc).

A lattice is a set with two semilattice operations A and V, which are related by the condition

that
aAb=a <== aVb=b.

A lattice may be partially ordered by defining a < b to mean a A b = a; with this order a A b

is the largest thing smaller than both a and b and aV 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
aA(b V c)= (a A b) V(a A c),

a V (b A c)= (aV 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 = aV a = a, which does not hold in arithmetic. There is more symmetry in the

equations defining distributive lattices than in ordinary arithmetic; formally "A" and "V" are

interchangeable, 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 n V, and

if (Ui) is a family of open subsets, then the union Ui Ui 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

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 supreme or infima while the maps are order preserving functions which

preserve said infima and supreme. P denotes the category of all posets and order preserving
maps.
The challenge is how one selects subsets which have infima and supreme. 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 f : A --+ B,

{f(S) : S E Z(A)} C Z(B).

Z-complete posets are posets A in which each set S E 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 f : A -* B, S C A, with

JAI < K implies If(A)I < K.

2. A subset S C A is (upward) directed if for each x, y E S, there exists u E S such that
x < u and y < u. The rule dir which selects all directed subsets of a poset is a subset
system: if f(x), f(y) E f(S) and S is directed, then there is u E S such that x < u
and y < u. Thus f(u) is a common upper bound for f(x), f(y).

3. A subset S C A is (upward) compatible if for any x, y E S there is u E A such that
x < u and y < u. "Compatible" differs from "directed" because for the former u E S,
while for the later we only require u E A. Similar arguments show that compact, which
selects compatible subsets is a subset system.

4. A subset of S C A is a chain if x, y E 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 S C A is (upward) self-bounded if there is s E S such that for all

x E 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 f : 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 Zo by

Zo(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.

P7 the category of all (j, m)-complete posets: that is, posets in which j-suprema and
m-infima exist and are preserved by all maps.

DP. the category of all completely distributive complete lattices.

DPF the full subcategory of PL containing objects which can be P--embedded in a
completely distributive complete lattice. DPJ) is discussed in Section 5.3.

M1 the full subcategory of PJ containing posets with an F structure, where F is

the monad defined in Section 5.3.

SplP, the full subcategory of IP- containing spatial posets. See Chapter 7.

8, the category of generalized spaces with (j, m)-complete families of distinguished
subsets: see Chapter 7.

F, the full subcategory of SplPJ, 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 M3, 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 monadicc 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.

M

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 Ml of F-algebras offers a (somewhat mysterious) generalization of the category

of distributive lattices. The subcategory DP3 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 DPW 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 Ml, DPI, and SppP3 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, PL and DPI,. 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

I

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 T1 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 familiarity with

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 A1 to A2 is denoted A(A1, A2).

A functor F : A -- 3 assigns each A E Obj(A) an object F(A) E Obj(3) and

each A-map f : A1 -+ A2 a 3-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 -- BP.

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

diagram

W f X

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 A1, A2 E A, F gives a function

from the horn-set A(A1, A2) into B(FAI, FA2) by

(f : A1 A2) F (Ff : Ai A2).

If, for each A1 and A2 this map is onto, then F is said to be full. If, for each A1 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 1' such that Obj(1B) C Obj(A) and

all f : A -- B with A, B E Obj(S) are 9-maps. 1B 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 f : A1 -- 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.

2. A map f : A2 -* A3 is mono, a.k.a. monic (in noun form, a monomorphism)

if whenever g and h are maps A1 -+ 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 f : A -> B is an isomorphism, if there is g : B A such that

idA = 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 = == 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 : Rd -* 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 4 from the trivially ordered set with two elements to the chain with

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].

Definition and Remarks 2.1.6. Consider a pair of maps f, g : A1 -- A2. A map i : Ao --

A1 right-identifies f and g if fi = gi. A map i : Ao -- A1 is called the equalizer of f and g

if:

(eql) i right-identifies f and g, and

(eq2) i has the feature that whenever j : Bo A1 right-identifies f and g, there
is a unique map e : Bo -- Ao such that j = ie.

The definite article is used for equalizers, because (eq2) implies that if i : Ao -- A and
i' : A' -- A are equalizers for f and g, then there is an isomorphism j : Ao A' such that
i' = ij. Notation: i = eq(f,g).

If there are f and g such that i : Ao -- A1 is the equalizer of f and g, then i is regular

mono. Regular monomorphisms are monomorphisms. If f is epi and regular mono, then f 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 f and g if

(coeql) e left-identifies f and g, and

(coeq2) e has the feature that whenever d: A2 B3 left-identifies f and g, there is
a unique map c : A3 -- B3 such that e = cd.

Notation: e = coeq(f, 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 f is extremal if whenever f = gh and g is mono, then g

is an isomorphism. Dually, a monomorphism f is extremal if whenever f = 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].

Definition 2.1.8. The map f : A B is split mono if there exists g : B A such that

idA = gf. The map g : B -- A is split epi if there exists f : A -- B such that idA = 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 isA.

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 E 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 c 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).

Definition and Remarks 2.2.1. Let A be a preordered set. Define the up- and down-

closures of x E A by

x = {a E A: a x}

and

T x = {a E A: x a}.

More generally, if S C A define

SS = U{ x:x E S} = {a A: 3s E S, a -< s}

and

SS = U{ :x E} = {a e A: 3s E S, s a}.

If S C A an upper (resp. lower) bound for S is a E A such that for all s E S, s i a (resp.

a -< s). Use the following notation for the set of upper bounds of S,

(S) = n{T: x :x E S} = {a E A : Vs E S, s a}

and similar notation for the set of lower bounds

B(S) = n{J x: x E S} = {a e A: Vs E S, a s}.

If a E A, S C A and f a = B(S), then a is a least upper bound, a.k.a. join, a.k.a.

supremum of S. An equivalent way to say this is

a< x <== VsES, s
If both a and a' are supreme of S, then a, a' E B(S). Thus a c a'. In a partially ordered

set, the (unique) supremum of S is written V S. If a E A, S C A and I a = B(S), then a is

a greatest lower bound, a.k.a. meet, a.k.a. infimum of S. An equivalent way to say a is a
least upper bound is

x < a = Vs E S, x < s.

In a partially ordered set, the (unique) infimum of S is denoted A S. (In using this notion
with a preordered set (A, -), one refers to the associated poset A/ L. For example, V
then denotes the equivalence class containing all supreme of S.) If a = V S or a = A S, a is
an optimum bound for S.

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 B to denote either operator B or B.

1. For x, y E A, x -y -= I x C1 y = T y CT x.

2. For S,T C A, S C T =- B(T) C B(S).

3. For S C A, B(S) = (1I S), B(S) = B(T S).

4. For (Si), l a family of subsets of A, B(US,) = nB(Si).

5. Iff : A -- B is order preserving, then f(BA(S)) 9 BB(f(S)).

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 f : A -- B and g : B -- A such that

(gcl) f and g are order reversing.

(gc2) For all a E A and b E 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 f and g, along with A and B is also a true
Note that f and g respect _. The notation f g means "for all a E A, f(a) g(b)."

1. g gfg: Suppose a E A. By (gc2), a f(g(a)) and g(a) g(f(g(a))). Using
a f(g(a)) and (gcl), g(f(g(a))) g(a).

2. f(A/ 2) is dually order isomorphic to {b E B : b c- fg(b)}: This follows from the
preceding statement. Since f 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 -< f(a): if a g(b), then b f(g(b)) f f(a). The converse is proved
similarly.

4. For any S C A, Bf (S) = g-1~(S). Calculate

x E B(f(S)) = Vs S, x f(s)

= Vs E S, s g(x)

g(x) E (S)

= x E6 g-'R(S)

5. If A and B are posets, the previous item implies A f(S) = f(V S).

It is helpful to rephrase condition 3 for posets: "g(b) is the largest a with b < f(a)."
In symbols,
g(b) = V{a E A: b f(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.

Definition and Remarks 2.2.4. Suppose A and B are preordered sets. A pair f : A -* B,
g : B -* A of order preserving functions is an adjoint connection between A and B if

(adl) For all a e 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. f fgf and g gfg

2. f(A/ c) = {b E B : b = g(f(b))} and g(B/ 2) = {a E A: a = f(g(a))}.

3. f(a) b <==> a g(b)

4. For any S C A, Bg(S) = f-'B1(S) and f (S) = g-'B(S)

5. If A and B are posets, f(A S) = A f(S) and g(V s) = V g(S).

Because of the asymmetry between f and g, and property 3, f is called the left adjoint
and g the right adjoint. Again there is an interpretation of 3 in words. "f(a) is the smallest
b such that a g(b); g(b) is the largest a such that f(a) b."
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 S C 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.

1. Suppose f : A -+ B is order preserving and A is complete. Then f is a left adjoint if
and only if f(V S) = V 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(A S) = A 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(V s) = A f(S) for all S C 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

g(b) = V{a E A: f(a) < b}.

Since f preserves supreme, f(g(b)) < b. Thus, g(b) is the largest a such that f(a) 5 b. O

In adjoint (resp. Galois) connections, f(a) and g(b) can be defined as supreme or
infima. 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 -* S be functors. A natural transformation a : F -+ G is
a rule which assigns a map aA : FA -* GA to each A E A such that if f : A -4 B is an
A-map the following diagram commutes.

FA F FB

GA GB
GA GB

Construction 2.3.2. Let F, G, H : X -- A be functors, a: F -- G and 3 : G -- H natural

transformations. (The situation is drawn in the diagram below.)

F

G 4
X 1

H 4
Then the assignment (3a)A = (/A)(aA) is a natural transformation.

Proof. Let f : A -* B be a map. Since a and / are natural, each square below commutes.
Ff
FA F FB

fa
GA -GB

0 Hf
HA HB
Therefore, the outside rectangle commutes; hence, fa is natural. O

Call 3a the vertical composition of a and 3.

Construction 2.3.3. If F : A -- and G, H : 3 -- e 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 : 3 -- e and G, H : A --+ 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 -- 3 and H, J : S -- e are functors, and a : F

G, 3 : H -- J are natural transformations. (The situation is drawn below.)
F H

A J e

G J

Then for each object A E Obj(A) the following square commutes.

FI3A
FHA FOA FJA

aHA aJA

GHA A GJA

The assignment (/3 a)A := (aJA)(FPA) = (G3A)(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 /A : HA -* JA. To prove the natural property of 3a,

we use the squares which define 3aA and /aB. Compare the corners of the squares using

maps obtained from f by application of the functors FH, GH, FJ, and GJ. The resulting

commutative cube shows 3a is a natural transformation. O

Call -3 a the horizontal composition of a and /.

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 23 are equivalent if there are functors F : A -- 3 and G : 3 -B A

such that FG is naturally equivalent to idB and GF is naturally equivalent to idA.

A and 3 are dual if there exist contravariant functors F : A -- 3 and G : 3 -- A

such that FG is naturally equivalent to idE and GF is naturally equivalent to idA.

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 f 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)

between them; for technical reasons one requires that if f, g E Map(D) and fg is defined
then fg E Map(D), and that for all A E Obj(D), idA E 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, {SA : S -* A: A E Obj(D)}) for D consists of
S E Obj(A) and maps SA : S -* A such that if f : 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, {eA : A E Obj(D)})
for D is the limit of D if whenever (S, {SA : A E Obj(D)}) is a source, there is a unique map
c: S -- L such that for each A E Obj(D), SA = eAC.
Dually, define a sink, or cosource for D to be an object S together with maps iA :
A -* S (for A E Obj(D)) such that for each f : A -- A' in Map(D), iA = iAf.
A colimit (C, {iA : A E Obj(D)}) for D is a sink for D, such that if (S, {iA : A E
Obj(D)}) is any sink there is a unique map c : C -- S such that for each A E Obj(D),
PA = CiA.
Note that limits and colimits of D are unique up to compatible isomorphism. To
prove this (for limits), suppose (L, eA) 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 = idj 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
(HX, rx).
Recall that 1l X contains all functions f : X -- U X,with the feature that f(X) E X
for all X E X. Such f are called choice functions, because they choose one member of each
X E X. The functions lrx : J X -- X are defined by 7ix(f) = f(X).

If there is a source (S, {sx : S X X E X}), there is a unique function c : S -- n X
making
SC S-- S

X
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
In Set, the colimit of the diagram containing all members of X and no functions,
is the disjoint union I X, with inclusion maps ix : X U IX. The colimit property is
satisfied by (U X, px) because if there is a sink (C, ix), the function c : [I X C defined
by c(x) = ix(x) for the unique X E 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 A-objects is "the A-object freely generated by U X."

Example 2.4.3. The [cojequalizer of f, g : A -* B is the [co]limit of the diagram containing
objects A and B along with maps f 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 f 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.

Example 2.4.4. Let us consider the diagram

A

B ---C.

(The diagram also contains identities for all objects, but for brevity these are omitted.) The
limit (B x cA, irA, TiB) of this diagram is called the pullback of f along g, or the pullback of g
along f. To be explicit, firA = rBg and if (Q, qA : Q -* A, qB : Q B) satisfies fqA = qBg,

then there is a unique map i : Q B x c A.
In Set,
B xcA = {(b,c) E B x A: g(b)= f(c)},

and the projection maps 7r 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 f : A -* C be any map and g : S -- C be a subset inclusion of S C C.

Note that (s, a) E S Xc A if and only if g(s) = f(a); suppressing mention of g, this reads
S xc A = {(s, b) : f(b) = s}. Thus, this pullback is canonically isomorphic to the preimage

of S under f. This example partially motivates the name "pullback."
Second, let f : A -* B be any map and consider the pullback of the diagram,

A

I

A B

i.e., the pullback of f along itself. Using the computation for pullbacks in Set, given above,

A XB A = {(a, a') E A x A: f(a) = f(a')}.

M

This relation on A is often called the kernel of f. Thus, one calls (A xB A, 7ruft, Tright) the

kernel pair of f. 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 1B. 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 E Map(D)}.
Let D be a diagram in A. A functor F : A -- 3 preserves the limit of D, if whenever
the limit (L, eA) exists in A, (DL, D(eA)) 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, {eA : A E Obj(D)}) is
the limit of D, then (GL, {GeA : GA -- GL}) is a sink for GD. If (GL, {GeA : GA -* GL})
is the colimit for GD, then G takes limits to colimits. Analogously, may G take colimits to
limits.

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 <- 3f : A-+ B.

Categories are more complex, because many maps could manifest 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 3 consists of contravariant functors F : A -- 3, G : B -- A, together with natural

transformations
ri: idA -* GF

and

e : ids FG

such that (Fr)A)(EFA) = idFA and (GeB)(riGB) = ideB for each A E Obj(A) and B E

Obj(B); these equations are the so-called triangle identities. (Alternate terminology: if
(F, G, rl, ) 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 rA (resp eB) is an
isomorphism are dual. [3, Section 4, Lemma 1]

2. There is a natural bijection A(A, GB) -3 S(B, FA) given by

(f: A GB) F(f)(EB) :B--FA.

The inverse map is

(g : B -- FA) H G(f)(r/A) : A -- GB.

There are two (identical) calculations required to check that the functions are mutually
inverse. One is summarized by the diagram below.

(GFf) GeB
GFA GFGB > GB

i7A I 77GBI

A GB

The square commutes because 7T is natural. The triangle commutes because of the
identity idac = (GeB)(rTGB). The reader may formulate and check what is meant by
"naturality" of the bijection.

3. F and G both take colimits to limits.

4. Each 77A has the following universal property: if f : A -- GB, there is a unique map

f = F(f)(eB) : B -- FA which makes the following diagram commute:

A -A GFA

GB
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 S consists of functors F : A -- 3 and G : S -- A and natural transformations

77: idA -4 GF and e : FG -- ids such that (GeB)(r)GB) = idGB and (eFA)(F7rA) = idFA;

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(r7) and fix(e) containing all objects such that r7A (resp eB) is an

isomorphism are equivalent.

2. There is a natural bijection A(A, GB) -- 'B(FA, B) given by

(f : A --GB)- (cB)F(f) : FA --+B.

The inverse map is
(g: B FA) G(f)(77A) : A -- GB.

3. F preserves colimits; G preserves limits. Borceux [6, Volume 1, 3.2.2]

4. Each r7A has the following universal property: if f : A -- GB, there is a unique map
f = (eB)F(f) : FA -- B which makes the following diagram commute:

A --- GFA

f 1Gf

GB
Each eB has the analogous universal property: if f : FA B, there is a unique
f = G(f)(77A): A -- GB such that

B --FGB

FA
commutes. Any functor for which there is a natural transformation t7 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 E S there is a set I and an I-indexed family of
maps fi : B -- GA, such that any map f : B -- GA can be written as h = (Gt)fi for
some i E I and t : Ai -- A.

CHAPTER 3

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

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

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, i, i) on A consists of a functor

T : A A,

a natural transformation

T : idA T,

and a natural transformation

such that the following identities (expressed by commutative diagrams) hold: the unit laws

T 2 T2 T

T
T
and the associative law -

T3 T 2 .

T2 A T

(In these diagrams, T" denotes the n-fold composite of T with itself.)

Intuitively, TA is the free object on A; rA is the "insertion of variables" map; pA

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 p.

Definition and Remarks 3.1.2. Let T be a monad on A. A T-algebra (A, a) consists of

A E Obj(A) and a : TA -, A (the so-called structure map) such that a(t7A) = idA (unit

law) and

T2A Ta TA

TA a A

TA ----- A

(associative law) commutes. A homomorphism f : (A, a) -, (B, b) of T-algebras is an A-map
f such that

TA -- TB

a b

A -A
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, iA)

and

FT(f) = Tf.

The associated natural transformations are

7T = 7 : idA ` UTFT = T

and
ET : FTUT idA : T(A, a) := a.

[21, VI.2.Theorem 1]

Limits in AT are "computed in A" in the following sense.

Proposition 3.1.3. [4, 3.3.4], [6, Volume 2, 4.3.1] Suppose (T, 7, I) is a monad onA. If D

is a diagram in AT such that UT(D) has a limit (L,pv(A)), then there is a unique structure

map : TL -- L such that each pu(A) is a T-algebra homomorphism.

Proof. Let D be a diagram in AT, such that UTD has a limit (L, fA). For each A E Obj(D),

name the structure map SA : TA -* A. The goal is to produce a structure map s : TL L

such that each eA : L A is a T-algebra map.

The requirement that each LA is T-algebra map amounts to: for each A, the diagram

below commutes.

FL FIA FA

18 IA

L A
L ---~--- A
Since (L, LA) 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) = idL: the diagram

A iA FA

L "FLA
L---FL

L

A
commutes using the definition of s, and that i is a natural transformation. Thus, for each

A, LA = LAs(iL). From the uniqueness of the map from the limit of D to any other source

for D, it follows that idL = s(iL).

A similar comparisonn of squares diagram" can be used to verify that the algebra

associative law holds. O

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 AT has coequalizers.

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

1. Let F : A --* G : B -- A, rj : idA GF, e : FG -+ idB be an adjoint connection.

Then (GF, 77, GF) is a monad on A.

2. If T = (T, 77, I) is a monad, then

GTFT = T,

T
77 =77,

and

GT T FT = 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

Example 3.2.2. Consider the category Grp of groups. The forgetful functor UGrp : Grp --

Set and the free group functor FGrp : Grp -* Set form an adjoint connection between Grp

and Set. Recall that FGrp(X) is the set of all reduced words

Z1 2 XnZ

where xi E X, si E Z; a word is reduced if for all i with 2 < i < n, xi_1 xi. The operation

on FGrp(X) is concatenation of words, followed by reduction. T = UGrpFGrp is the functor

part of the induced monad; 7 : idset T is the natural transformation whose component at

X sends x E X to word x'; if w~ w'n E T2(X) and for each i, wi = xz1 ,x) then
(X)(w/ w") =. bf',1 .. =b 'm(1al ... br1 ,l ... bn,,(n)a
.. ,1 .. 1,r(1) ..." (xln,l 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 SetT. (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, 7, and p of a monad generally have the

same roles as in this example.)

For example, consider the category Top of topological spaces and continuous maps. The

forgetful functor

UTop : Top -- Set

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 -4 'B, G : 3 -- A, 7 : idA -- GF, and e : FG -- ids

(T = GF, 7t, GpF). 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 3 is monadic over A omitting mention
of G.

These examples illustrate the qualitatively different behaviors of monadic and non-
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 7r is natural, the following equations hold:

(nat) (Ta)(77TA) = (rA)a,

unitt) (IA)(77TA) = idTA,

(unit2) a(r7A) = idA, and

(assoc) a(Ta) = a(yA).

These equations imply that a = coeq(pA, Ta): for if b : TA -* B right-identifies tpA and Ta,
then b(qA) : A -- B and

b(7rA)a = b(Ta)(7rTA) by (nat)
= b(IA)(77TA) since b right identifies
= b by (unitl).

One verifies that f = b(77A) the unique map f : A B with fa = b; for if fa = b, then
f = fa(71A) = b(7A).
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.

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 = see,

idA = 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 --r G : 3 -+ A, r7, e) be an adjoint connection, and T = GF,

77 = 77, and p = GEF be the associated monad. The following conditions are equivalent:

2. If f, g : A -- B E Map(B) and the pair (Gf, Gg) has an absolute coequalizer e', then
e = coeq(f, g) exists and Ge = e'.

3. If f, g : A -- B E Map(1B) and the pair (Gf,Gg) has a split coequalizer e', then

e = coeq(f, g) exists and Ge = e'.

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:

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.

Lemma 3.3.6. [6, Volume 2, Corollary 4.5.7] Suppose U : 3 -+ A, V : e -- A, and

Q : 3 -- e are functors. If U = VQ, U and V are monadic, and B has coequalizers, then Q

3.4 Distributive Laws

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, r7T, YT) and S =

(S, 7S, /~s) 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: S acts on functions by

Sf(xiX2 ." Xn) = f(xI)f(x2) *' f (X).

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 definitions
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

satisfying the compatibility conditions:

T S

ST e TS ST T TS

and

SST se STS es TSS

ST TS

STT TST -TTS
If there is a distributive law e : ST TS of S over T, then the composite monad is TS =

(TS, rl7 :s= r7T7S, T := j TS(TeS)). In the definitions of 77TS and prTS, juxtoposition

denotes horizontal composition of natural transformations. The verification that TS actually

In the case when "S=free monoid" and "T=free abelian group," the natural trans-

formation e : ST -* TS expresses a product-of-sums as a sum-of-products in the usual

way:

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."

Definition 3.4.2. T has a lifting into As, if there is a monad T such that TUs = UST,
US3T' = 7TUs and USpT = pTUS. To sketch the situation, T is a lifting of T onto As if

As ; As

SUs lUs
ALS---T---^A1

A T A
commutes.

Theorem 3.4.3. There is a bijective correspondence between lifting of T to As and dis-
tributive laws of S over T.

The proof is outlined in Beck [5]. The correspondence is defined as follows: if t is a
distributive law, and (A, a) is an S-algebra, then T is defined as follows:

T(A, a) = (TA, (Ta)(A)),

T(A,a) = A : (A, a) -T T(A,a),

and

7'(A, a) = MTA : TT(A, a) T(A, a).

Consider the following diagram:

STA

SITA

l SA TSA

as (T) jT(as)

A ) TA
the square commutes because 77T is a natural transformation; the triangle commutes because
e is compatible with ?s. Since the perimeter of the diagram commutes, 7TA is an S-algebra

map, A similar diagram, which uses the compatibility between f and PT, shows ,1T is an

S-algebra map. Since the underlying maps are defined in A and obey the monad laws, 7T

and /1 also obey the monad laws. If T is a lifting of T over As, e is defined to be the
following composition:

ST ST STS = STUSFs = UsFsUSTFS USESTFS USTFs = ST.

(In the above equation, ES is the counit of the adjunction (Fs, Us, Ls, 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 t : 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 S-structure as : SA -- A, T-structure aT : TA A, and S-structure

t : STA -- A for TA such that the following diagram commutes.

STA saT SA

t as

TA A

Given a TS-algbra (A, a), A has a T-structure aT map defined by

aT = a(Tr7SA) : TA -- A

and an S-structure map as defined by

as = a(r7TSA) : SA -* A.

The following diagram which expresses the distributivity of the structure maps commutes.

STA LA TSA

SaT Tas

SA TA

aS
/aT

A

Thus we have a map ATs -- (As)T, given by,

(A, a) E ATS (A, aT, as, Tas(eA))

which is functorial, because both rlT and 7rs are natural.

The inverse functor (As)T -+ ATS, maps (A, aT, as, t) to asS(aT) = art. The verifi-

cations for these assertions is given in Beck [5].

CHAPTER 4

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, 77, iz) is a monad on a reasonable

category, and that f : 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, (rlA)(A) C F(A) i.e., that for each A and a E A, FA contains the
"constant polynomial with value a". F is not generally "closed under composition"; i.e., it

is not generally true that (pA)(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 ?1 : E -- F such that each component is a subset inclusion. The

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 f, 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 f = m'e' is

another way of writing f 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

{fx : 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 fA = mi.

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

aAVS= V{aAs:s 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.

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 cl and c2 such that s = tci and t = sc2. So t = tc1c2 and s = sc2c1. Since

c2c1 shows s C s, c2c1 = ids. Similarly, cIc2 = idT. One identifies extremal subobjects s and
t if s C t and t 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 = tcb = sb, so (since s is mono) a = b; 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 n and

join operation U. In general, joins in Sub(A) are not disjoint unions; in general, U Ai is
computed using

U A = n{A' :Vi, Ai C A'}.

(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 E Sub(A), use the (epi, extremal mono) factorization to obtain a unique

s' E Sub(B) such that

A f B

s f+'(S)
commutes. The notation f+l(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) -

(s' : = f+(S) -- B) is the left adjoint. The right adjoint f-1 : Sub(B) -- Sub(A) is
obtained by taking the pullback of s : S -* B along f. Thus,

f+(S) C T S C f-(T),

f +(U S,) = U f+'(s),

and

f-,(n si = n f-'(s,)

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 E Obj(X) such that if
f : X -- Y, then

(Ff)+'(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)+'(EX) C EY.
Since eY : EY -, FY is mono, Ef is uniquely determined by the condition that

FX Ff FY

eX eY

EX -E EY
commutes. Thus, (Ff)+'(EX) C EY implies that Ef can be defined to make the above
square commute.

The converse also holds: if there is a map Ef such that the square commutes, then
(Ff)+'(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)+'(EX) -- FT]

and (Ff)+'(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
El c E2 4 VA E Obj(A), El(A) E2(A).

If E1 C E2, then, for each A, there is a unique extremal monomorphism cA such that
elA = (e2A)(cA). One easily verifies that cA are the components of a natural transformation
c: E1 -* E2; in fact, c: E1 -- E2 is a subfunctor.

The following constructions show that the class Subfun(F) behaves very much like
Sub(A).

Construction 4.1.6. Subfun(F) is complete. If {E, : i E I} is any class of subfunctors,
there is a supremum U Ei and infimum nEi. The supremum is given by (U Ej)(A) =
U Ei(A). The infimum is given by (f E)(A) = Ei(A).

Proof. Since A is extremally wellpowered, for each A, the class {E2(A) : i E I} has a
representative set. Thus, the object-by-object definitions for U Ei and f E, make sense. It
is obvious that if U Ei and n Ei are subfunctors that they are the optimum bounds in the
subfunctor lattice.
Let f : X -* Y be any map. A fni Ei(A) is a subfunctor because for each i E I,

(Ff)+'(nEi(A)) 9 (Ff)+'(Ei(A)) 9 Ei(B),

so (Ff)+I(ni E(A)) c ni E(B).
A H- Ui E1(A) is a subfunctor because,

(Ff)+'(UE(A)) = U(Ff)+'(E(A)) C UE,(B).
i i

Construction 4.1.7. Suppose a : F -- G is a natural transformation. Then there is an

a+ : Subfun(F) + Subfun(G) : a-1

The left adjoint a+1 is defined by (ar+E)(A) = (aA)+'(EA). The right adjoint a-1 is defined
by (a-'E)(A) = a-'(EA).

Proof. The order on Subfun(F) is defined "object-by-object", and whenever

f : A -- B,

f+1 and f-1 form an adjoint connection between Sub(A) and Sub(B). So it suffices to check
that a+IE and a-tE define subfunctors.

49

Let E E Subfun(F), f : X -+ Y be a map, and consider the following diagram.

(aA)+'(EA) (aB)+'(EB)

e'A
e'B

GA Gf GB

(iii) (i)

zA aA aB zB
(iv)

FA F -FB

eA (ii) eB

EA EB

The trapezoids (i) and (iii) are obtained by factoring (aB)(eB) and (aA)(eA), respectively;

in each case z 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+lE 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 f = 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-'E is a subfunctor

of F whenever E E Subfun(G). The missing map (aA)-'(EA) (aB)-'(EB) is obtained

from the pullback property of (aB)-1(EB). O

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+'E F be the subfunctor described in Construction 4.1.7. There is a
natural transformation z : a+E -- F such that for each object A, (aA) = (e'A)(zA) is the

(epi, extremal mono)-factorization of aA.

Proof. One defines e' and z as in the preceding proof. Examining the comparison of squares
diagram used to produce a+lEf, shows that z and e' are natural. O

Construction 4.1.9. Assume E, F, G, H are functors A --+ A. Suppose that

1. a: E -* F and : G -* H are subfunctors,

2. Either E or F preserves extremal monomorphisms.

then the horizontal composition pa : 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/A) = (FPA)(aGA),

by definition of horizontal composition (see Construction 2.3.5). O

4.2 Meseguer's Lemmas

Lemma 4.2.1. Suppose A is an SF-category, (T, p, r7) is a monad on A, and F is a sub-
functor ofT, such that (77A)+*(A) C FA. There is a smallest subfunctor F ofT such that
F C F C T and for all A E Obj(A)

(A) (T2A) C -PA.

If the equation
(IA)(G2A) c GA

holds for G, we say G is closed under p.

Proof. Let 9 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 = fl7 exists.
Since S i-4 (zA)(S) is an order preserving map Sub(T2A) Sub(TA), so for each A

(ItA)(72A) C FA. O

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 l7 and p 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 4.2.1.) There are natural
transformations F : idA -- T and f : F -+ F which make (F, n, m) a monad.

Proof. One defines four sequences of natural transformations:

1. fx : F -+ T the subfunctor generated at stage A,

2. n\ : ida F\ a natural transformation obtained from q7 by suitably modifying the
domain and codomain,

3. mx : F,2 F,+, a natural transformation obtained from A by suitably modifying
domain and codomain, and

4. c\ : F\ F,+I the inclusion.

Define Fo = F; use the notation fo : Fo -- T. mo and co are defined according to the same
pattern that defines later "m"s and "c"s, which is described below.

Defining f : FA T (A > 0), mA : FA2 FA+1, and cA : FA FA+I. Assume
that fA has been previously defined. Consider the following diagram:

F,2A A TA2A

(pA)+'(F\) U FAA

Given fA and pA, the square is obtained by (epi, extremal mono)-factorization of the com-
posite (pA)(f.A). All maps in the triangle are extremal monomorphisms, obtained by
comparing subobjects of TA. Define

FA+A = (jpA)+'(F\A) U FAA

mA : F)A -- FA+lA

to be the map shown on the left side of the diagram. It follows from Constructions 4.1.7 and
4.1.6 that fA+1 : FA+I T is a subfunctor. Note that

(fA+i)(mA) = (MA)(f.A)

and mA is natural. For bookkeeping purposes, let us call

CA : FA FA+I

the subfunctor which exhibits FA C FA+1.
If K is a limit ordinal, define

F = U{Fx : A < K};

f, : F, T is a subfunctor by 4.1.6.
Defining the sequence of "nx"s: The following definition of nA is not recursive; n\

can be defined once we know f\, but the definition of fA does not involve nx at all. Consider

the following commutative diagram of functors and natural transformations.

id^a > (77idA)+1 e T

The natural transformations z and e are defined by condition that r7 = ez is the (epi, extremal

mono)-factorization of rf, as described in 4.1.8. By assumption and the construction of the

sequence (FA),

(rqidA)+1 C Fo C FA.

Let ix denote the natural transformation such that e = fix; ix exists because e C i. Define

n\ := i\z. Evidently,
77 = f\nx

and

nA+ = Cn\.

What happens when the sequence terminates: Since A is extremally wellpow-

ered, for each A, the sequence (FAA)x of subobjects of TA eventually terminates, say when

A = Kn. Again using extremal wellpoweredness there is an ordinal, say K2, such that the

sequence (FAF,(A))A of subobjects of TF,,A terminates at K2. Define n = n,,, f = f,

and m = m,2. These assignments give natural transformations; to check this, one considers

a map f : 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 (FAA)x
terminates at FA.

Verifying that (F, n, m) is a monad. To prove the unit laws, it suffices that for
all A,
(mxA)(FxnA) = (mAA)(nxF\A) = c\A

(because, once the sequence terminates, the "cxA"s become identity maps on F\A). Fix any
ordinal A. Naturality of fx and nx implies

(Tn7 A)(fAA) = (AFAFA)(FxnA)

and
(nxTA)(fxA) = (FAAfA)(nxFAA);

Note that r7 = fAnX and, by definition of horizontal composition,

fA = (TfXA)(fXAFA) = (fATA)(FxfA).

Calculating, one finds
(fA)(FnxA) = (fTA)(FfxA)(FxnxA) def. noted above
= (fATA)(FxrlA) noted above
= (TrA)(fxA). fx is natural
Thus,
(f2A)(FnxA) = (T77A)(fxA)

and (by a similar computation)

(f,\A)(n FAA) = (rlTA)(fAA).

To show (m\A)(F\xnA) = c\A, consider the following diagram:
FxA \Fx(nxA) FA

fxA -f FfA

TA T- A T2A ---- FfA

TA <- Fx+lA
A

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, rl, p). Reading
the perimeter of the diagram, one finds

fxA = (fA+iA)(mAA)(FxnAA).

By uniqueness of maps manifesting inequalities between subobjects,

cAA = (mAA)(FAnAA).

The diagram needed to prove
(mxA)(nAFAA) = cAA

is similar and omitted. This establishes the unit laws.
To prove the associative law for (TF, i,), one may choose sufficiently large A, then
draw a diagram comparing the associative squares for T and F, using appropriate powers of

f\ to compare the corners. The comparison squares commute because of the definitions of
m, and horizontal composition. O

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 -- U Obj(D). Since U 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 occurring 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

extremal subobject, which by our notation is U Ai. 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' = U A. This depends on the delicate issue of whether
epimorphisms are surjective. If epis are not necessarily subjective, then U 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, (T, 77, y) is a monad, and F is
a subfunctor of T which contains r+'1idA. A partial algebra (A, a) is an object A equipped
with a map a : FA -- A such that a(noA) = idA.
Let pAlg(Fo, no) denote the category of all partial algebras, with maps f : (A, a)

(B, b) such that f : A -+ B is an A-map, and fa = b(Fof).

Lemma 4.2.6. Continue with situation and hypotheses from 4..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 i : FA -* A if and only if for each ordinal
A there is a unique a\+l making the diagram below commute.

F (A) a F (A)

FA(A) a+ A

Proof. To prove a map : FA -- A exists, use transfinite induction. The hypothesis gives a\
at successor ordinals. To construct a\, when A is a limit ordinal use the coproduct property
of FxA = Uc,< Fc(A).
Inductively one shows that

(ext) for each A, a\ = ax+1(cAA) (each a\x+ extends the preceding aA),-

(unit) for each A, ax(nxA) = idA,

If (unit) holds for successor ordinals, then it holds for all ordinals. Suppose

K is a limit ordinal. The map a, is epi, because it is a limit of epimorphisms. Note that a,

and a,,na,, are both compatible maps FA .- A; the definition of colimits implies that

a, = anKaK,;

because a, is epi, idA = anr.

The algebra associative law holds for (A, ). If a = ax, then a\ = a,+1, so by

the hypothesis regarding the existence of a, such that a\+ (mxA) = a (FAaA) proves the

associative law.

If (unit) holds for A, then (ext) holds for A. Note that the equation ax+l(m\A) =

ax(F\ax) defines aA+l. By induction hypothesis (unit),

(FVax)(F nAA) = idFAA;

therefore precomposing both sides of the defining equation for a\+l with (F\n\A) shows

a\ = aA+l(mAA)(FAnAA) = ax+l(cAA).

The second equality follows from (mA)(F\xnA) = (cAA), 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.

F),A ,A A

FxnxA

FA mA F,+1A

-A+

FA aA
The left triangle commutes because (unit) holds for A; the bottom trapezoid commutes by

definition of a,+l; the top trapezoid commutes because

(nxA)(FxnxA)(mxA) = (nA)(cxA) = n,+iA.

The first equality holds by proof of the unit law for F; the second equality holds because of

the compatibility between "nx"s. Thus,

ax+l(nA+1) = a+1 (mAA)(FAnAA)(nAA)

= aA(Fxa) (F xnA)(nA)

= a,(nxA)

= idA;

this establishes the unit law for (A, a). O

Use the notation Alg(Fo, 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 m\ is epi. The map

(A, a) i-* (A, U) induces an equivalence of categories; the functors involved are

E Alg(Fo, no) A',

defined by extension of structure and

R A" -- Alg(Fo, no)

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, no) map

0: A --+ B,
FAA FB

A _'B

commutes for each ordinal A. By definition of Alg(Fo, no), the square commutes for A = 0.

Suppose it commutes for A; note that

a m+l1(mA) = 0ax(F ax) def. ax+l
= bx(FA,)(Fxa) ind. hyp.
= b,(F\b) (F\q) ind. hyp.
= bx+I(mAB)(Fx.) def. bA+l
= bx+l(FA+i1)(m A) naturality mx.

Because m\ is epi, one concludes fax+1 = b\+l(F\+lb). At limit ordinals, properties of

colimits insure that the diagram commutes. By transfinite induction, and the definition of
the extended structure, any map which preserves (Fo, no)-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, ao) Obj(Alg(Fo,no)), RE(A, ao) = (A,ao).

The construction of the extended structure shows the restriction of any F-structure to Fo
uniquely determines the T-structure; hence

ER(A,a) = (A,a).

D

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 a)+l such that

ax+lmA = aA(FAaA) 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, ,, p) has the following parts:

1. Given a set X, TX consists of all words with variables in X. A magma word is any
expression formed by finitely many applications of the rules:

(i) If x E 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 f : X Y, to compute Tf we apply f to all members of
X in a given word, leaving the tree and circle structure unchanged.

3. The "insertion of variables" map XX : X TX, (r7X)(x) = 0.

4. The "semantic composition" map ItX : T2X TX sends a word s E T2X of words
to a word in TX, by removing the circles around each element of TX used in making
s. p 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 E TN is the word

and t=ss (N) is the word
and t1 = ss E T2(N) is the word

/ S

then (/AN)(ti) 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 (/IN)(tl) are in F(N), but

is not, because dep(57) = 1 and dep(1) = dep(4) = 2. Evidently, F is a subfunctor, and for
any X, (77X)+'(X) C F(X). But, for any non-empty X, (pX)(F2(X)) is not contained in
F(X). One readily verifies that P = 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, 7 : idA -- F) is a map
a : FA -* A satisfying idA = a(rA). Define n : F(N) -- N on 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(pN)(tl) = 4.

In order for n : F(N) -- N to extend to n : Fi(N) -+ N there must be a function ni
making
F2 (N) F F(N)
F ( ------ ---- F(N))

1N n

F(N) n" N
commute. No such al can exist, for (MN)(tl) is

which equals (p/N)(t2), where t2 is

Finally, note that (Fn)(ti) is

so n(Fn)(ti) = 1. Note that (Fn)(t2) = (pN)(t2), so n(Fn)(t2) = 4. Thus we have elements

tl, t E F2N, that pN identifies and n(Fn) does not identify, so there is no function nl such
that ni(/pN) = n(Fn).

CHAPTER 5
FREE ALGEBRAS

This chapter explores a generalization of complete distributivity for IP-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; lD is the monad for complete

join semilattices, U is the monad for complete meet semilattices.

The gist of section 5.2 is that there is a lifting of U over D (and a lifting of D over U).

Therefore, UTD and D)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 DUl 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 U1). Meseguer's

Lemmas are applied to this subfunctor to produce a monad F. Any P -objects which

is P,-embeddable in a completely distributive complete lattice is an F-algebra. Any F-

algebra has a natural P,, 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": S1 E T(A), S2 E TP2A S2 is a family of sets, S3 E P3A 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, 1) (resp. (U, i,/ )). The description of D is well known; for
example, Meseguer [25] uses it. The reader will have noticed that p is used as a name for

two different natural transformations; this would normally be horrible notation, but in this
case, the formula for Cp is the same. Thus, our notational economy should cause no confusion.
The functors U and 0D act on a poset A by

U(A) = SCA:x> yES = E S}

- the set of increasing subsets of A ordered by reverse inclusion- and

D(A) = {S CA: x y E S = xE S}

-the set of decreasing subsets of A ordered by inclusion.
Given monotone f : A -* B, we define

JDf : D(A) D-* (B) : S {b E B : 3s E S, b < f(s)}

and

Uf : U(A) U(B) : S {b E B: 3s E S, b > f(s)}

The following facts will be of later use.

1. U and D are functors: trivially they respect identity arrows. Observe that for any
S E A and montone functions f: A --+ B, g : B -- C,

D(gf)(S) = {c : sS, c = {c C: 32f(s) E Df(S), c gf(s)}
= (Dg)(Df)

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. U(A) is a complete lattice, with
infimum given by union and supremum given by intersection.

3. For each S2 C ID(A),

D (Us, ) = U {(ID (S)): S S2}.

In particular, this holds if 6 is empty. So Df preserves all supreme. Similarly, if
S2 C U(A), then
Uf(Us)= U {(f)(s): S G}

so that Uf preserves all infima.

Define

dA: A -- D(A) : a H z- x A : x
iA : A U 1(A) : a t{x E A : x > a}, and

pA: D2(A) 'D(A) : S2 E 2(A) -+ US2.

It may be puzzling that VD(A) is ordered by subset inclusion and U(A) is ordered by
reverse subset inclusion. The fact that

x < y < dA(x) C dA(y) =4- iA(y) C iA(x)

motivates the choice, because one wants iA to be order preserving. Moreover, one wants Uf
to preserve all infima, which only happens if the infimum operation in UA is union.
The reader may check that i, d and yu are natural transformations. The monad
assocative law
93 -D2

VD2 D--
holds for both D and U, because if S3 E P3(A), then

U US3 =U { US2: S2 e S .

D D ;- 2 d D2

hold for D, because if S E D(A), then

S=U {(dA)(x) : x e S} = U {T e D(A) : T C S.

Similar computation shows the unit laws hold for U.
Now to describe algebras over these monads:

Lemma 5.1.1. Let T C D(A), such that for all x E A, (dA)(x) E T. Then the following
are equivalent:

1. there is an order preserving map a : T -+ A such that idA = a(dA);

2. each S E T has a supremum;

3. dA: A -* DA has a right adjoint.

The analogous conditions involving T C UA are also equivalent.

Proof. We give the proof for T C DA, leaving the upside-down argument for T C UA to the
(2 = 1) Define a(S) = V S and compute.
(1 == 2) a(S) is an upper bound for S, because if x E S, then (dA)(x) C S,
whence x = a(dA)(x) < a(S). Suppose for all x E S, x < u. Then (dA)(x) C (dA)(u), so
S = U{(dA)(x) : x S} C (dA)(u), therefore a(S) < u.
(2 4=4 3) By definition of V s, x C S ==* x < V s. o

Proposition 5.1.2. P' is equivalent to the category of D-algebras. P, is equivalent to the
category of U-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 1 S. By the definition D-algebra homomor-
phisms coincide with order preserving functions which preserve all supreme of decreasing
sets. It is easy to see that a map preserves supreme of all decreasing sets if and only if it
preserves supreme of all sets.
Conversely, if A E PB it has structure map

:D(A) A: S-VS.

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. O

Lemma 5.1.3. For any S2 E D2(A),

V {x E A: 3SE S2, x < Vsi} = VUS2-
Proof. To verify this equality, one notes that for all x E U S2, 3S1 E S2 such that x E S1,
so x < V S1. Thus the left hand side dominates the right. If u is an upper bound for U S2,
and 3S1 E S2, x < V S1, then x < u. So the left hand side is the least upper bound of U 6;
this establishes the lemma. O

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 U over D (and similar arguments
show that there is a lifting of D over U). Explicitly this means:

1. For any U-algebra A, there is a U-structure on D~(A).

2. U 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 : U2 -* U preserve all joins; similarly, d and

Ip: D2 --+ D preserve all meets.

Lemma 5.2.1. For any poset A, both U(A) and D(A) are complete lattices. Thus, U(A)
and D(A) are both U- and D- algebras.

Lemma 5.2.2. For any poset A,

1. dA : A -> 2(A) prserves all existing infima.

2. iA: A -* U(A) prserves all existing supreme.

Proof. Suppose S C A and u = A S exists. Since dA is order preserving ,

(dA)(u) C fIdA(x) : x S}.

For the reverse inequality, suppose t E ({dA(x) : x E S}, i.e., f is a lower bound of S. Then
e < u, so t E dA(u). The proof for uA is upside-down, but otherwise identical. O

Lemma 5.2.3. If f : A -> B preserves all supreme, then Uf : U(A) -* U(B) also pre-
serves all supreme. Similarly, if f preserves infima, so does D(f). Therefore, for any order
preserving map f, both UIDf and )Uf preserve all infima and supreme.

Proof. A proof of the first fact is given, the second is very similar, but notationally easier. Use
the fact that U(A) is a complete sublattice of T(A), and therefore completely distributive.

69

Let (SA)~EL be an indexed subset of U(A); for each S\, choose a family (x,\ : a E K) such
that
SX =T (X,K :~ E K).

Recall that:

joins in UA are intersections;

T V yA, = (iA)(V y\) = n T Y,
A

for any indexed family (yA) C A.

One computes as follows:

vf(ns\) = uf(nux Ax)
A A K
= Uf (U{n t-,c(A): c:L- })
= U f({ x,,c(,): c: L K})
= U{t (V X.)): c:-L K

= U{ V f(x,,(A)) : c: L- K
= U{nTf(x,,c): c:L- K
A
= Nff(s.\)

Corollary 5.2.4. For any poset A,

1. p : D2(A) D(A) preserves all infima.

2. t : U2(A) -+ U(A) preserves all supreme.

Proof. Suppose S2,, E V2(A), for A E L : claim 1 amounts to

Un S2, = nU s2, .
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 S1,\ E S2,x with x E S1,A. Then

S1 = n S1,\
A
is in each S2,A because each S2,A is downward closed relative to the inclusion order in V2(A).
Since x E Si, this proves that x is in the right hand quantity, and the desired equality of
sets holds.
The proof for the U works similarly. O

The preceding lemmas show that there are lifting of U over D (and vice versa), so
we have:

Corollary 5.2.5. UD = (U, r7, v), where

r7 :idp -+ UD : a -* {S E IA :J a C S}

and
v : UDUD -+ UD : S4 i--UlS4,

is a monad over P 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, 77 is the horizontal composition i d. To
compute v : UDUVD -, UD one uses the fact that vA is the structure map for the free algebra
UDA. Consider the situation in light of the discussion following Theorem 3.4.5. (Here U
plays the role of S, while D plays the role of T.) The relation a = aT(Tas) applied to the
object UDA implies

(A)(S4) = U(VFn)(s4)

= U{TeDA:3S2e S,Tc SN ,

= Uns.

The assertion about algebras follows from Theorem 3.4.5. 0

Remark 5.2.6. UD is a monad because there is a lifting of U over P'. One could use

the correspondence between lifting 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 f : A B factors as A --

f(A) -- 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 E A,

{a} E 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] Pj and Pm are monadic.

Proof. Given a subset system j

Jo(A)= {S: SE j(A)}

defines a subfunctor of 2D. This extends to a subfunctor J of D such that (j3, d, ) is a monad.

The natural transformations d and ji are obtained from d and 1p by suitably modifying the

domains and codomains. The precise definition of these maps is contained in the proof of

Lemma 4.2.6.

If (A, ao) has a Jo-structure, Lemma 5.1.1 shows that

ao(S) = Vs

for all S E Jo(A). Thus, to define al one must show some map J1A -- A makes the following
diagram commute.
Jo(V)
J2A o(V) JoA

U V

J1A >A
Lemma 5.1.3 shows that V S is defined for all S E J1A and, moreover, that for S2 E J2A,

vuS2 = V (JOV) (2)-

Identical arguments show show that for each A, if

a,(S) Vs

that
a+l1(S)= VS.

It follows that any poset, A, in which each member of JoA has a supremum, each member of
J(A) has a supremum. Moreover, each map preserving Jo-suprema also preserves J-suprema.
Thus, one obtains an equivalence of categories between Pi and P-.
An upside-down version of this argument shows that Pm is a category of algebras. O

Given subset systems m and j, F := MoJo forms a subfunctor of

T := U'D.

By Corollary 5.2.5, (T, v, r7) 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.

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. D], denotes
the full subcategory of P3 containing objects such that there is a P3 map : A -- B, with
4 P-extremal mono, and B E Obj(DP").
Ml denotes the category of F-algebras, where F is the monad described above.
It will be shown that
SpP" C DFP'_ C M3_.

To see the first inclusion, one notes that any family T of subsets of a set X which is also a
P3 object is in DP,; the inclusion 7 C P(X) is a P--embedding and T(X) is a completely
distributive complete lattice. Proposition 5.3.7 shows the second inclusion.

Let A E DP(, and 0 : A -- B be given as in the definition of DPW ; 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 UTD-algebra associativity diagram for B. From this
point on, the notation of Lemma 4.2.2 is adopted with slight modification; v : T2 -* T is
the monad multiplication; vA : F. -- F\+1 is defined inductively (and plays the role that
was played by m\ in the proof of 4.2.2); 7 : idp -- T is the unit; 77 : idp -- F are defined
inductively, and play the role that was played by n\ in 4.2.2. The natural transformations

f\ : F\ -- T and c\ : FA -- F+I1 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(S2)= A{ V : S, e S.

Since 4 is a P -map, the following diagram commutes.

FA FB TB

Ao bB

A -B B

I

The left square expresses that 4 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 ao(S2) = A(U V)(S2); this map satisfies aor/o = idA.

Proof. Let us begin by more explicitly calculating ao. Using the definition of U on maps we
find that

ao(S2) = A(uV)(s2)

= A[xeA:BSiES2,Vs, = AI {VS s, 1 V ,2}

This definition makes sense; each S E Jo(A) so each V S1 exists because A E P",. (u V) is
an order preserving map, and S2 E mojo(A), which implies that

(U V)(S2) E Mo(A).

Thus, the meet defining ao(S2) exist.
Recall 7(x) = {S E DA :J x C S}. Now one calculates

ao(,7(x)) = A{VS:SE DA, x S}
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 A, v\A is surjective. Therefore vxA is P-epi.

Proof. Because of the structure of P, unions of subposets actually are set theoretic unions.
Let S E Fx+l = FA U (vuA)+1(F2A). If S E (vXA)+1(F2A), then there is an S2 E FIA such
that (vxA)(S2) = S. Since cAA = (vAA)(rxFAA) so for any S1 E FAA, T2 := (rjAFAA)(S) is a
member of F2A such that (vAA)(T2) = S1.
This lemma holds in any category where unions of subobjects actually are set-theoretic
unions. O

After noting that extremal monomorphisms in P are injections f : A -- B, where

f(A) has the order induced as a subset of B, one easily verifies:

Lemma 5.3.6. The functors U, ~D, 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 E Obj(DPJ), B E Obj(DP.), 4 : A -* B as in the definition of
DPn. Then

(ex) for each A, there is a unique map

ax+l : F,+iA A,

such that

ax(Fax) = ax+lvA;

(cor) for each A, fa\ = b(fB)(F,\).

Thus, DP3 C Ml

Proof. Remark 5.3.3 established (corn) for A = 0. If (corn) holds for successor ordinals,

properties of unions insure that (corn) holds at all ordinals.

Suppose (com) holds at stage A; we will establish (com) and (ex) hold for
stage A + 1. Consider the following diagram.

T2B Tb TB

F2B (Fxb)(FxfxB) F
F IB FxB

F:A -- F, FAA

vB vYB v\A IA b(fxB) b

Fx+lA A

Fx+lB B

TB -B
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 vA; 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)(f TB) = (fAB)(Fxb). By definition of horizontal composition

f,\B = (f TB)(FA AB).

Thus,

(Tb)(f2B) = (Tb)(fxTB)(FjfB)

= (fB)(Fb)(FAf.B)

This proves that the top outer square commutes.
Consider the (epi,extremal mono)-factorization of

b(fA+lB)(F(+l1);

say it factors as

FxA+A -a A' C B.

Since v\A is epi, the (epi, extremal mono)-factorization of

q := b(fA+lB)(FA+j1)(vuA)

is

F:A "0) A' C B.

The proof of Lemma 4.2.6 shows that each existing a\ is split epi, which implies both a\ and

(Fa\) are epi. Since
q = Oa,(F\a\),

the (epi, extremal mono)- factorization of q is also:

FAA aF) ACB.

Uniqueness of factorization implies there is an isomorphism i : A -- A' compatible with the
factorizations. So a\A+ = ia is a map making (com) hold for A + 1.
Reading off the diagram one finds that:

OaA(Fa,) = b(Tb)(fB)(FA0)

= b(vB)(f\B)(Fx)

= b(fA+IB)(FA+1 )(VuA)

= Oax+i(v\A)

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+l is the only map which makes the inner square commute because v\A is epi, by
Lemma 5.3.5. 0

Corollary 5.3.8. DP' is a full subcategory of M,.

Proof. By Lemma 5.3.3, any IPj-map preserves the F-structure of a DPI object. Lemma

4.2.7 implies that any P -map extends to a Ml map. O

Corollary 5.3.9. The forgetful functor DP -* P has a left adjoint.

Proof. First note that FA E Obj(DPL) 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

f : A B is an order preserving map from A E Obj(P) to B E Obj(DPlv) C Obj(PF), then
there is a DP -unique map f* : FA -- B. This proves that 7 has the universal property

described in 2.5.2.3; thus, F is the left adjoint to the forgetful functor DP3 -- P. O

Corollary 5.3.10. The forgetful functors DP' -- DP' and DP. -- M3 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.) O

There is some, rather limited, information about members of Mj.

Proposition 5.3.11. If A Obj(ML), then A e Obj(Pm).

Proof. Define JA, MA, a, 3, -y, and J, by the requirement that the squares below are

pullbacks.

JA *--f FA
JA JA

DA D UDA

MA ----FA

UA dA
I A U" IDA

By assumption, j(A) and m(A) contain singletons, therefore

(i)A)+'(JoA) C MoJoA C FA,

and
(UdA)+'(MoA) C MoJoA C FA.

It follows that
JoA C JA = (iDA)-FA

and
MoA C MA = (UdA)-FA

Again using the fact that j and m contain singletons,

(dA)+(A) C JA

and

(iA)+1(A) C MA,

so corestrictions dAj : A -- JA and iA| : A -- MA satisfying a(dA)j = dA and y(iA) = iA
exist.
If a : FA -+ A is an F-algebra, then a/3(dAI) = idA and a6(iA|) = idA. 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|) = idA if, and only if, y(S) = VS for each S E 7
(and analogously for subsets of UA). Thus, we conclude a/3 = V and a6 = A. O

Remark 5.3.12. If j = m = w the subset system which selects all finite subsets, then
SpP = 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 Ml 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 A(U V)S2, but the author cannot presently substantiate such
hopes.

Remark 5.3.14. The author knows no examples of objects in Obj(Ml) \ Obj(D1 ,). In
light of Corollary 5.3.10, any DPW object is freely embedded in a completely distributive
lattice. In fact, A E Obj(Mj) is in Obj(DP3) 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 f : 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 = 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 E Obj(MW)
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 [S] 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 IP-meets. Li defines IS(P) to
be the family of all decreasing sets in TP, ordered by S T if, and only if, there is an
indexed chain (Si)iE with I = Q n [0,1] such that So = S, S1 = T and whenever i < j,
S. n [P \ Sj] = 0. The map u : P -- ISP is defined by u(x) = {U e UA : U -< x}.

Nonetheless, DP3, it is a fairly nice category. In fact, it is complete.

Proposition 5.3.15. DPJ, is complete.

Proof. Note that a poset A is a P, object if and only if there is an order preserving map
a: JoA x MoA -- A such that a(doA x ioA) = idA.
Suppose (Ai)iE is a family of P, objects and for each i E I,

0i: Ai Bi

is an embedding of A, into a completely distributive lattice Bi. The proof of Lemma 3.1.3
implies that the poset product 1I A, is a P3-object. Moreover, the product-induced map

0 : RI,, Ai Ri,, Bi is an embedding which preserves (j, m)-optimum bounds. (Essentially
what is going on is that meets and joins are computed coordinate-by-coordinate.)

If f, g : A -- B are P, maps, then the proof of Lemma 3.1.3 implies eq(f, g) is
a (j, m)-complete subset of A. If A E Obj(DP~), then A can be (j, m)-embedded into a
completely distributive lattice, so eq(f, g) may also be so embedded.
Thus, any set of objects in DP, has a product, and any pair of DI~ maps has an
equalizer. By Borceux [6, Volume 1, 2.8.1], this implies DP, is complete. O

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 f is onto then f is epi. For the
converse, suppose b E B \ f(A). Define

Si =1 (f(A)n 1 b)

and
S2 =1 b.

Evidently, both S1 and S2 have the same intersection with f(A), but b E S2 \ S1. Therefore
characteristic functions of B \ Si and B \ S2 are distinct maps (say

c, c2 : B- 2

are respectively the characteristic functions of B \ S1 and B \ S2) such that clf = C2f. Both

c, are order preserving, because S1 and S2 are decreasing sets. O

Definition and Remarks 6.1.2. The following construction is paraphrased, following Mese-
guer [25] pp 73-74.
I
If f : A B is any order preserving map, we may define a preorder on A by
f
a a2 4 f(a,) f(a2)

This relation is obviously reflexive and transitive, but -because f may not be injective -

may not be anti-symmetric.

Suppose (A, <) is a poset and is a preorder strengthening <, i.e.,

al <_ 2 ==* al < a2

Then we have an order preserving map a : A -* A/ :. (Recall that A/ is the set of

equivalence classes

{x E A:x < a and a -< }

partially ordered by <.)

Define maps fi, f2 : A -- B to be equivalent, if there is an isomorphism, i : B1 B2

such that

f2
A fiB1

B2

commutes.

One verifies that:

I
1. The maps (f : A -- B) H- and (-) a : A -- A/ are mutually inverse

correspondences between

the set (modulo equivalence) of surjective maps f with domain A.

the set of preorders on A which strengthen <.

I
2. For any f : A -- B, f = ca, where a : A -- A/ f and c(a(a)) = f(a).

3. Given surjections fi, f2 : A Bi, there is a c: B1 -- B2 if and only if

fi f2
al -< a2 == al a2.

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 S generates <, we can describe

_ explicitly: ao an if

a0 = an, or

there is a finite sequence al, a2, an-l, such that for all i with 0 < i < n 1,

(a, ai+i) E 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 E A}.

Lemma 6.1.5. Let f : A -- B be a subjective order preserving map. The following are

equivalent.

1. f is the coequalizer of some pair g, h : Ao -- A.
f
2. There exist a poset Ao, and maps g, h : Ao -- A such that the preorder on (A, <) is

generated by 5 and C(g, h).

I
3. The preorder : on (A, <) is generated by g and K(f) = {(x, y) E B x B : f(x) = f(y)}
I
4. The preorder 3 on (A, <) is generated by < and some equivalence relation.

Proof. (1 e= 2) is evident from 6.1.4. (3 == 4) is trivial, because K(f) is an equivalence

relation. We have
C(gh)IK(f)

because fg = fh and

(x, y) E K(f) = f f(x) 5 f(y) and f(y) < f(x).
fI
Since C(g, h) and I
For (4 ==* 1), let be generated by <: and the equivalence relation E C A x A

with the trivial order. Then the projection maps r1, r2 : E -- A, given by

7Tr(xai, X2) = i

are order preserving. The coequalizer of 71r and ir2 is f. C

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 f : 2flat -* 2 from the (trivially
ordered) two point set onto a two-point chain. Any such map f 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 P,,.

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 E A x A: 7r,(S) e 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 P3-subobject of A x A.

Proof. We show that a(Vxi) = Va(x() for any {x( : i E I} E j(A). Clearly a(Vxi) is an
upper bound of {a(xi) : i E I}. Since is a P.-subobject of A x A, {xi : i E I} E j(A) and
for all i, xi < a,

V{(xi, a) : i I} = (VW, a)

is a member of <; Vxi < a. Thus, any upper bound of {a(x,) : i E I} dominates a(Vxz).
(The proof for meets is identical, so omitted.)
Conversely, if a preserves j-joins and m-meets, and (for example)

{x : i E I}, {yi: i E I} e j(A)

and for all i E I, xi yi, then Vizi Viyi. So is a Pi-subobject of A x A. O

Definition and Remarks 6.2.3. Suppose j and m admit congruences. Let (A, <) E

Obj(P1). If is a preorder strengthening < and simultaneously a PL-subobject of A x A,
then we say 1 is a (j, m)-pocongruence.
Pocongruences of Pm-objects behave very much like preorders of posets.
f
1. The maps (f : A -- B) -* and ( a) a : A -+ A/ are mutually inverse
correspondences between

the set (modulo equivalence) of surjective maps f, which preserve (j, m)-optimum
bounds and have domain A.

the set of (j, m)-pocongruences on A.
f
2. For any f : A -> B, f = ca, where a : A -* A/ and c(a(a)) = f(a). If f preserves

(j, m)-optimum bounds, so do c and a.

3. Given (j, m)-optimum bound preserving surjections f, f2 : A -- Bi, there is c: B1
B2 if and only if
fi f2
al a2 == al a2.

Consideration of Lemma 6.2.2 shows that if f : A -+ B is surjective and S E j(A)
(resp. S E m(A)) then f(S) has a supremum (resp. infimum) in B. Moreover, sets of the
form f(S) for S E j(A) (resp. S E 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.

Definition and Remarks 6.2.4. Let Z be a subset system. We say Z preserves surjections
if whenever f : A -- B is a poset surjection,

Z(B) = {f(S): S E Z(A)}.

Similarly, we say Z preserves regular epis if whenever f : A -- B is a poset regular
epimorphism,
Z(B) = {f(S) : S E 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 f is split epi (in Set). Hence, there is s : B -> A (not necessarily order
preserving) such that idB = 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 tl

I ~I..

Evidently N is j-join complete.

Consider the function f : N -- N, where N = {0, 1,2, } with the usual order,
defined by f(b1) = i (for i > 0) and f(ti-1) = i (for i > 1). Speaking roughly, "f stacks
I
the two point chains." f is regular epi as a map of posets. Moreover, is a j-subobject of
N x N, because f 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 PJ. The only thing
stopping map a : B -- B/ (described below) from actually being a coequalizer in Pi is
that B/ : is generally not (j, m)-complete.

Construction 6.2.6. Suppose f, g : A (B, <) are P, maps, and C(f, g) is defined as in
6.1.4. Let -< be the smallest pocongruence containing < and C(f,g). Then a : B -* B/ -
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/ 1 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 PI,. In particular, if j = K and m = A are cardinality subset systems, then a is the
coequalizer.

Corollary 6.2.8. DP2 has coequalizers.

Proof. The preceding shows that any pair f, g : A -- B E P has a coequalizer. Suppose
B is completely distributive. Since a : B coeqpo (f, g) is a surjection which preserves all
meets and joins, coeqp (f, g) is completely distributive. In particular if both A and B are
completely distributive, then coeqp. (f, g) is also the coequalizer in DP.. O

6.3 Factorization of Meetsemilattice maps

Several simplifications occur describing coequalizers in P3 if the objects have a meet-
semilattice structure, i.e., m > w.

Lemma 6.3.1. Let f : A -- B be a meetsemilattice map. f is P-regular epi if and only if f

is subjective.

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

I
x y =* f(xA y) = f(x).

Thus we have the sequence x, x A y, y, where f(x) = f(x A y) and x A y < y. By Lemma

6.1.5, this proves f is regular epi. 0

I
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 > w and j and m preserve regular epis. Let f : A -* B be a

P, map. The following are equivalent.

1. f is subjective,

2. f is a quotient by some congruence K,

3. f is P regular epi,

4. f PJ-regular epi,

5. f is PL-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 f : A -- A/K. Since K C A x A, we have projection
maps i7r, 7r2 : K -, A. f is plainly the coequalizer of 7rl, 7r2.

(5 == 2) holds because we may factor any map f through its associated pocogruence.

If f is extremal epi, then f must coincide with the quotient by its associated pocongruence.
O

Corollary 6.3.3. Let j and m be subset systems which preserve regular epis. Let m > w.

The forgetful functors Up : P -* P and Uset : P -* Set preserve and reflect regular epis.

Lemma 6.3.4. The forgetful functor UDI, : DP -- Set reflects kernel pairs.

Proof. One verifies that the kernel pair of a map f : A -* B (calculated in DPF,) consists of

the projections (from A x A) restricted to the set

{(x,y) E 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 > w.

The forgetful functor Uset :DIPJ -- Set is monadic.

6.4 Coequalizers in DPI

Theorem 6.4.1. For any pair g, h : A -- B E DP~, the coequalizer coeq(g, h) exists.

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.

;/

A -B B
h

Eo
7 DM( ) E (7
C

7C D(C) E(C)

9(Bl~) E(BI ~~
D(Bl ~~

--/

Let a : B -- B/ be the map given by Construction 6.2.6. Let f : B -- C be any
DPI, map such that fg = fh. The existence and uniqueness of f is given by 6.2.6.
To construct the leftmost square, we apply the free completely distributive complete
lattice functor D = tUD; 77 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/ -)

such that k(rl(B/ ))a preserves all (j, m)-optimum bounds. Similarly, define

i: D(C) -- E(C)

to be the smallest quotient such that (rCC)f preserves all (j, m)-bounds. Since

WD(f)(7(B/ _))a = t(rC)fa = e(77C)f,

preserves all (j, m)-bounds we have the induced map E(f).
Define Eo to be the smallest DPU-subobject of E(B/ -) through which k(r7(B/ -))
factors. (By Proposition 5.3.15, the intersection defining Eo exists.) The maps j : B/ -*
Eo and Eo -- E(B/ -) are obtained by factoring (7l(B/ -))a) through Eo.
By construction, the map e(rqC) : C -- E(C) has the universal property that any map
(j, m)-optimum bound preserving q : C B, with B E Obj(DP-) factors as = te('C) for
a uniquely determined map t. Since C E Obj(DPm), there is such a 0 which is P-extremal
mono. Thus, e(?C) is P-extremal mono. Therefore, the P-(epi, extremal mono) factorization
of t(77C) produces the factorization (rlC) = 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 DP'-subobject of E(C) through which te(7C) factors.
We claim that coeq(g, h) = Eo via j : B -- E0. It suffices to show the existence
of a unique compatible map : Eo -- C. The commutativity of the largest rectangle

92

in the diagram above implies that k(77(B/ d)) factors uniquely through the DP--object
E(f)-'(C) E(B). Because Eo is the smallest DP -object through which k(rl(B/ -))
factors, Eo C E(7)-'(C). This insures the existence and uniqueness of *. O

CHAPTER 7
(j,m)-SPACES

This chapter studies (j, m)-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 DP3 and S Section 7.2 develops a convenient description of epimorphisms

in 8~, 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 Sm; Section 7.5

gives the flat spectrum (co)reflection on spatial objects which is equivalent to a reflection

on S~,. This reflection on spaces is a generalization of the Tl-reflection of topological spaces.

Last, but not least, Section 7.6 partially describes the epicomplete S8- objects; the

description is complete for To-spaces. Epicomplete To-spaces are chains with the specializa-

tion order. Products of epicomplete To-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 S, untouched.

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) := {( : 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

(obl) an underlying partially ordered set, denoted A,

(ob2) a family of designated subsets 3A, such that V S exists for all
S E A,

(ob3) a family of designated subsets MA, such that A S exists for all
S E 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 q : A -* B is a function A B such that:

(mapO) 0 is monotone, i.e., a < b E A => ((a) < 4(b).

(mapl) 4[2A] := {(S) : S E 3A} C 3B, and VS E OA, O(V S) = V O(S);

(map2) O[MA] C MB, and VS E MA, O(A S) = A 4(S).

The category AS has objects (X, D(X), E(X), A(X)) where X is a set, D(X) is a family of
subsets of X, and E(X), A(X) are families of subsets of D(X), such that (0(X), E(X), A(X))
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
f-1 : (Y) -* (X) is a APos-map.

Definition and Remarks 7.1.2. We describe the functorial Galois connection between AS
and APos.