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SUBSET SYSTEMS AND GENERALIZED DISTRIBUTIVE LATTICES By ERIC R. ZENK A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY UNIVERSITY OF FLORIDA 2004 ACKNOWLEDGMENTS I would like to thank  * Jorge Martinez. Sometimes words are 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 "fRings and Ordered Algebraic Structures" conferences in Gaines ville and Nashville during my time in graduate school. Because of these conferences, I feel like I am joining a family of researchers, rather than just "getting a degree". It is an honor to know them. * Those who helped proofread this document: constructive comments were made by Jorge Martinez, Scott McCullough. and Pham Tiep. * Anyone taking the time to read these words: a dissertation, like any book, is meant to be read. TABLE OF CONTENTS 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 2.5 Adjoint Functors ............................... 23 3 ALGEBRAS OF A MONAD ............................ 28 3.1 Categories of Algebras ............................ 28 3.2 Adjoint Connections induce Monads ................ .... 32 3.3 Detecting Categories of Algebras ................ ...... 34 3.4 Distributive Laws ................... ........... 37 4 GENERATING SUBMONADS .......................... 42 4.1 Subfunctors . . . .. 42 4.2 Meseguer's Lemmas ................... .......... 50 4.3 A Partial Algebra Which Does Not Extend . .... 59 5 FREE ALGEBRAS ................... ............. 63 5.1 Complete semilattices ............................ 64 5.2 Completely Distributive Complete Lattices . ... 68 5.3 Some categories of algebras . ..... .. .... 71 6 COEQUALIZERS ................... ............. 82 6.1 Epis and Equalizers in P ................. ........ 82 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 Falgebra extends to an Falgebra. 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 Falgebras. 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 pointset topology. Assuming suitable restrictions on j and m, subspaces can be defined. Assuming these restrictions, there are wellbehaved 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). Zcomplete 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) selfbounded if there is s E S such that for all x E S, x < s; a selfbounded set contains a maximum element. The rule sb, which selects all selfbounded sets, is a subset system. Note that any order preserving map preserves joins of upward selfbounded sets. 6. A nonexample: An antichain is a set of pairwise incomparable elements. The rule ac which selects antichains is not a subset system, because there is an order preserving surjection f : D + 2, where D is the twopoint antichain 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 jsuprema and minfima 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 Pembedded 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 Falgebras. M In Chapter 5, Meseguer's Lemmas are brought to bear upon monads for free complete semilattices and free completely distributive complete lattices. Given any subset systems j and m, there is a submonad F of the free completely distributive complete lattice monad. The category Ml of Falgebras 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 jjoin preserving surjective image of a jcomplete poset need not be jcomplete. The results of Chapter 7 predate 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 junions and mintersections. 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 Tostyle 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 general topology is helpful. Also some comfort with category theory is assumed. Namely, the reader can fill in the blanks in the following informal definitions. A category A consists of a class of objects Obj(A) and maps Map(A), such that each object has an identity map, and there is an associative notion of composition of maps. The set of Amaps 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 Amap f : A1 + A2 a 3map 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 hornset 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 onetoone, 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 9maps. 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 torsionfree 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 i1 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 41 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 rightidentifies f and g if fi = gi. A map i : Ao  A1 is called the equalizer of f and g if: (eql) i rightidentifies f and g, and (eq2) i has the feature that whenever j : Bo A1 rightidentifies 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 leftidentifies f and g if ef = eg. A map e : A2  A3 is the coequalizer of f and g if (coeql) e leftidentifies f and g, and (coeq2) e has the feature that whenever d: A2 B3 leftidentifies 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 splitA. 2. f is regularA. 3. f is extremalA. 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 boundoperators 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 statement about Galois connections. 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) = g1~(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 wellknown 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 Amap 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 categorytheoretic product of a set X of objects (in any category) as the limit of the diagram containing all members of X and no maps. In most categories of "sets with structure" products (exist and) look like products in Set, with suitable structure added. In Set, the colimit of the diagram containing all members of X and no functions, is the disjoint union 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 Aobjects is "the Aobject 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 indepth 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. 2.5 Adjoint Functors There are several useful concepts of adjoint connections between categories. There is a strong analogy between preordered sets and categories; any category may be preordered by A B < 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 socalled 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) :BFA. 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 categories 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 socalled 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 Iindexed 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 ALGEBRAS OF A MONAD Monads (a.k.a. triples, a.k.a. standard constructions) and their (EilenbergMoore) algebras provide a concise formulation of many important categorical aspects of universal algebra. The category of algebras for a monad has special properties, which are summarized in Section 3.1; a category of algebras is often complete and cocomplete, and always has "free objects." Each (functorial) adjoint connection induces a monad; the correspondence between adjunctions and monads is discussed in Section 3.2. In Section 3.3, the question of when an adjoint connection connects A to a category of algebras is addressed. In Section 3.4, the question "when does the composite of two monadic adjunctions yield a monadic adjunction?" is discussed. The results in this chapter are reasonably well known. MacLane [21, Chapter VI], Borceux [6, Volume 2, Chapter 4], Barr and Wells [4, Chapters 3 and 9], Manes [23] and the introduction to the seminar notes [1] contain good expositions of monads from various perspectives. 3.1 Categories of Algebras Definition 3.1.1. A monad T = (T, 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 nfold 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 Talgebra (A, a) consists of A E Obj(A) and a : TA , A (the socalled 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 Talgebras is an Amap f such that TA  TB a b A A commutes. The category of all Talgebras and Talgebra 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 Talgebra 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 Talgebra map. The requirement that each LA is Talgebra 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 LFL 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. 3.2 Adjoint Connections induce Monads The following proposition gives a correspondence between monads and adjoint con nections. It should be emphasized that the correspondence is not bijective. Each monad gives rise to a unique adjoint connection, but in general many adjunctions induce the same monad. Proposition 3.2.1. Correspondence between monads and adjoint connections 1. Let F : A * 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 over the induced monads. 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 Talgebra; the group multiplication and inversion give a map from the free group on the underlying set of G to G. Conversely, each Talgebra 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.) Example 3.2.3. Some (nontrivial) adjunctions involving Set induce the trivial monad. For example, consider the category Top of topological spaces and continuous maps. The forgetful functor UTop : Top  Set has a left adjoint FTop : Set  Top, which sends a set X to the discrete topological space with underlying set X. One easily checks that T = GTopFTop is the identity functor on Set, and that all associated natural transformations are identities. Definition 3.2.4. Suppose F : A 4 'B, G : 3  A, 7 : idA  GF, and e : FG  ids forms an adjoint connection. By Proposition 3.2.1, the adjoint connection induces a monad (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 monadic adjoint connections. Monadic categories are determined by the combinatorial struc ture of the free algebra functor FT. Most "relational" categories like Top, P, the category of graphs, etc. have free functors, but these free functors do not add any structure to the underlying set; they merely attach the "most discrete" possible relation to the given set. 3.3 Detecting Categories of Algebras Because of the special properties of AT, the question of when an adjoint connection is monadic has great practical importance. Remark 3.3.1. Let (A, a) be a Talgebra. 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 rightidentifies 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 Talgebra (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: 1. The adjunction (F, G, 7, E) is monadic. 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'. Recognizing monadic adjunctions is complex because compositions of monadic func tors are not generally monadic. Example 3.3.4. The category Ab of abelian groups is monadic over Set, and tfAb (see Example 2.1.3) is monadic over Ab, because any reflection is monadic. Each free abelian group is torsion free, so the monad on Set induced by the free abelian group functor is the same as the monad induced by the "free torsion free abelian group functor." Thus, if tfAb were monadic over set, then tfAb and Ab would be equivalent categories both equivalent to a suitable category of monad algebras. The categories tfAb and Ab are obviously not equivalent. (The preceding example is summarized from Borceux [6, Volume 2, Example 4.6.4].) If G : A  Set, the hypotheses of Beck's theorem can be reformulated to make it easier to check whether G is monadic. Theorem 3.3.5. [11, Theorem 4.2] Let G : A  Set be a functor with a left adjoint. Suppose that A is complete and has coequalizers. The following are equivalent: 1. G is monadic. 2. G satisfies the following conditions G preserves and reflects regular epimorphisms; if f : GA  X is an isomorphism, then there is a unique map g : A  B such that Gg = f; G reflects kernel pairs. The following lemma provides useful information concerning when a functor between two categories of algebras is monadic. 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 is monadic. In particular, Q has a left adjoint. 3.4 Distributive Laws Compositions of monadic adjunctions are not generally monadic; when a composition of monadic adjunctions is monadic, it indicates a distributive law between the two structures. The following section summarizes the results later needed from Beck's [5]; this exposition follows Beck's notation, except that composition of functions here will read righttoleft. 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 defines a monad is omitted; for more information see Beck [5]. In the case when "S=free monoid" and "T=free abelian group," the natural trans formation e : ST * TS expresses a productofsums as a sumofproducts 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 ALST^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 Salgebra, 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 Salgebra map, A similar diagram, which uses the compatibility between f and PT, shows ,1T is an Salgebra 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 Aobject, A, along with an Sstructure as : SA  A, Tstructure aT : TA A, and Sstructure t : STA  A for TA such that the following diagram commutes. STA saT SA t as TA A Given a TSalgbra (A, a), A has a Tstructure aT map defined by aT = a(Tr7SA) : TA  A and an Sstructure 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 GENERATING SUBMONADS In this chapter, a technique for creating monads is laid out; before proceeding formally, let us consider a rough outline of the technique. Suppose (T, 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 extremallywellpowered. 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 SFcategory. ("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 SFcategory. It follows that P, Set, Top and practically any reasonable category of topological spaces, is an SFcategory. 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 SFcategory. Locale is extremallywellpowered, but not wellpowered. The author's interest in locales motivated him to use the given definition (which only requires extremal wellpoweredness) for SFcategory rather than defining SFcategory 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 pointset 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 SFcategory 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 f1 : 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 objectbyobject 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 (order theoretic) adjoint connection a+ : Subfun(F) + Subfun(G) : a1 The left adjoint a+1 is defined by (ar+E)(A) = (aA)+'(EA). The right adjoint a1 is defined by (a'E)(A) = a'(EA). Proof. The order on Subfun(F) is defined "objectbyobject", and whenever f : A  B, f+1 and f1 form an adjoint connection between Sub(A) and Sub(B). So it suffices to check that a+IE and atE 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 SFcategory, (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 i4 (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 boldface. 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 SFcategory, (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 Amap, 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 Falgebra 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 Amaps 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 Fstructure. 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 Fstructure to Fo uniquely determines the Tstructure; hence ER(A,a) = (A,a). D 4.3 A Partial Algebra Which Does Not Extend The category Set is an SFcategory 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 freemagma 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 nonempty 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 Talgebra 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 IPobjects. 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 Falgebra. Any F algebra has a natural P,, structure. Because of computational difficulties, no exact algebraic characterization of Falgebras 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 = (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 . The monad unit laws 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 upsidedown argument for T C UA to the reader. (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 Dalgebras. P, is equivalent to the category of Ualgebras. Proof. The first assertion is proved, leaving the second to the reader. By Lemma 5.1.1, any Dalgebra 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 Dalgebra 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: SVS. 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 Ualgebra A, there is a Ustructure on D~(A). 2. U maps Dalgebra maps to Dalgebra 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 upsidedown, 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; iA preserves supreme, which reads 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 iUlS4, 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 SFcategory 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 settheoretic 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 nontrivial 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 Jostructure, 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 Josuprema also preserves Jsuprema. Thus, one obtains an equivalence of categories between Pi and P. An upsidedown 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 Pextremal mono, and B E Obj(DP"). Ml denotes the category of Falgebras, 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 Pembedding 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 Tstructure for B. The verification of the hypothesis of 4.2.6 proceeds by comparing the diagram to be completed with the UTDalgebra 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 Fstructure; the right square expresses that the Fstructure on B is the restriction of the Tstructure. 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, 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 Pepi. 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 settheoretic 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 Talgebra 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 IPjmap preserves the Fstructure 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 Falgebra 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 Falgebra, then a/3(dAI) = idA and a6(iA) = idA. Lemma 5.1.1 shows that for any subset 3 C DA containing each principal downsegment 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 Fstructure maps. One might hope that each Fstructure 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 Pextremal 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 SPsets 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 SPjoins. Let TP denote the class of increasing sets, which are closed under IPmeets. 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 P3object. Moreover, the productinduced 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 coordinatebycoordinate.) 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 7374. 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 antisymmetric. 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, anl, 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 Pregular epi. An example of this situation is any map f : 2flat * 2 from the (trivially ordered) two point set onto a twopoint 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 P3subobject 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 jjoins and mmeets, 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 Pisubobject 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 PLsubobject of A x A, then we say 1 is a (j, m)pocongruence. Pocongruences of Pmobjects 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. Pregular 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 twopoint chains. (See drawing below for help visualizing, and to fix notation.) to tl I ~I.. Evidently N is jjoin 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(ti1) = 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 jsubobject of N x N, because f preserves all existing joins. But N is not jjoin 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 Pregular epi if and only if f is subjective. Proof. Since Pregular 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 PJregular epi, 5. f is PLextremal 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 rightoutside 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 DPUsubobject 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 Pextremal mono. Thus, e(?C) is Pextremal mono. Therefore, the P(epi, extremal mono) factorization of t(77C) produces the factorization (rlC) = qm. By construction, m is both Pepi and Pextremal mono; so m is a Pisomorphism. 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 DPobject 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 Tostyle 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 Tospaces. 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 Tlreflection of topological spaces. Last, but not least, Section 7.6 partially describes the epicomplete S8 objects; the description is complete for Tospaces. Epicomplete Tospaces are chains with the specializa tion order. Products of epicomplete Tospaces are not epicomplete, so there is no functorial epicompletion in the category of Tospaces. 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 APosmap 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 APosobject, 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 f1 : (Y) * (X) is a APosmap. Definition and Remarks 7.1.2. We describe the functorial Galois connection between AS and APos. 