The Hunting of the Hopf Ring Andrew Stacey Sarah Whitehouse November 23, 2007 Abstract We provide a new algebraic description of the structure on the set of all unstable cohomology operations for a suitable generalised cohomology theory, E*(-). Our description is as a graded and completed version of a Tall-Wraith monoid. The E*-cohomology of a space X is a module for this Tall-Wraith monoid. We also show that the corresponding Hopf ring of unstable co-operations is a module for the Tall-Wraith monoid of unstable operations. Further examples are provided by considering operations from one theory to another. ____________________________ 2000 Mathematics Subject Classification. Primary: 55S25; Secondary: 55N20, * *16W99 The authors acknowledge the support of the EPSRC, grant no.: GR/S76823/01. 1 1 Introduction In this paper we provide a new algebraic description of the structure on the se* *t of all unstable cohomology operations for a suitable generalised cohomology theory, say E*(-). The bigraded set of unstable operations for E*(-) is identified by the usual Yoneda lemma argument with the bigraded set of cohomology groups of the representing spaces, {Ek(E_l)}. This has considerable structure and it is natural to ask how to best describe it. To date, the most comprehensive work in this area is due to Boardman, Johnson, and Wilson [BJW95 ]. They provide the following descriptions of this structure. 1. The unstable operations of a suitable cohomology theory define a monad on the category of complete, Hausdorff, filtered, graded, commutative, unital E*-algebras. 2. The unstable operations of a suitable cohomology theory are dual to the enriched Hopf ring of the corresponding co-operations. The Hopf ring part - i.e., ignoring the enrichment - of the second answer is, of course, a well-established and important notion in algebraic topology. Since the work of Ravenel and Wilson [RW77 ] the language of Hopf rings has been widely used and the structures of the Hopf rings associated to many important cohomology theories have been calculated; see, for example [CMS02 , RW77 , RW96 , Wil84]. A useful introduction to Hopf rings, with further references, can be found in [Wil00]. In considering a Hopf ring, we have not yet taken into account one of the most obvious pieces of structure: that operations may be composed. A Hopf ring does not include any structure which dualises to composition of operations. The enrichment in the second answer encodes the dual of composition. The enriched Hopf ring structures of several important cohomology theories are described in [BJW95 ]. The first answer, describing operations as a monad on a suitable category, certainly includes the composition as a fundamental part of the structure. How- ever, this answer does not explicitly describe the internal structure on the set of operations; instead it specifies the action of operations on some category. In particular, this approach does not lend itself to specifying the structure of operations via generators and relations. Of these descriptions, that of an unenriched Hopf ring has proved to be more amenable to further study than either that of an enriched Hopf ring or that of a monad on a suitable category of algebras. Our description of operations may be thought of as a monoidal reinterpre- tation of the first answer. It is algebraic in nature and we employ the language of general or universal algebra to express it. One advantage of this approach is that it describes the structure of unstable operations, including compositio* *n, directly. Another is that it allows for descriptions of unstable operations via generators and relations. Such descriptions are expected to shed light on the relationships between stable, additive, and unstable operations. For the Morava K-theories and related cohomology theories, these relationships were studied in [SW06 ]; the results of this paper provide the foundations for a very explicit description, via generators and relations, of the splitting of stable operations 2 from unstable operations given in [SW06 ]. We expect to develop this point of view for familiar examples in future work. Roughly speaking, our answer is that the unstable operations of a suitable theory have the structure of a graded, completed Tall-Wraith monoid; a term that we shall now explain. Let V be a variety of algebras, in the sense of general or universal algebra* *. A Tall-Wraith V-monoid is a set with precisely the algebraic structure required f* *or it to act on V-algebras. To make this precise, one considers the category of co* *-V- algebra objects in V. This is equivalent to the category of representable funct* *ors from V to V and so it has a monoidal structure corresponding to composition of functors. A Tall-Wraith V-monoid is then defined to be a monoid in this category. One example is very familiar: a ring is a Tall-Wraith V-monoid for V the category of abelian groups. The case originally considered by Tall and Wraith in [TW70 ], under the name biring triple, was for V the category of commutative unital rings. Recently, Borger and Weiland [BW05 ] rediscovered this and extended it to the case where V is the category of commutative unital k-algebras, for a commutative unital ring k. They adopted the term plethory in that situation; thus a plethory is that-which-acts-on-algebras. This is clea* *rly relevant to our purposes as unstable cohomology operations of multiplicative cohomology theories act on the cohomology algebras. It is also clear that there remains work to be done, because the cohomol- ogy theories that we are considering are graded and topologised. The grading introduces no technical difficulties: varieties of graded algebras have been st* *ud- ied for almost as long as varieties of ordinary algebras; they are also known as many-sorted algebras in the literature. We arrive at Tall-Wraith V*-monoids for V* a variety of graded algebras. These are naturally bi-graded, as are unstable cohomology operations. The main work of this paper comes in the extension of Tall-Wraith monoids to a suitably topologised context. For E*(-) a multiplicative cohomology theory and X a CW -complex, E*(X) is given the pro-finite topology; that is, the filtration topology for the filtration by the ideals {keri*: E*(X) ! E*(Xf)} for all inclusions of finite subcomplexes i: Xf ! X. We therefore develop a theory of filtered Tall-Wraith monoids so that our description of unstable operations takes into account the pro-finite filtrations. While there are vario* *us notions of filtered objects in a (suitable) category in the literature, we have* * not been able to find a set-up suited to our needs. Therefore we introduce a suitab* *le definition of filtered objects in a category to model the topology; in this set* *ting we introduce iso-filtrations as the generalisation of complete,-Hausdorff!space* *s.-! This allows us to formulate the notion of a Tall-Wraith V*-monoid, where V* denotes the category of iso-filtered objects in the variety of graded algebras,* * V*. Once we have established the general theory of graded filtered Tall-Wraith monoids, the applications to suitable generalised cohomology theories are strai* *ght- forward. We can now state precisely our monoidal reformulation of the first description of unstable operations. We adopt standard notation related to a cohomology theory E*(-), so the representing spaces are denoted by E_k(for k 2 Z), the corresponding homology theory is denoted by E*(-), and the coef- ficient ring by E*. Theorem A. Let E*(-) be a graded, commutative, multiplicative, cohomol- ogy theory. Let V* be the variety of graded, commutative, unital E*-algebras. 3 Suppose that E*(E_k) is free as an E*-module for each k. Then E*(E_*) is a -!* Tall-Wraith V -monoid. As noted in [BJW95 ], the freeness hypothesis of theorem A is satisfied for HFp, BP , MU, K(n), and KU. There is a natural notion of a module for a Tall-Wraith monoid. The coho- mology of spaces provides examples of modules for the Tall-Wraith monoid of unstable operations. Theorem B. Let E*(-) and V* be as in theorem A. Let X be a topological space. The natural morphisms of sets Ek(X) ! V*(E*(E_k), E*(X)) ___ make the completed cohomology of X, E*(X), into a module for the Tall-Wraith -!* * V -monoid of unstable operations E (E_*). It turns out that the Hopf ring of co-operations is also a module for the Tall-Wraith monoid of operations, and it is this extra structure that is encoded in the term enriched. Further examples come from considering operations and co-operations from one theory to another. Theorem C. Let E*(-) and V* be as in theorem A. Let F *(-) be an- other graded, commutative, multiplicative cohomology theory. Suppose that each F*(E_k) is free as an F *-module. Then the following statements are true. 1. The bigraded set F *(E_*) of unstable operations E*(-) ! F *(-) has the -!* * structure of a module for the Tall-Wraith V -monoid E (E_*). 2. The Hopf ring F*(E_*) is a module for E*(E_*). 3. Let X be a space such that F*(X) is free as an F *-module. Let CF* be the category of F *-co-algebras. Then the natural morphisms of sets Ek(X) ! CF*(F*(X), F*(E_k)) extend to a morphism of modules for E*(E_*). This paper is organised as follows. Section 2 covers background material from general algebra, in both the ungraded and graded contexts. The main aim of this section is to establish the necessary conditions to consider Tall-Wraith monoids and certain important related concepts. Section 3 is concerned with setting out the necessary details of the theory of filtered objects in a catego* *ry. In this section we consider several types of filtrations and the relationships bet* *ween them. As we shall want to apply the constructions of general algebra in such categories we are also concerned with establishing the categorical properties of these categories of filtered objects. The main technical work of this paper is in this section and concerns functors between categories of filtered objects. Section 4 brings together the work of the preceding sections by considering Tall-Wraith monoids in the filtered context. Section 5 applies the results to t* *he examples arising in algebraic topology from suitable generalised cohomology theories, thus proving the theorems stated above. 4 The reader more interested in the results than the method by which they are demonstrated may prefer to read this paper in reverse. Finally, we wish to acknowledge the work of Boardmann, Johnson, and Wil- son in understanding unstable operations. Although this paper does not depend on [BJW95 ] mathematically, it was an invaluable resource as a guide to deter- mine the form of our final answer. 5 2 General Algebra In this section we shall expand a little on the basic constructions of general algebra. The results quoted in this section are all standard results from that field. For ungraded algebra objects in Set, these results can be found in any good introduction to the subject, for example [Ber98]. The more general cases can be found in the wider literature, for example in [KPMS82 ]. We record these results here without proofs to establish notation and as a quick reference for the rest of the paper. For those initiates of the deeper secrets of general alg* *ebra we mention that we are only considering algebras of finite arity and so we can assume that our identities are specified by finite sets. We start by summarising the results that we need in the arena of ungraded algebras, also known as single-sorted or homogeneous algebras. We shall then explain how this generalises to graded algebras, also known as many-sorted or heterogeneous algebras. 2.1 Ungraded Algebra Definition 2.1 Let D be a category with finite products. 1. A type is a pair (| |, n) where | | is a set and n: | |! N0 is a morphism of sets called the arity morphism. 2. An -algebra object, H, in D consists of an object in D, |H|, together with, for each ! 2 | |, a D-morphism !H :|H|n(!) ! |H|; these mor- phisms are called the operations of the -algebra object in D. A mor- phism of -algebra objects in D is a morphism of the underlying objects in D which intertwines the operations. 3. An -algebra is an -algebra object in Set. 4. We denote the category of -algebra objects in D by D and the category of -algebra objects in Set by . We refer to the functor D ! D which assigns to an -algebra object in D the underlying object in D as the forgetful functor. We write the underlying object in D of an -algebra object, H, in D as |H|. We trust to context to distinguish between the type and the resulting category of algebras. One of the key initial results in general algebra is the following character* *i- sation of -algebra object structures. Proposition 2.2 To give an object, |H|, in D the structure of an -algebra ob- ject is equivalent to giving a lift of the contravariant hom-functor D(-,_|H|):* * D ! Setto a functor D ! . |__| If H is an -algebra object in D and D is an object in D we shall write D(D, H) for the corresponding -algebra with underlying set D(D, |H|). The operations on D(D, H) are straightforward; let ! 2 | |be an element of arity n and let 6 !H :|H|n ! |H| be the operation on V . Then the corresponding operation on D(D, H) is the D-morphism n f1x...xfn !H D --! Dn ------! |H|n --! |H| where n :D ! Dn is the n-fold diagonal morphism. -algebras and -algebra objects simply have operations, they are not con- strained to satisfy any particular identities. To consider identities, we need * *to know about free -algebras. Proposition 2.3 Let D be a category with finite products which is closed under filtered co-limits and such that finite products commute with filtered co-limit* *s. Then the forgetful functor D ! D has a left adjoint, F :D ! D , which is_ called the free -algebra functor. |__| For identities, we only need to know about free -algebras in Set. Definition 2.4 Let be a type. An identity for -algebras is a triple (X, u, v) where X is a (finite) set and u, v 2 |F (X)|. Let H be an -algebra object in D. An identity for -algebras, (X, u, v), induces natural D-morphisms uH , vH :|H|X ! |H|. These are defined as follows. The canonical projections |H|X ! |H|in D define a set morphism X ! D(|H|X , |H|). As H is an -algebra object in D, the right hand side of this is the underlying set of an -algebra. Using the adjunction we therefore have a morphism of -algebras F (X) ! D(|H|X , H) and thus the elements u, v 2 |F (X)| define elements in the underlying set of the -algebra D(|H|X , H) which is D(|H|X , |H|). Thus we have the required D-morphisms uH , vH :|H|X ! |H|. Definition 2.5 An -algebra object, H, in D is said to satisfy the identity (X, u, v) if the two induced D-morphisms uH , vH :|H|X ! |H|are the same. Definition 2.6 A variety of algebras, V, is specified by a type, , and a set of identities for -algebras, J. It is the full subcategory of consisting of * *all -algebras which satisfy all of the identities in J. The pair ( , J) is a presentation of V. Let D be a category with finite products, V a variety of algebras with prese* *n- tation ( , J). The category of V-algebra objects in D, DV , is the full subcate* *gory of D consisting of all -algebra objects in D which satisfy all of the identi* *ties in J. As is well-known, presentations are not unique. Proposition 2.2 holds in the presence of identities. Proposition 2.7 To give an object, |H|, in D the structure of a V-algebra object is equivalent to giving a lift of the contravariant hom-functor D(-, |H|):_D ! Setto a functor D ! V. |__| 7 To get free V-algebra objects we need to know that we can "impose" iden- tities on an -algebra object. Theorem 2.8 Let D be a complete category with finite products. Suppose that D is an (E, M) category for some classes E of epimorphisms and M of monomor- phisms, that E is closed under taking finite products, and that D is E-co-well- powered. Then the inclusion functor DV ! D has a left adjoint, D ! DV ,_ called imposition of identities. |__| Corollary 2.9 Under the conditions of theorem 2.8 and proposition 2.3, the forgetful functor DV ! D has a left adjoint, FV :D ! DV , which is called_the free V-algebra functor. |__| Dual to V-algebra objects are co-V-algebra objects. Definition 2.10 Let V be a variety of algebras, D a category with finite co- products. A co-V-algebra object in D is a V-algebra object in Dop. A morphism of co-V-algebra objects in D is a morphism in D which intertwines the co- V-algebra structures. We denote the category of co-V-algebra objects in D by DVc . The morphisms are chosen such that there is an isomorphism of categories DVc ~=(Dop V)op and there is a natural forgetful functor DVc ! D. The analogue of proposi- tion 2.2 is the following. Proposition 2.11 To give an object, |G|, in D the structure of a co-V-algebra object is equivalent to giving a lift of the covariant hom-functor D(|G|,_-): D* * ! Setto a functor D ! V. |__| We shall use similar notation: D(G, D) will denote the V-algebra with un- derlying set D(|G|, D). The main tool of our analysis is the link between representable functors and functors with adjoints. This result is a standard one from general algebra, although one of its corollaries is perhaps the best known result. Theorem 2.12 Let D be a category that has finite products, is co-complete, is co-well-powered, is an (E, M) category where E is closed under finite products, and is such that its finite products commute with filtered co-limits. Let V be a variety of algebras. Let F be a category with co-equalisers. Let G: F ! DV be a covariant functor. Then the following statements are equivalent. 1. G has a left adjoint, G!. 2. The composition |G|:F ! D of G with the forgetful functor DV ! D has __ a left adjoint, |G|!. |__| The relationship between the two adjoints is that there is a co-equaliser sequence in F, natural in R, rR // pR |G|!(|FV(|R|)|)___sR//_|G|!(|R|)//_G!(R). Working in the opposite category we obtain the corresponding result on co-algebra objects. 8 Corollary 2.13 Let D be a category that has finite products, is co-complete, is co-well-powered, is an (E, M) category where E is closed under finite products, and its finite products commute with filtered co-limits. Let V be a variety of algebras. Let F be a category with equalisers. Let G: F ! DV be a contravariant functor. Then the following statements are equivalent. 1. G is one of a mutually right adjoint pair. 2. The composition |G|:F ! D of G with the forgetful functor DV ! D is __ one of a mutually right adjoint pair. |__| Since a functor from a co-complete category into Setis representable if and only if it has a left adjoint, the following standard result of general algebra readily follows from proposition 2.11. Corollary 2.14 1.Let F be a co-complete category, V a variety of alge- bras. For a covariant functor G: F ! V, the following statements are equivalent. (a)G has a left adjoint. (b)G is representable by a co-V-algebra object in F. (c)|G| is representable by an object in F. 2. Let F be a complete category, V a variety of algebras. For a contravariant functor G: F ! V, the following statements are equivalent. (a)G is one of a mutually right adjoint pair. (b)G is representable by a V-algebra object in F. (c)|G| is representable by an object in F. |___| We shall need to know various categorical properties of V. Theorem 2.15 As a category, V is complete, co-complete, well-powered, ex- tremally co-well-powered, and is an (extremal epi, mono) category. A morphism is an extremal epimorphism if and only if the underlying mor- phism of sets is an epimorphism. Moreover, all extremal epimorphisms are_ regular epimorphisms. |__| 2.2 Graded Algebras We turn now to graded algebras. A graded algebra has components indexed by some (fixed) set and its operations go from components to components rather than being globally defined. The theory of graded algebras is very similar to that of ungraded algebras. We fix some (non-empty) grading set Z. We shall regard this both as a set and as a (small) discrete category. We write DZ for the category of functors Z ! D. As Z is a discrete category, there is no distinction between covariant and contravariant functors from Z. An object, D, in DZ represents both a covariant and a contravariant functor D ! SetZ via 0 D*(D0) = z 7! D(D(z), D ) , 0 D*(D0) = z 7! D(D , D(z)) . 9 To define a graded algebra, we first need to define the graded version of a type. Definition 2.16 A Z-graded type, *, is a triple (| *|, i, o) where | *|is a set, and a i: | *|! Zm and o: | *|! Z m2N0 are morphisms of sets. For an operation !, we call i(!) the input and o(!) the output of !. We define the arity function, n: | *|! N0 by the composition a a | *|-i! Zm ! {*} ~=N0. m2N0 m2N0 ` We think of m2N0Zm as the set of finite ordered sets of elements of Z and so we interpret the element in Z0 as representing the empty subset of Z. Under the assumption that D has finite products, for an element (z1, . .,.zm ) and an object, D, in DZ , we write mY D(z1, . .,.zm ) = D(zj) j=1 with D(;) = TD , the terminal object in D. Definition 2.17 Let D be a category with finite products. 1. An *-algebra object, H, in D consists of an object, |H|, in DZ together with, for each ! 2 | *|, a D-morphism !H :|H|(i(!)) ! |H|(o(!)). A morphism of *-algebra objects in D is a morphism of the underlying objects in DZ which intertwines the operations. 2. An *-algebra is an *-algebra object in Set. 3. We denote the category of *-algebra objects in D by D * and the category of *-algebra objects in Set by *. We refer to the functor D * ! DZ which assigns to an *-algebra object in D the underlying object in DZ as the forgetful functor. We write the underlying object in DZ of an *- algebra object, H, in D as |H|. All of the results for ungraded algebras have their counterparts in graded algebras. Proposition 2.18 To give an object, |H|, in DZ the structure of an *- algebra object is equivalent to giving a lift of the contravariant hom-functor_ D(-, |H|): D ! SetZ to a functor D ! *. |__| As before, if H is an *-algebra object in D and D is an object in D we shall write D(D, H) for the corresponding *-algebra with underlying object in SetZ, z 7! D(D, |H|(z)). Free *-algebras exist under suitable circumstances. 10 Proposition 2.19 Let D be a category with finite products which is closed under filtered co-limits and such that finite products commute with filtered co-limit* *s. Then the forgetful functor D * ! DZ has a left adjoint, F * :DZ ! D * , __ which is called the free *-algebra functor. |__| Identities are defined by modifying the ungraded definition in the obvious way. Definition 2.20 Let * be a graded type. An identity for` *-algebras is a triple (X, u, v) where X is an object in SetZ such that z2ZX(z) is a finite s* *et and u, v 2 |F *(X)|(z) for some z 2 Z. Let (X, u, v) be an identity for *-algebras with u, v 2 |F *(X)|(z1). Let H be an *-algebra object in D. Consider the object in SetZ Y X(z) z0 7! D |H|(z) , |H|(z0) . z2Z ` As z2ZX(z) is finite, the product on the left is finite. For each z0 2 Z, the* *re is an obvious projection morphism in D Y |H|(z)X(z)! |H|(z0)X(z0) z2Z and thus for x 2 X(z0) we have a canonical projection morphism in D Y |H|(z)X(z)! |H|(z0). z2Z This yields a Setmorphism Y X(z) X(z0) ! D |H|(z) , |H|(z0) z2Z and thus a natural transformation of functors, equivalently a SetZ-morphism, Y X(z) X ! D |H|(z) , |H|. z2Z The same adjunction argument as in the ungraded case now produces D-mor- phisms Y uH , vH : |H|(z)X(z)! |H|(z1). z2Z Definition 2.21 An *-algebra object, H, in D is said to satisfy the identity (X, u, v) if the two induced morphisms uH , vH are the same. Definition 2.22 A variety of graded algebras, V*, is specified by a graded type, *, and a set of identities for *-algebras, J. It is the full subcategory of * * * consisting of all *-algebras which satisfy all of the identities in J. The pair ( *, J) is a presentation of V*. Let D be a category with finite products, V* a variety of graded algebras wi* *th presentation ( *, J). The category of V*-algebra objects in D, DV* , is the full subcategory of D * consisting of all *-algebra objects in D which satisfy all* * of the identities in J. 11 Proposition 2.18 holds in the presence of identities. Proposition 2.23 To give an object, |H|, in DZ the structure of a V*-algebra object is equivalent to giving a lift of the contravariant hom-functor_D(-,_|H|* *): D ! SetZ to a functor D ! V*. |__| The same conditions as in the ungraded case allow us to impose identities and so get free V*-algebra objects. Theorem 2.24 Let D be a complete category with finite products. Suppose that D is an (E, M) category for some classes E of epimorphisms and M of monomorphisms, that E is closed under taking finite products, and that D is E-co-well-powered. Then the inclusion functor DV* ! D * has a left adjoint,_ D * ! DV* , called imposition of identities. |__| Corollary 2.25 Under the conditions of theorem 2.24 and proposition 2.19, the forgetful functor DV* ! DZ has a left adjoint, FV* :DZ ! DV* , which is_ called the free V*-algebra functor. |__| Dual to V*-algebra objects are co-V *-algebra objects. Definition 2.26 Let V* be a variety of graded algebras, D a category with finite co-products. A co-V *-algebra object in D is a V*-algebra object in Dop.* * A morphism of co-V *-algebra objects in D is a morphism in DZ which intertwines the co-V *-algebra object in D structures. We denote the category of co-V *- algebra objects in D by DV* c. The analogue of proposition 2.18 is the following. Proposition 2.27 To give an object, |G|, in DZ the structure of a co-V *-alge- bra object is equivalent to giving a lift of the covariant hom-functor_D(|G|,_-* *): D ! SetZ to a functor D ! V*. |__| We shall use similar notation: D(G, D) will denote the V *-algebra with underlying object D(|G|, D) in SetZ. Theorem 2.12 easily generalises to the graded case. Theorem 2.28 Let D be a category that has finite products, is co-complete, is co-well-powered, is an (E, M) category where E is closed under finite products, and its finite products commute with filtered co-limits. Let V* be a variety of graded algebras. Let F be a category with co-equalisers. Let G: F ! DV* be a covariant functor. Then the following statements are equivalent. 1. G has a left adjoint, G!. 2. The composition |G|:F ! DZ of G with the forgetful functor DV* ! DZ __ has a left adjoint, |G|!. |__| We have the same relationship between the two adjoints as in the ungraded case: there is a co-equaliser sequence in F, natural in H, rH // pH |G|!(|FV*(|H|)|)___sH//_|G|!(|H|)//_G!(H). The graded version of corollary 2.13 follows immediately. To get the graded version of corollary 2.14 we need to understand the relationship between ad- junctions and representability in the graded case. Lemma 2.29 Let D be a co-complete category. A covariant functor G: D ! SetZ has a left adjoint if and only if it is representable by an object in DZ . 12 Proof.Suppose that G has a left adjoint, say H: SetZ ! D. We extend this to a functor HZ :(SetZ)Z ! DZ in the obvious way. Let I be the object in (SetZ)Z defined by _ ( 0 ! z 7! z07! {*} ifz = z . ; otherwise Then for an object, X, in SetZ we have isomorphisms in SetZ Z SetZ (I, X)= z 7! Set (I(z), X) ~= z 7! Y Set(I(z)(z0), X(z0)) z02Z Y ~= z 7! Set({*}, X(z)) x Set(;, X(z0)) z06=z ~= z 7! X(z) x Y {*} z06=z ~= z 7! X(z) ~= X, all natural in X. Hence for D an object in D there are natural isomorphisms G(D) ~=SetZ(I, G(D)) ~=D(HZ (I), D) and so G is represented by the object, HZ (I), in DZ . Conversely, suppose that G is represented by the object, G, in DZ . Let X be an object in SetZ. We have the following natural isomorphisms of sets SetZ(X, G(D)) ~=SetZ(X, D(G, D)) ~=Y Set(X(z), D(G(z), D)) z2ZY ~= D(G(z), D)X(z) z2ZY a ~= D G(z), D z2Z X(z) ~=D a a G(z), D . z2ZX(z) Therefore we define the functor H: SetZ! D on objects by a a H(X) = G(z) z2ZX(z) and in the obvious way on morphisms. This is the required left adjoint. |__* *_| As a corollary we deduce the graded version of corollary 2.14. Corollary 2.30 1.Let D be a co-complete category, V* a variety of graded algebras. For a covariant functor G: D ! V*, the following statements are equivalent. 13 (a)G has a left adjoint. (b)G is representable by a co-V *-algebra object in D. (c)|G| is representable by an object in DZ . 2. Let D be a complete category, V* a variety of graded algebras. For a contravariant functor G: D ! V*, the following statements are equivalent. (a)G is part of a mutually right adjoint pair. (b)G is representable by an V*-algebra object in D. (c)|G| is representable by an object in DZ . |___| The categorical properties of V* are the same as those of V. Theorem 2.31 As a category, V* is complete, co-complete, well-powered, ex- tremally co-well-powered, and is an (extremal epi, mono) category. A morphism is an extremal epimorphism if and only if the underlying mor- phism of objects in SetZ is an epimorphism. Moreover, all extremal epimor-_ phisms are regular epimorphisms. |__| The work of the following sections can be viewed simply as applications of corollaries 2.14 and 2.30. 2.3 The Tall-Wraith Monoidal Structure The categories VVc and V* V*c have a monoidal structure corresponding to composition of (representable) functors. The first trace of this that we have discovered in the literature is [Fre66] where it is referred to as the tensor p* *roduct of algebras. The first study of the corresponding monoids that we have found is [TW70 ] which deals with the category of commutative, unital rings. As we are similarly interested in the monoids, we have elected to call them Tall-Wraith monoids. For consistency, and because the terminology of tensor products is already somewhat overused, we name the monoidal structure on VVc and V*V*c the Tall-Wraith monoidal structure. Although, as we have just said, the ideas in this section go back at least to [Fre66], we have not been able to find a reference which covers all that we wish to do; in particular, theorem 2.34 and the extensions to graded algebras. On the other hand, these results are not central to this paper but rather are a guide * *to what to expect in the filtered context and so we have not included their proofs here. The missing proofs can be found in [SW07b ]. In addition to [Fre66] and [TW70 ] mentioned above, similar ideas occur in [BW05 ] and [BH96 ]. Theorem 2.32 Let V be a variety of algebras. The category VVc has a monoidal structure which we write as (VVc , , I). The functor VVc ! CovFun (V, V) given by sending a co-V-algebra object in V to the covariant functor that it represents, is strong monoidal. The underlying object in V of the unit,_I, is isomorphic to FV({*}). |__| We shall not give a full proof of this result here; the idea can be found in [Fre66] and a full proof is in [SW07b ]. We shall give a description of the pr* *oduct pairing as this will be important later. For a co-V-algebra object, B, in V let us write B*: V ! V for the covariant functor that it represents. By corollary 2.14, this functor has a left adjoint 14 which we denote by B!. Now co-V-algebra objects in V are objects in V with extra structure; this extra structure involves morphisms from the underlying object in V to iterated co-products of it. As B!is a left adjoint, it preserves* * co- products and thus lifts to a functor Bc!:VVc ! VVc. The assignment B 7! Bc! is functorial in B. The pairing on VVc is, up to natural isomorphism, given on objects by (B1, B2) 7! B2c!(B1). It has the property that we have natural isomorphisms V(B1, V(B2, B3)) ~=V(B1 B2, B3). Theorem 2.32 readily adapts to the following situations. Proposition 2.33 Let D be a co-complete category, V and W varieties of al- gebras. There are products VVc x DVc ! DVc , (VVc )opx DV ! DV , VWc x VVc ! VWc , all compatible with the monoidal structure of VVc and with composition of_rep- resentable functors. |__| We write all of the pairings using the notation - -. There are two things to note from this generalisation. Firstly that there are two pairings which involve VVc and V. The first views V as SetV and so comes from the middle pairing above with D = Set; in terms of functors we have (B V )*(X) = Set(X, B V ) ~=V(B, Set(X, V )). The second views V as VSetcand so comes from the third pairing with W = Set; in terms of functors we have (V B) *(V 0) = V(V B, V 0) ~=V(V, V(B, V 0)). This latter pairing was the one considered in [TW70 ] with V the category of commutative, unital rings. The second thing to note from this generalisation is the annoyance of having a partially contravariant pairing. Providing D is sufficiently structured this * *can be countered; again, the details can be found in [SW07b ]. Theorem 2.34 Let D be a category satisfying the conditions of theorem 2.12. Then there is a pairing VVc x DV ! DV , (B, R) 7! B " R, which is covariant in both arguments and satisfies DV (B " R, R0) ~=DV (R, B R0) naturally in all arguments. |___| In a monoidal category it is natural to consider monoids. Definition 2.35 Let V be a variety of algebras. A Tall-Wraith V-monoid is a monoid in VVc . We write the category of such monoids as VVc T . 15 These were discussed briefly in [BH96 , chs 63, 64], though without explicit reference to the underlying monoidal structure on VVc . With a monoid one can consider modules for that monoid. Since the monoidal category VVc acts on other categories we can consider modules that are not co- V-algebra objects in V. That is, if P is a Tall-Wraith V-monoid and D is a co-complete category then we can consider co-V-algebra objects, G, in D for which there is a DVc -morphism P G ! G satisfying the obvious coherences. In [BH96 ] the authors show that the category of objects in V with an action of a Tall-Wraith V-monoid is again a variety of algebras. Extending this, we easily see that a V-algebra object in D or co-V-algebra object in D is a module for a Tall-Wraith V-monoid if and only if the corresponding functor D ! V factors through the category of objects in V with an action of the Tall-Wraith V-monoid. Two remarks are worth making at this juncture. Firstly, if W is another variety of algebras then the structure of a P -module on a co-W-algebra object in V does not have such an interpretation since a co-W-algebra object in V represents a functor out of V. Secondly, due to the variance shift, a P -module in DV is better thought of as a P -co-module as the required morphism is R ! P R. We can surmount this using the product " since the adjunction turns a co-action as above into a more normal-looking action. That is to say, if a V-algebra obje* *ct, R, in D is a P -co-module for with co-action morphism R ! P R then it is a P -module for " with action morphism P " R ! R. The adaptation of all this to the graded situation is straightforward. Theorem 2.36 Let V* be a variety of graded algebras. The category V*V* chas a monoidal structure which we write as (V* V*c, , I). The functor V*V* c! CovFun (V* , V*), given by sending a co-V *-algebra object in V* to the covaria* *nt_ functor that it represents, is strong monoidal. |__| As before we shall give a description of the product. A co-V *-algebra objec* *t, B, in V* represents a functor B*: V* ! V*. By corollary 2.30 this functor has a left adjoint B!:V* ! V*. We extend this to a functor BZ!:V*Z ! V*Z in the obvious way. Co-products in a graded category are formed by taking component- by-component co-products, whence BZ!preserves co-products because B!does. Hence it lifts to a functor Bc!:V*V*c ! V*V*c. This has the property that the adjunction isomorphism lifts to an isomorphism of V*-algebras V*(B1, V*(B2, V )) ~=V*(B2 c!(B1), V ). Thus there is a natural isomorphism of co-V *-algebra objects in V* B1 B2 ~=B2c!(B1). 16 The other part of the structure that needs a word of explanation is the representing object for the unit of the monoidal structure. We saw in the proof* * of lemma 2.29 that the identity functor SetZ! SetZis representable by an object in (SetZ)Z, labelled I in that proof. The free V*-algebra on the components of this object in (SetZ)Z represents the identity on V*. Proposition 2.37 Let D be a co-complete category, V* and W* varieties of graded algebras. There are products V*V*cx DV* c ! DV* c, (V* V*c)opx DV* ! DV* , V*W* c x V*V*c ! V*W* c, all compatible with the monoidal structure of V*V* c and with composition_of representable functors. |__| We write all of the pairings using the notation - -. We remark that the varieties of graded algebras, V* and W* , could be graded by different indexing sets. This allows us to take, for example, W* = Setand so get the obvious pairing V* x V*V*c ! V*. We can remove the variance switch in the middle product by means of the graded analogue of theorem 2.34. Theorem 2.38 Let D be a category satisfying the conditions of theorem 2.12. Then there is a pairing V*V* cx DV* ! DV* which is covariant in both arguments and satisfies DV* (B " H, H0) ~=DV* (H, B H0) naturally in all arguments. |___| Definition 2.39 Let V* be a variety of graded algebras. A Tall-Wraith V*- monoid is a monoid in V* V*c. We write the category of such monoids as V*V* cT . The remarks regarding modules (and co-modules) for a Tall-Wraith V-monoid carry over to the graded case. 17 3 Filtered Categories The purpose of this section is to introduce a categorical version of a very spe* *cific type of topology. What we wish to generalise is the following situation: we have a topological space whose topology is the projective topology for a family of maps into discrete spaces. This particular case is easy to put into a general categorical situation and we do not need any of the usual machinery used to meld topology and category theory. In the first part we introduce the basic idea: filtered objects. To give a s* *et, X, a topology in this fashion it is sufficient to give a family of maps with so* *urce X. Putting this into a categorical context leads to projectively filtered objec* *ts in an arbitrary category. We shall also define inductively filtered objects sin* *ce we shall need to consider how contravarient functors transform filtered objects in one category into filtered objects in another category. In the example of topological spaces, we only need to consider surjective morphisms and we can reduce an arbitrary filtration to one in which all the morphisms are surjective. We cannot mirror this reduction in all categories and, moreover, the condition that a functor preserve epimorphisms is more restrictive than we wish to impose. However, in certain categories there is a reduction functor and we shall examine the extra features of the theory that this introduces. When we can consider these reduced filtrations it makes sense to consider variations on the themes of being complete and being Hausdorff. Completion is not a purely topological concept, rather it is a notion from the theory of uniform spaces. The correct generalisation of these two notions to reduced filtered objects involves examining the projective limit of the filtration. The* *re is a morphism from the underlying object to this limit and we can ask whether this morphism is a monomorphism, epimorphism, or isomorphism. Being a monomorphism corresponds to the topology being Hausdorff whilst being an epimorphism generalises the notion of completeness. We start by introducing the most general form of filtrations before moving on to the reduced version. Once we have that then we can consider the projective limit. 3.1 Projective Filtrations We start with the general case of a filtration on an object in a category. Definition 3.1 Let D be a category, D an object in D. We define D# to be the quasi-ordered class whose elements are D-morphisms with source D and whose ordering is given by d1 d2 if there is a D-morphism h such that hd1 = d2. A projective filtration, Q, on D is a non-empty, saturated, directed subclass of D#. We say that Q1 is stronger than Q2 if Q2 Q1. If we are given a projective filtration Q on an object in D without having specified the underlying object in D we shall write it as |Q|. An element of Q * *is a D-morphism which we shall write as q :|Q| ! Qq. In this context, saturated means that if d1 d2 and d1 2 Q then d2 2 Q. Let Q be a projective filtration on D. Let f :D0 ! D be a D-morphism. The family of all elements of D0#of the form df for d an element of Q is a 18 non-empty, saturated, directed subclass of D0#and hence a projective filtration on D0. Definition 3.2 We refer to this filtration as the pull back of Q by f and write it as f*(Q). This construction is strictly associative. Lemma 3.3 Given D-morphisms D1 f-!D2 g-!D3 and a projective filtration Q on D3, the projective filtrations f*(g*(Q)) and (gf)*(Q) on D1 are the_same. |__| With these notions we can define a category of projective filtrations on ob- jects in D. Definition 3.4 We define the category of projectively filtered objects of D, .K* *D. Its objects are projective filtrations on objects in D. A .KD-morphism f :Q1 ! * *Q2 is a D-morphism |f|: |Q1| ! |Q2| with the property that Q1 is stronger than f*(Q2). By construction, the obvious functor .KD! D is faithful. Any projective filtration is completely determined by an initial subclass, which per force is directed, and any non-empty directed subclass of D# de- termines a projective filtration by saturation; that is, if d1, d2 are in D# wi* *th d1 d2 and d1 is in the specified class then we include d2. It is clear that t* *he original directed class is initial for the resulting projective filtration. If * *D has finite products then any subclass of D#, directed or not, determines a projecti* *ve filtration: first we include all finite products and then we saturate it. Ther* *e- fore we could choose to work with directed subclasses of D#, or even arbitrary subclasses, but the above formulation of saturated subclasses is most directly analogous to a topology on a set. The correspondences are: projective filtra- tion and topology, directed subclass and a basis of the topology, subclass and a subbasis of the topology. We shall find it useful to have a characterisation of when a D-morphism lifts to a .KD-morphism in terms of choices of initial subclasses of the projec- tive filtrations involved. Let Q1 and Q2 be projective filtrations in D and let f :|Q1| ! |Q2| be a D-morphism on the underlying objects in D. Suppose that we have initial subclasses of Q1 and Q2 indexed by 1 and 2 respectively. Then f lifts to a .KD-morphism if and only if for each ~2 in 2 there is a ~1 in 1 and a D-morphism f~1,~2:Q1,~1! Q2,~2such that the following diagram of D-morphisms commutes |Q1|___f___//|Q2| q1,~1|| |q2,~2| fflffl|f~1,fflffl|~2 Q1,~1_____//Q2,~2. Lemma 3.5 The assignment D ! D.Kunderlies a 2-functor of 2-categories Cat ! Cat. .K .K Proof.For a covariant functor G: D ! E we define a covariant functor G :D ! .KEin the obvious way: for Q an object in D.K, G.K(Q) is the saturation of the 19 non-empty, directed subclass of G(|Q|)# consisting of G(q) for q 2 Q. For a .KD-morphism f :Q .K .K 1 ! Q2, G (f) has underlying E-morphism G(|f|). That this is an E-morphism is obvious. This construction is compatible with composition ...K .K.K in that GH = GH . It is obvious that identity functors map to identity functors. Similarly, if :G ! H is a natural transformation of covariant functors .K .K .K D ! E, we define a natural transformation ..K:G! H. For an object, Q, in D , the .KE-morphism ..KQhas underlying E-morphism |Q|. Again, this construction is obviously compatible with composition and identity natural transformations._ |__| An important consequence of this is the following result. Proposition 3.6 Let G: D ! E be a covariant functor and suppose that it has .K .K __ a left adjoint, say H: E ! D, then H is left adjoint to G . |_* *_| If D has a terminal object then the class of all projective filtrations of a* * fixed object in D is a (large!) complete lattice. Its top and bottom elements provide adjoints to the forgetful functor .KD! D. Proposition 3.7 The forgetful functor .KD! D has a left adjoint D: D ! .KD. If D has a terminal object then the forgetful functor .KD! D has a right adjoint I: D ! .KD. For an object, D, in D, D(D) is D# whilst I(D) is the subclass of D# con- sisting of all D-morphisms from D to terminal objects in D. Proof.Clearly the descriptions given of D(D) and I(D) do produce projective filtrations on D and if Q is another projective filtration on D then we have I(D) Q D(D); the second inclusion by definition and the first as Q is non-empty. From this, it is clear that if f :D1 ! D2 is a D-morphism then it underlies .KD-morphisms D(D .K 1) ! D(D2) and I(D1) ! I(D2). As the forgetful functor D ! D is faithful, this is sufficient to define D and I on morphisms. Clearly, if we apply either D: D ! .KDor I: D ! .KDand the forgetful functor .KD! D then the resulting composition is the identity functor on D. Finally, from above we see that the identity on D lifts to morphisms D(D) ! Q ! I(D). These provide the required natural transformations for the adjunc-_ tions. |__| Definition 3.8 For an object, D, in D we refer to D(D) as the discrete (pro- jective) filtration on D and I(D) as the indiscrete (projective) filtration on D (assuming that D has a terminal object). These two functors are very simple examples of a more general type of func- tor. Definition 3.9 To filter a category is to give a functor D ! .KDwhich is right inverse to the forgetful functor. We call such a functor a projective filtration functor. Examples 3.10 1. The first example is of a pro-finite filtration. Let F be a non-empty full subcategory of D which is closed under finite products. We refer to objects in F as finite objects. 20 Let D be an object in D. We define a projective filtration on D as follows. We start with the subclass of D#consisting of all D-morphisms with target a finite object. Our assumption on F ensures that this is directed. We saturate it to produce a projective filtration. It is straightforward to show that the assignment to an object in D of its pro-finite filtration is functorial. We therefore have the pro-finite filt* *ration functor on D. 2. The second example of a category that can be filtered is that of a filtered category. We shall define a filtration functor .KD! .K.KD. Let Q be an object in .KD. We start by observing that for q 2 Q, the D- morphism q :|Q| ! Qq is the underlying D-morphism of a .KD-morphism Q ! D(Qq). Let us write .Kq:Q ! D(Qq) for the resulting .KD-morphism. The subclass of Q# consisting of the elements .Kqis directed, as the origi- nal projective filtration was directed, and hence saturates to a projective filtration. It is clear from its construction that this is functorial in Q. 3.2 Reduced Filtrations Having looked at general filtrations, we now turn to a particular type of filtr* *a- tion. Let us consider the example of a topology on a set defined by a projective filtration. The structure of the category of sets means that we can ensure that each of the maps in the filtration is a surjection. This has certain advantages which we wish to mirror in our more general filtered categories. Although the definition below does not depend on any additional properties of the underlying category, in order to do anything useful with it we need to assume that this category is an extremally co-well-powered (extremal epi, mono) category. We also want to know that the forgetful functor .KD! D has a right adjoint; for th* *is we need to know that D has a terminal object. Definition 3.11 Let D be a category. A projective filtration Q on an object in D is said to be reduced if Q has an initial subclass consisting of extremal epimorphisms. We write DK for the full subcategory of D.Kconsisting of reduced projective filtrations. Let S: KD! .KDbe the inclusion functor. We could broaden our definition by first choosing a reasonable class of epi- morphisms and considering those filtrations with morphisms in that class, but we shall only be interested in extremal epimorphisms and so we confine our attention to those. K .K Under the right conditions, the inclusion functor S: D ! D has a right adjoint. Proposition 3.12 Let D be an (extremal epi, mono) category. Then there is a reduction functor R: .KD! KDwhich is faithful. The composition RS: KD! KD is the identity functor. The functor R is right adjoint to S. 21 Proof.Let Q be a projective filtration in D. We define another projective filtration with the same underlying object in D as follows. Each element q 2 Q is a D-morphism q :|Q| ! Qq. By assumption on D this has a factorisation as mqqK where qKis an extremal epimorphism and mq a monomorphism. Let us write KQqfor the target of Kq. We claim that the class of morphisms consisting * *of these Kqis directed. This follows from the diagonal property of an (extremal ep* *i, mono) category. Suppose that q1 q2 in Q. Then there is some D-morphism f :Qq1! Qq2such that q2 = fq1. Thus the following is a commutative diagram in D. 9Kq1 K mq1 |Q|_____//_Qq1____//Qq1 "" 9Kq2|| """ fflffl| """ K "" Qq2 ""f" "" mq2|| """ ff"""""lffl| Qq2 The diagonal property of the (extremal epi, mono)-factorisations now implies the existence of a D-morphism QKq1! QKq2which fits into the above diagram. Hence in |Q|# we have 9Kq1 9Kq2. Thus as Q is directed, the class { Kq} is al* *so directed. Its saturation is thus a projective filtration which, by construction* *, is reduced. Let us write the result as QK. We define R on objects by R(Q) = KQ. To define R on morphisms we need to examine its interaction with pull backs. Let Q be a projective filtration on an object in D, let D be an object in D, and let f :D ! |Q| be a D-morphism. We wish to compare f*(QK) with 9 9 9K K f*(Q). We obtain an initial subclass for f*(Q ) by taking the extremal epimor- phisms appearing in the (extremal epi, mono)-factorisations of elements of Q and composing with f. That is, it consists of the D-morphisms qKf :D ! 9KQq where q :|Q| ! Qq is an element of Q with (extremal epi, mono)-factorisation mqqK and intervening object, 9KQq, in D. On the other hand, an initial subclass 9 9 9K of f*(Q) consists of the extremal epimorphisms appearing in the (extremal epi, mono)-factorisations of the D-morphisms qf for q in Q. For q in Q we therefore have the diagram f 9Kq 9K D _____//_|Q|___//Qq | |e| |mq| fflffl| m fflffl| Dq ______________//Qq where me is the (extremal epi, mono)-factorisation of qf. As D is an (extremal epi, mono) category there is a D-morphism Dq ! 9KQqwhich fits into this dia- 9 9 9K gram. Hence f*(QK) f*(Q). The proof that the defining subclass of KQis directed easily extends to show that if Q1 and Q2 are projective filtrations on the same underlying object in D with Q1 Q2 then 9KQ1 9KQ2. Putting this together with the above, we see 22 9 9*9 9K 9K that if f :Q1 ! Q2 is a .KD-morphism then |f|*(Q9K2) |f|(Q2) Q1 and so |f| also underlies a DK-morphism 9KQ1! 9KQ2which we denote by R(f). As f and R(f) have the same underlying D-morphism, this assignment clearly respects composition and identities whence we have a functor R: .KD! KD. As (extremal epi, mono)-factorisations in D are unique up to canonical iso- morphism, the composition DK ! D.K! DK is clearly the identity (saturation ensures that it is actually the identity, rather than just isomorphic to the id* *en- tity). The obvious inclusion Q QKprovides the other natural transformation in the adjunction. Since both forgetful functors .KD! D and DK! D are faithful, and since the reduction functor .KD! KDcovers the identity on D, the reduction functor_must be faithful. |__| The discrete filtration functor, D: D ! .KD, factors through KDwith no modif* *i- cation since D(D) contains the initial subclass {D 1-!D} and every isomorphism is an extremal epimorphism. This provides us with a left adjoint to the forgetf* *ul functor DK! D. The indiscrete filtration functor, I: D ! .KD, (assuming that D has a terminal object) is not so fortunate. It is not even true that the termin* *al morphism tD :D ! TD is always an epimorphism. Thus we need to define K K .K K I:D ! D as the composition of I with the reduction functor D ! D. This is right adjoint to the forgetful functor DK! D. 3.3 Iso-Filtrations In this section we consider those filtered objects that are analogous to Hausdo* *rf spaces and to complete uniform spaces. In order to work with these objects we need to make additional assumptions on our underlying category, namely that it be complete and extremally co-well-powered. As these filtered objects are a subclass of the reduced filtered objects we also retain the assumptions of the previous section. Thus we assume that D is a complete, extremally co-well- powered, (extremal epi, mono) category. Note that completeness implies the existence of a terminal object. We start by defining another functor KD! D by taking the limit of a reduced filtration. We need to be in KDrather than .KDto ensure that the definition mak* *es sense. We start with the most general definition. Let D be a category and Q an object in DK. We define a category (Q, D) by Objects elements of Q, Morphisms a morphism q1 ! q2 is a D-morphism, d, from the target of q1 to the target of q2 such that dq1 = q2. An alternative description of this, which explains the notation, is as the full subcategory of the comma category (|Q|, D) with class of objects the same as Q. Note that the quasi-ordered class Q when viewed as a category is a quotient of (Q, D) under the relation f ~ g if f and g have the same source and target. 23 There is an obvious functor (Q, D) ! D which sends an object of (Q, D) to its target, and which regards a morphism in (Q, D) as a D-morphism. Although the category (Q, D) is usually large, the functor (Q, D) ! D still might have a limit. By abuse of notation, we shall refer to the limit of the functor (Q, D) * *! D as the limit of Q and write it as lim-Q. When this limit exists, it is obvious that there is a natural D-morphism |Q| ! lim-Q. The standard properties of limits show that if f :Q1 ! Q2 is a DK-morphism and Q1 and Q2 are such that both of the appropriate limits exist, then there is a corresponding D-morphism f0: lim-Q1 ! lim-Q2 compatible with the above natural morphisms. Definition 3.13 If Q is an object in DKsuch that lim-Q exists and the natural morphism |Q| ! lim-Q is an isomorphism then we say that Q is an iso-filtration on |Q|. We write -!Dfor the full subcategory of DKconsisting of all such objects in DK. We write B: -!D! KDfor the inclusion functor. Under reasonable conditions on D we can show that, in fact, every object in DKhas a limit. Proposition 3.14 If D is an extremally co-well-powered complete category, then every object in DKhas a limit. Proof.To prove the required result, we observe that if Q is an object in DK, it has an initial subclass consisting of extremal epimorphisms with source |Q|. As D is extremally co-well-powered, we can take this subclass to be small. It is, * *per force, directed. We claim that this is an initial full subcategory of (Q, D). L* *et q1 and q2 in Q be extremal epimorphisms. Then as q1 is an epimorphism, there can be at most one D-morphism with the property that dq1 = q2. Comparison of the ordering on elements of Q with the definition of morphisms in (Q, D) now shows that (Q, D)(q1, q2) has at most one element and it has precisely one element if and only if q1 q2. Hence our initial subclass of Q is a full subcategory of (Q, D). It is clearly initial in (Q, D). As D is complete we can therefore find a limit of the functor (Q, D) ! D by taking a limit of its restriction to our small initial subclass of Q. This is u* *nique up to canonical isomorphism. It also depends functorially on Q since if we have a DK-morphism Q1 ! Q2 then the pull back of Q2 is contained in Q1, whence __ we get a canonical D-morphism on the limits, as they exist. |__| By the remarks preceeding definition 3.13, these limits fit together to form a functor KD! D and there is a natural transformation of functors KD! D from the forgetful functor to this limit functor. Definition 3.15 We shall refer to the functor constructed above as the projec- tive limit functor and write it as L: KD! D. Using the fact that a category of filtered objects is itself naturally filte* *red, we obtain the following important construction. Proposition 3.16 Let D be a complete, extremally co-well-powered (extremal epi, mono) category. Then there is a natural lift of the projective limit funct* *or 24 L: KD! D to a functor C: KD! -!Dwhich is left adjoint to the inclusion functor B: -!D! DK. Moreover, the composition-CB:!-!D! -!Dis naturally isomorphic to the identity functor on D . Since the forgetful functor-is!naturally isomorphic to the projective-limit! functor when restricted to D we do not need to specify which functor D ! D we are lifting along. Proof.Let Q be an object in DK. Let ': |Q| ! L(Q) be the canonical D-mor- phism. Let q be an element of Q. By the definition of a limit, there is a canon* *ical D-morphism -!q:L(Q) ! Qq such that q = -!q'. We claim that the subclass of L(Q)# consisting of the morphisms -!qwhich arise in this fashion is a projective filtration. It is obviously non-empty and directed. To show that it is saturated, let d: L(Q) ! D be a D-morphism and suppose that it factors through -!qfor some q in Q, say d = f-!q. Then d' = fq whence d' is in Q. The D-morphism f defines a morphism in (Q, D) from q to --! d'. Hence, by the definition of a limit, the canonical morphisms (d'):L(Q) ! D --! -! --! and -!q:L(Q) ! Qq satisfy (d')= f q, whence (d')= d and so d is in our chosen subclass. Hence this subclass is saturated and we have a projective filtration. This construction clearly depends functorially on Q. Let us show that this is, in fact, a reduced projective filtration. We need * *to show that it has an initial subclass of extremal epimorphisms. It certainly has an initial subclass consisting of elements -!qwhere q :|Q| ! Qq is an extremal epimorphism. Since q is an epimorphism, so is -!q. Suppose that we have a factorisation -!q= mf where m is a monomorphism; let D be the intervening object in D. As D is an (extremal epi, mono) category, the morphism f': |Q| ! D has an (extremal epi, mono)-factorisation, say f' = m0e0. Then mm0e0 = mf' = -!q' = q. As both m and m0 are monomorphisms, their composition is still a monomorphism whence as q is an extremal epimorphism, mm0 is an isomorphism. The monomorphism m is therefore a retraction, with right inverse m0(mm0)-1, and so is an isomorphism. Hence -!qis an extremal epimorphism and so the projective filtration that we have defined on L(Q) is reduced. Let Q0denote this reduced projective filtration on L(Q). We now claim that Q0 is an iso-filtration. Consider the categories (Q, D) and (Q0, D). As |Q0| = --! -! L(Q) is the limit of (Q, D) ! D, the assignment q 7! -!qsatisfies (fq) = f q and thus we have a covariant functor (Q, D) ! (Q0, D) which on objects is q 7! -!qand which leaves morphisms alone (when viewed as D-morphisms). The D-morphism ': |Q| ! L(Q) defines a covariant functor (Q0, D) ! (Q, D). Since -!q' = q, the composition is the identity functor on (Q, D). The argument above which showed that the family {-!q} is saturated shows that the composition in the other direction is the identity functor on (Q0, D). Since these functors do not change the targets of the objects when viewed as D-morphisms and do not change the morphism sets it is clear that this isomorphism intertwines the two functors (Q, D) ! D and (Q0, D) ! D. Hence as L(Q) is the limit of (Q, D) ! D it is also the limit of (Q0, D) ! D and thus Q0is an iso-filtration. We therefore have a functor C: KD! -!Das required. 25 It is clear from this construction that if we start with an object in -!Dthen all we do is replace the underlying object in D by an isomorphic one (with a specified isomorphism), whence the composition CB is naturally isomorphic to the identity functor. It is also clear from the construction that the natural transformation from the forgetful functor to L underlies a natural transformation from the identity on DKto the composition BC. It is obvious that these natural transformations produce the adjunction_as stated. |__| Definition 3.17 We shall refer to the functor C: KD! -!Das the filtered pro- jective limit functor. We have already noted that the discrete filtration functor, D: D ! .KD, fact* *ors through DK; it is equally easy to see that it factors through -!D. For comparis* *on, K K -! I: D ! D does not factor through D , and there is little point in considering K the composition CI. 3.4 Categorical Properties We wish to determine the categorical properties of the various categories of fi* *l- tered objects. Certain results on .KD, KD, and -!Dhave depended on the categori* *cal .KK properties of D. We shall want to work with categories such as D and so we need to know whether the various properties of D lift to, say, DK. We shall not K.K consider categories such as D where the second type of filtration is more restr* *ic- tive than the first. We also want to be able to apply the results of section 2 and therefore we need to know other categorical properties to ensure that these apply. In summary, we want to know the following. 1. Conditions on D to ensure that DKis an extremally co-well-powered (ex- tremal epi, mono) category with a terminal object. 2. Conditions on D to ensure that -!Dis a complete extremally co-well-pow- ered (extremal epi, mono) category. 3. Conditions on D to ensure that -!Dis co-complete. 4. How to form products and (finite) co-products in each of DKand -!D. Let us state all the conditions on D that we need so that they are collected in one place. We assume that D is 1. complete, 2. co-complete, 3. an (extremal epi, mono) category, and 4. extremally co-well-powered. 26 Let us illustrate the various functors that we have. We denote the forgetful functor .KD! D by U, though we shall still use the notation |Q| for U(Q). We shall also find it useful-to!have a notation for the forgetful functor -!D! D so we denote this by Y: D ! D. Let us write the discrete filtration functor D as a functor into -!Drather than .KD. We have the following (non-commuting!) diagram. -!D___B____//_OOKS____//_.K oo______D_goo______D_gOOOO OOOOOCOOO R OOOO|OL|OOOO |||| |OOOOYOOOI |U| | OOOOOO || fflffl|OOO''fflffl|OOOD| D D We have the following identities and adjunctions RS = 1, CB ~=1, YD = 1, UI = 1, Y = USB USR = U, LBC ~=L, D --pY, C --pB, S --pR, SBD --pU, U--pI. The functors B, S, D, and I are fully faithful; U is faithful, whence also R and Y are faithful. The category -!Dis a reflective subcategory of DK. We shall use some results from [HS73 , xX] to transfer results from DKto -!D. These results refer to reflective subcategories that are also full subcategories and closed under isomorphisms. These conditions are satisfied by -!Dinside DK. We note in passing that in what follows we are essentially proving that KDis topological over D. This is mildly reassuring since our intention was to model a specific type of topological space. Let us start by defining push forward filtrations. Lemma 3.18 Let {QI} be a class of objects in DK. Let (D, fI) be a sink for the underlying class of objects in D. Then there is a reduced projective filtratio* *n, say Q0, on D such that K 0 1. each D-morphism fi:|Qi| ! D lifts to a DK-morphism fi:Qi! Q , and 2. if Q00is an object in DKand h: D ! |Q00| is a D-morphism then h lifts to 9K 0 00 00 a DK-morphism h :Q ! Q if and only if each hfi:|Qi| ! |Q | lifts to a DK-morphism. In the case that the class of objects in DKhas only one element, say Q, and so the sink is just f :|Q| ! D then we shall refer to the resulting object in DK as the push forward of Q along f and write it as f*(Q). When the class of objects in DKis empty, clearly the discrete filtration has the required properties. Proof.We define Q0 as follows: it consists of all D-morphisms, g, with source D such that for each i 2 I, gfi is in Qi. Firstly, Q0 is not empty as D has a terminal object and so the terminal morphism from D to this is in Q0. 27 Secondly, Q0is directed. To see this, suppose that g1: D ! D1 and g2: D ! D2 are in Q0. Consider the D-morphism (g1 x g2) : D ! D1 x D2 (which exists as D is complete). If we can show that this lies in Q0 then we are done as it preceeds both g1 and g2. Let i 2 I. As Qi is directed and saturated, (g1fix g2fi) is in Qi. By the functorality of products, this is (g1 x g2) fi. As this holds for all i, (g1 x g2) is in Q0. Thirdly, Q0is saturated. To see this, suppose that g1 is in Q0and g2 = kg1. Then for i 2 I, g1fi is in Qi and so as this is saturated, kg1fi is in Qi. Hence g2 is in Q0. We therefore have a projective filtration on D. We shall now show that it is reduced. Let g :D ! D1 be in Q0. As D is an (extremal epi, mono) category it has an (extremal epi, mono)-factorisation g = me with intervening object, D2, in D , say. Let i 2 I. The D-morphism gfi:|Qi| ! D1 is in Qiand so since Qiis reduced it factors through an extremal epimorphism in Qi, say gfi= hiei with intervening object, D3, in D and ei in Qi. We therefore have a commutative diagram in D. |Qi|__fi__//D__e__//D2 __ |ei| |g|__m__ fflffl|hifflffl|""__ D3 _____//_D1 As D is an (extremal epi, mono) category, we can find a D-morphism D3 ! D2 which fits into this diagram. Hence efi is in Qi. Thus e is in Q0 which is now shown to be reduced. K By construction the D-morphisms fi:|Qi| ! D lift to D -morphisms Qi ! Q0. Let h: D ! |Q00| be a D-morphism. Let q00be an element of Q00. Then q00h is in Q0if and only if q00hfi is in Qi for all i 2 I. Hence h*(Q00) Q0if and_* *only if (hfi)*(Q00) Qi for all i 2 I. |__| We already have the notion of pull back filtrations in the not-necessarily- reduced case and it is easy to see that this generalises. Lemma 3.19 Let {QI} be a class of objects in DK. Let (D, fI) be a source for the underlying class of objects in D. Then there is a reduced projective filtra* *tion, say Q0, on D such that K 0 1. each D-morphism fi:D ! |Qi| lifts to a DK-morphism fi:Q ! Qi, and 2. if Q00is an object in DKand h: |Q00| ! D is a D-morphism then h lifts to 9K 00 0 00 a DK-morphism h :Q ! Q if and only if each fih: |Q | ! |Qi| lifts to a DK-morphism. In the case that the class of objects in DK has one element we obtain the reduced pull back filtration, R(f*(Q)). If the class of objects in KDis empty, * *we obtain the reduced indiscrete filtration on D. Proof.Each D-morphism fi:D ! |Qi| defines a pull back filtration fi*(Qi) on D. The union of these is a subclass of D#. As D is complete, we can find a smallest projective filtration containing this subclass: first we include all f* *inite 28 products to ensure that it is directed and non-empty (via the empty product) and then we saturate it. Considered as an object in .KD, this clearly has the required properties. We then apply R to this object in .KD. The properties then follow from the fact that R is right adjoint to the inclusion S: KD! .KDand that both of these_ functors cover the identity on D. |__| As the forgetful functor DK! D is faithful and has both a left and a right adjoint, it reflects and preserves monomorphisms and epimorphisms. Extremal epimorphisms are easy to characterise. Corollary 3.20 A DK-morphism f :Q1 ! Q2 is an extremal epimorphism if and only if |f| is an extremal epimorphism in D and Q2 = |f|*(Q1). Proof.Let us show the "only if" part first, so that we suppose that f :Q1 ! Q2 is an extremal epimorphism. We need to show two things: that |f| is an extremal epimorphism and that Q2 = |f|*(Q1). As the forgetful functor KD! D has a right adjoint, namely the reduced indis- crete functor, |f|: |Q1| ! |Q2| is an epimorphism. Suppose that |f| = mg with m a monomorphism. Let D be the intervening object in D. We put the reduced pull back filtration on D via m. Then by lemma 3.19, m lifts to a DK-morphism mK:R(m*(Q2)) ! Q2 and g lifts to a DK-morphism gK:Q1 ! R(m*(Q2)) with mKKg= f. As the forgetful functor is faithful, mK is a monomorphism. Since f is an extremal epimorphism, mK is thus an isomorphism. Hence m = |mK| is an isomorphism. Thus |f| is an extremal epimorphism. From lemma 3.18, the identity on |Q2| underlies a DK-morphism |f|*(Q1) ! Q2 and so f factorises as Q1 ! |f|*(Q1) ! Q2. As the forgetful functor is faithful, the DK-morphism |f|*(Q1) ! Q2 is a monomorphism. Hence as f is an extremal epimorphism, it is an isomorphism. As it covers the identity on |Q2|, |f|*(Q1) and Q2 must in fact be the same projective filtrations. Now let us show the "if" part. Let f :Q1 ! Q2 be such that |f| is an extremal epimorphism and Q2 = f*(Q1). As the forgetful functor is faithful, f is per force an epimorphism. Suppose that we have a factorisation of f as mg with m a monomorphism and intervening object, Q0, in DK. We therefore have a factorisation of |f| as |m||g|. As the forgetful functor has a left adjo* *int, |m| is a monomorphism. Hence, as |f| is an extremal epimorphism, |m| is an isomorphism. Consider the D-morphism |m|-1|f|: |Q1| ! |Q0|. This simplifies to |g| which lifts to a DK-morphism. Hence, by lemma 3.18 since Q2 = |f|*(Q1), |m|-1 lifts to a DK-morphism and thus, as the forgetful functor is faithful,_m * *is an isomorphism. Hence f is an extremal epimorphism. |__| This characterisation helps us prove the required extremallity properties of DK. Corollary 3.21 DK is an (extremal epi, mono) category. Proof.Let f :Q1 ! Q2 be a DK-morphism. The D-morphism |f| has an (ex- tremal epi, mono)-factorisation, say |f| = me. We can lift this to a factorisat* *ion 29 of f as Q1 ! e*(Q1) ! Q2. This is an (extremal epi, mono) factorisation by corollary 3.20 and as monomorphisms lift to monomorphisms. To show uniqueness it is sufficient to show that we have the diagonal prop- erty. That is, suppose that we have a commutative square in DK, Q1 _____//Q2 |e| |m| fflffl| fflffl| Q3 _____//Q4 with e an extremal epimorphism and m a monomorphism. The underlying square in D has the same properties and thus there is a (unique) D-morphism h: |Q3| ! |Q2| which fits into the corresponding diagram in D. Then he lifts to a DK-morphism so as Q3 = e*(Q1), h lifts to a DK-morphism and thus DKhas the (extremal epi, mono)-diagonalisation property. Thus KDis an (extremal_epi, mono)-category. |__| Corollary 3.22 DK is extremally co-well-powered. Proof.Let Q be an object in KD. From the characterisation of extremal epimor- phisms in DKwe see that the forgetful functor DK! D defines a bijection from the class of isomorphism classes of extremal epimorphisms in DKwith source Q to the class of isomorphism classes of extremal epimorphisms in D with source |Q|. Hence the property of being extremally co-well-powered lifts from D to DK. |___| From lemmas 3.18 and 3.19 we can deduce that DK is both complete and co-complete. Proposition 3.23 DK is complete and co-complete. Proof.This is a standard proof. We form limits and co-limits in DKby forming the limit or co-limit first in D and then putting the appropriate reduced filtr* *ation on the resulting object: the pull back filtration for the limit and the push_fo* *rward for the co-limit. |__| We therefore have all our required properties of DK. We now turn to -!D. Completeness and co-completeness follow directly from proposition 3.23. Corollary 3.24 -!Dis complete and co-complete. Proof.It is a reflective, full subcategory of KDwhich is closed under isomorphi* *sm. Hence by [HS73 , corollaries 36.14,18], both completeness and co-completeness descend from DKto -!D. |___| Note that co-limits in -!Dare not simply the co-limits of the corresponding family in KDand therefore do not necessarily project down to the corresponding co-limit in D. Rather we form the co-limit in DKand then apply the functor C to the resulting object. 30 Extremal epimorphisms in -!Dare-more!complicated than in DK and so we need to work harder to prove that D is an (extremal epi, mono) category and is extremally co-well-powered. Proposition 3.25 -!Dis an (extremal epi, mono) category. Proof.The proof of the factorisation property is an adaptation of the standard proof that every morphism in a complete well-powered category is (extremal epi, mono)-factorisable. Let f :K1 ! K2 be a -!D-morphism. We consider the class of all factorisa- tions f = mh with m a monomorphism. This is not empty as it contains the factorisation (1, f). This is a quasi-ordered class with (m1, h1) (m2, h2) if there is a -!D-morphism from the source of m1 to the source of m2 making the obvious diagram commute. If this morphism exists, it is obviously unique and a-monomorphism.!There is an obvious functor from this quasi-ordered class to D . We wish to show that this functor has a limit. We shall do this by showing that the class has a small initial subclass. Let |f| = mD eD be the (extremal e* *pi, mono)-factorisation of |f| in D with intervening object, D, in D. Let f = mh be-a!factorisation of f with m a monomorphism and intervening object, K, in-! D . Then |f| = |m||h| is a factorisation of |f|. As the forgetful functor D ! D has a left adjoint (the discrete filtration functor) it takes monomorphisms to monomorphisms and so |m| is a monomorphism. Hence as D is an (extremal epi, mono) category there is a D-morphism g :D ! |K| making the following diagram commute. |h| |K1| _____//_|K|<< yy |eD|gyyyy |m||| fflffl|yyyfflffl|mD D _____//_|K2| We pull back and reduce the projective filtration K2 on |K2| via mD to one on D; let us write this as Q. The above diagram then lifts to DKwith Q in the lower left corner. Via the adjunction C --pB, the DK-morphisms with source Q factor through the natural morphism Q ! BC(Q). We therefore have the diagram in DK B(h) B(K1) __________________//B(K) -!g)t99t |9KeD| B( tttt |B(m)| fflffl|' tttB(--!mDfflffl|) Q _______//_BC(Q)____//B(K2), where ' is the canonical morphism. We claim that -!gis a monomorphism. It is necessary and sufficient to show that |-!g| is a monomorphism as the forgetful functor is faithful and has a left adjoint. Thus let d1, d2: D0! |C(Q)| be D-morphisms such that |-!g|d1 = |-!g|d2. As |C(Q)| is the underlying object in D of an object in -!D, it is a limit and * *so d1 and d2 are completely determined by their compositions with the morphisms into the appropriate family. This family is the projective filtration Q and for* * q in Q we have a D-morphism "q:|C(Q)| ! Qq such that "q|'| = q. 31 The projective filtration Q was defined as the reduction of the pull back of K2 via mD . It therefore has an initial family as follows: for each k in K2 the fact that D is an (extremal epi, mono) category implies the existence of a commutative diagram, unique up to canonical isomorphism, mD D _____//_|K2| kD|| |k| fflffl|mkfflffl| Dk _____//_Kk, with kD an extremal epimorphism and mk a monomorphism. The family kD is initial for Q. We therefore have the following commutative diagram. |'| m|-!g| D _____//E|C(Q)|___//|K2| EE EEE "kD| |k kD EE""Efflffl|| fflffl|| Dk __mk__//_Kk Thus d1 and d2 are completely determined by the compositions "kDdi for k in K2. Now as they satisfy |-!g|d1 = |-!g|d2, for each k in K2 we have km|-!g|d1 = km|-!g|d2 whence mk"kDd1 = mk"kDd2. As mk is a monomorphism, we therefore have "kDd1 = "kDd2 and thus d1 = d2. Hence |-!g| is a monomorphism and thus so is -!g.-! The D -morphism --!mDis equal to m-!gand so is a monomorphism. Let -!eD:K1 ! C(Q) be the -!D-morphism which, under the inclusion -!D KD, maps to 'e9KD. The factorisation of f in -!Das --!mD-!eDis thus in our class of fact* *orisations and it preceeds the factorisation f = mh. The key property of this factorisation is that the morphism -!eDis obtained * *by applying the functor C to a KD-morphism 9KeD:B(K1) ! Q such that |e9KD| = eD ; note that eD depends only on f and not on the factorisation that we were trying to dominate. This factorisation is therefore completely determined by the reduced projective filtration Q on D. As D is extremally co-well-powered, the class of all reduced projective fil- trations on a specified object in D is actually a set. To see this, observe that a reduced projective filtration is completely determined by its subclass of ex- tremal epimorphisms. Moreover, this subclass is closed under isomorphism and so is a union of equivalence classes of extremal epimorphisms, whence a reduced projective filtration is completely determined by an element of the power class of the class of equivalence classes of extremal epimorphisms emanating from the original object in D. As D is extremally co-well-powered, the class of equiva- lence classes of extremal epimorphisms is actually a set and so its power set is also a set. Hence the class of reduced projective filtrations on a given object* * in D is a set. -! Thus-our!class of factorisations of the D -morphism, f, has an initial set a* *nd so, as D is complete, has a limit. The proof that this limit is an (extremal epi, mono)-factorisation of f proceeds exactly as in the analogous proof for a morphism in a complete well-powered category. See, for example, [HS73 , 17.8,17.16]. 32 The proof that -!Dis in fact an (extremal epi, mono) category now follows __ since it is complete. See, for example, [HS73 , 34.1]. |_* *_| It is worth pointing out that even if D were, in fact, a (regular epi, mono) category then it would not necessarily be true that -!Dwas a (regular epi, mono) category. Buried within the above proof are all the necessary pieces to prove the final property that we want. Proposition 3.26 -!Dis extremally co-well-powered. Proof.It is easy to see from the proof that -!Dis an (extremal epi, mono) categ* *ory that every extremal epimorphism is obtained by applying C to a DK-morphism of the form B(K) ! Q with underlying D-morphism an extremal epimorphism (this is a necessary, but not sufficient, condition). To specify the isomorphism class of an extremal epimorphism in the category of iso-filtered objects of D it is therefore sufficient to specify the isomorphism class of the corresponding extremal epimorphism in D and a-reduced!projective filtration on the target. Hence for an object, K, in D , the class of isomorphism classes of extremal epimorphisms emanating from K injects into a K Iso(D D) Iso(ex epid:|K|!D) where Iso(ex epid: |K| ! D) is the class of isomorphism classes of extremal epimorphisms with source |K| and Iso(DKD) is the class of isomorphism classes of the fibre category of DK! D at D; that is, the class of reduced projective filtrations on D. As D is extremally co-well-powered, all of the classes-in!this co-product are small. Hence the co-product is small and thus D is extremally co-well-_ powered. |__| 3.5 The Canonical Filtration Functor * *-!-! In-this!section we shall construct a right adjoint to the forgetful functor Y: * *D ! D . The indiscrete filtration functor, and its reduction, provide right adjoint* *s to the forgetful functors .KD! D and DK! D but in general the forgetful functor -!D! D does not have a right adjoint. In the specific case -!-!D! -!D, however, we are able to construct one. It is a straightforward adaptation of the filtrat* *ion functor for a filtered category as described in example 3.10(2). We assume that D has the properties of section 3.4. It is simple to adapt the definition of example 3.10(2) to define a functor -!D! ..K-!D. For an object, Q, in .KDthe canonical filtration was defined by ta* *king the projective filtration on Q with initial subclass the family of .KD-morphisms ..Kq:Q ! SBD(Qq) (recall that we now regard D as a functor into -!D). Similarly, ..K-! we define a functor -!D!-D!by taking-the!projective filtration on K with initial subclass the family of D -morphisms k :K ! D(Kk). 33 -!-! Proposition 3.27 This functor factors through D . Proof.We need to show first that the filtration defined above is reduced and then that it is an iso-filtration. ..K Let K be an object in -!Dand let P be the resulting object in -!D. By -! construction, an initial subclass for the filtration P is given by taking the D* * - morphisms -! k :K ! D(Kk) for k in K. It is obvious that we may refine this further and take k in an init* *ial subclass of K. In particular, we can take those-k!which are extremal epimorphisms. Let k be one of these. We wish to show that k is an extremal-epimorphism.!Since k is an epimorphism-and!the forgetful functor is faithful, k is an epimorphism. Now let k = mf be a-factorisation!with m a monomorphism. Let K0 be the intervening object in D . By applying the forgetful-functor!we obtain a factorisation of k as |m||f|. The forgetful functor D ! D has a left adjoint, namely the discrete filtration functor, so preserves monomorphisms. Hence |m| is a monomorphism and thus, as k is an extremal epimorphism, |m| is an isomorphism. Its inverse is a D-morphism Kk ! |K0| and hence lifts to a -!D-morphism D(Kk) ! K0. This-lift!is inverse to m because the forgetful functor -!D! D is faithful. Hen* *ce k is an extremal epimorphism and so P is a reduced projective filtration. To show that it is an iso-filtration we need to show that the limit of P is isomorphic to-K!via the canonical morphism. This follows from the description of limits in D : they are formed by taking the underlying limit in D and putting the reduced pull back filtration on the resulting object in D. In our case, the resulting object in D is (naturally isomorphic to) |K| and it is obvious that_t* *he_ reduced pull back filtration of the family |K| ! |D(Kk)| is again K. |__| Definition 3.28 We shall refer to the functor-defined!above as the canonical filtration functor and denote it by X: -!D! -!D. Let us now show that this functor is the required adjoint. -!-! Proposition 3.29 The canonical filtration functor-X:!-!D! D is right adjoint and right inverse to the forgetful functor Y: -!D! -!D. -!-! -! Proof.In this proof, Y will refer exclusively to the forgetful-functor!D ! D and we will use the notation |-| for the forgetful functor D ! D. By construction, YX is the identity functor on -!D. This provides the natural transformation which will-be!the co-unit of the adjunction. Let L be an object in -!D. Both L and XY(L) are iso-filtrations on the same underlying object in -!D, namely Y(L). We shall show that XY(L) L as projective filtrations on Y(L). By construction, XY(L) is the projective filtration on Y(L) with initial subclass -! k :Y(L) ! D(Y(L)k) 34 for k in Y(L). Now L is an iso-filtration on Y(L) and so the canonical morphism Y(L) ! lim-Ll l is-an!isomorphism of objects in -!D, where the limit is over l in L. Limits in D are formed by taking the corresponding limit in D and then putting the reduced pull back filtration on the resulting object. Thus an initial subclass of Y(L) consists of the extremal epimorphisms coming from the (extremal epi, mono)-factorisation of D-morphisms of the form 0 |Y(L)| |l|-!|Ll| l-!Ll,l0 for l in L and l0 in Ll. We can assume that l and l0 are themselves in initial subclasses of their respective filtrations and so we can assume that they are extremal epimorphisms in their respective categories. Let us show that the composition l0|l| is itself an extremal epimorphism. It* * is an epimorphism because l0and |l| are both epimorphisms. Let l0|l| = me be the (extremal epi, mono)-factorisation in D of l0|l| with intervening object, D, in* * D. By the above, e is in Y(L) and thus lifts to a -!D-morphism "e:Y(L) ! D(D). The D-morphism m: D ! Ll,l0lifts to a -!D-morphism D(m): D(D) ! D(Ll,l0). This is again a monomorphism-as!the forgetful functor is faithful. Since l0 is * *in Ll it also lifts to a D -morphism "l0:Ll ! D(ll,l0). As lifts of D-morphisms to -!D-morphisms are unique, we therefore have the following commutative diagram in -!D. Y(L) ___l___//_Ll "e|| |"l0| fflffl|D(m) fflffl| D(D) _____//D(Ll,l0) Since -!Dis an (extremal epi, mono) category there is a -!D-morphism g :Ll ! D(D) which fits into the above diagram. Applying the forgetful functor we see that l0 = m|g|. As l0 is an extremal epimorphism we see that m is an isomorphism. Hence l0|l| is isomorphic to e and thus is an extremal epimorphism. Thus an initial subclass for Y(L) is the family of D-morphisms 0 |Y(L)| |l|-!|Ll| l-!Ll,l0 with l in L and l0in Ll. Thus an initial subclass of XY(L) consists of the fami* *ly of -!D-morphisms "l0 |L| l-!Ll-! D(Ll,l0). As each of these factors through l it is in L. Hence XY(L) L and so the -!-! identity -!D-morphism on Y(L) lifts to a D -morphism L ! XY(L). These lifts fit-together!to define a natural transformation of functors from the-identity!on -!Dto XY: all the necessary diagrams commute because they do in D . This will be the unit of our adjunction. 35 Our functors, Y and-X,!are both lifts of the identity functor on -!Dalong the forgetful functor -!D! -!D, one lifting the source and the other the target. The natural transformations, YX ! I and I ! XY, are both lifts of the -! identity natural transformation I ! I on -!D. Therefore-for!an object, L, in -!D and object, K, in -!Dthe forgetful functor Y: -!D! -!Dinduces a commutative diagram of morphisms of hom-sets. -!D(Y(L), K)_____//-!-! ____//_-! D (L, X(K)) D (Y(L), K) | | =|| Y|| |=| -! fflffl|= -! fflffl|= -! |fflffl D (Y(L), K) _____//D(Y(L), K)____//_D(Y(L), K) As the forgetful functor is faithful, the morphisms in the upper line are_isomo* *r-_ phisms and hence X is right adjoint to Y. |__| 3.6 Lifts of Functors In this section we shall examine certain lifts of functors involving filtered c* *ate- gories. The two lifts that we shall consider are described in the next definiti* *on. Recall that for a functor G: D ! E we defined, in section 3.1, a corresponding .K .K .K functor G :D ! E. The conditions that we impose on our categories-in!the following theorems are not minimal. Recall that we write Y: -!D! -!Dfor the forgetful functor. Definition 3.30 Let D and E be complete, co-complete,-extremally!co-well- powered, (extremal epi, mono) categories. Let G: D ! E be a covariant functor. -! -! -! We define G : D ! E by -! .K -! X -!-!B 9K-!S ..K-!.KG.KRK C -! G : = CRG SBX: D -! D -! D -! D -! E-! E-! E. -! -! -! Let H: E ! -!Dbe a covariant functor. We define H : E ! D by -! .K -! B K S .K.KH..K-!R9K-!C-!-!Y-! H : = YCRH SB: E -! E-! E-! D -! D -! D -! D . We trust that there will be no confusion with using the same notation for two different constructions. Note that in these definitions-we!have two differe* *nt instances-of!various9functors.KIn the definition of G the inclusion functor B * *is -! -! K from -!Dto -!Dwhereas in the definition of H it is from E to E. We trust that this also will not cause confusion. We wish to prove two results about these constructions. The first gives a condition whereby the first construction is associative. The second relates to adjunctions. Theorem 3.31 Let D, E, and F be complete, co-complete,-extremally!co-well--! powered, (extremal epi, mono) categories. Let G: D ! E and H: E ! F be covariant functors. Then-if!H-preserves!monomorphisms, there is a natural isomorphism of functors D ! F --!-! -!-! HG ~=H G 36 satisfying the obvious coherence for triples. Proof.Let us expand out the two sides to make clear what we have to prove. --!-! ....K-! .K..K-! HG = CRHG SBX = CRH G SBX, -!-! .K .K .K -! H G = CRH SBXCRG SBX = CRH SBXG . ..K-! -! From this it is clear that the first step is to compare G SBX with SBXG . ..K-! ..K-! -! These are functors -!D! E . Using the fact that |G (-)| = G (|-|) we see that ..K-! -! -! ..K-!-! both G SBX and SBXG are lifts of G along the forgetful functor E ! E-.! To compare these lifts we need to examine the resulting filtrations on G (K) for an object, K, in -!D. Firstly, let us establish some notation. Applying R to an object, Q, in .KDd* *oes not change the underlying object in D, it merely alters the filtration. An init* *ial class for R(Q) is given by taking the extremal epimorphisms which come from the (extremal epi, mono)-factorisations of the D-morphisms in Q. As before, for q :|Q| ! Qq in Q let us write 9Kq:|Q| ! 9KQq for the corresponding extremal-epimorphism.!-! Let K be an object in D . Let us examine G (K). An initial subclass of the filtration X(K) is given by the family K ! D(Kk) for k in K. Here and henceforth we will suppress the label for the morphism as the notation rapidly becomes unwieldy; in each case it will be the obvious morphism derived from k. .K An initial subclass of the filtration GSBX(K) is thus given by the family G(K) ! GD(Kk) (3.1) .K and of RG SBX(K) by 9 9 9 9 K G(K) ! GD(Kk) . -! To get G (K) we apply C which replaces the source of these morphisms by the -! appropriate limit. This produces an initial family for G (K) consisting of the ..K-! E -morphisms -! 9 9 9 9 K |G (K)| ! GD(Kk) . -! We can read off from this an initial family for SBXG (K). It consists of the -!E-morphisms -! 9 9 9 9 K G (K) ! D(GD(Kk) ). -! By applying the above as far as (3.1)with G in place of G we can also read off ..K-! -! an initial family for G SBX(K). It consists of the E -morphisms -! -! G (K) ! G D(Kk). 37 -! It is obvious-from!the construction that G D = DGD. Thus we can rewrite the above E -morphisms as -! G(K) ! DGD(Kk). 9 9 9 9 K The objects, GD(Kk) , in E are defined (up to canonical isomorphism) by the (extremal epi, mono)-factorisations 9 9 9 9 K G(K) ! GD(Kk) ! GD(Kk). -! From the construction of G as a limit (via the functor C) we see that we can -! -! replace G(K) by |G (K)| in this. Thus, as projective filtrations on G (K), ..K-! -! G SBX(K) is contained in SBXG (K). From this we deduce that the identity -! ..K-! on G (K) lifts to a E -morphism -! ..K-! SBXG (K) ! G SBX(K). -! -! As this is a lift of the identity on G it defines a natural transformation SBXG* * ! ..K-! G SBX. We claim that this becomes a natural isomorphism-after!reduction. To prove this claim, consider the following diagrams in E . -! _______// G (K) DGD(Kk)88r | rrrrr | rrr fflffl|rr 9 9 9 9 K DGD(Kk) -! ______//_99 9 9 9_K___// G (K) DGD(Kk) DGD(Kk)99s | ssss | sss fflffl| sss 9 99999999KK ssss DGD(Kk) sssss | ssss | sss fflffl|ss 9 9 9 9 K DGD(Kk) The second is derived from the first by taking the (extremal epi, mono)- factorisations of the vertical and horizontal morphisms. The diagonal arrow in each diagram is a monomorphism since it is D applied to a monomorphism and D preserves monomorphisms. Thus by the diagonalisation property of an (extremal epi, mono) category, there is an isomorphism 999999999KK9 9 9 9 9 K DGD(Kk) ~=DGD(Kk) fitting in to the second diagram. 38 Thus the natural transformation -! ..K-! SBXG ! G SBX induces a natural isomorphism -! ..K-! RSBXG ! RG SBX. This is still not quite what is needed as there is an occurence of the funct* *or R which is not present in the expansions of the two functors that we wish to compare. What we shall now see is that we could easily insert R at the appropriate juncture without changing the result; actually we insert SR. There is a natural transformation SR ! 1 coming from the adjunction .K .K .K S --pR. Applying RH yields a natural transformation RH SR ! RH . We ..K-! wish to show that this is a natural isomorphism. Let Y be an object in E. Using notation as above, we have the following initial classes of the projective filtrations 9 9 9K .K H(|Y |) ! H(Yy) for RH (Y ), 9 999KK .K H(|Y |) ! H(Yy) for RH SR(Y ) and the natural transformation is the identity on H(|Y |). This natural transfo* *r- mation comes from the fact that the diagonalisation property of an (extremal epi, mono) category allows us to add in the required morphism on the inner diagonal of this diagram: H(|Y |)____//999KH(Yy)//_H(Yy) <