IIsbell conjugacy and the reflexive completion
Tom Avery Tom Leinster ∗ Abstract
The reflexive completion of a category consists of the
Set -valued func-tors on it that are canonically isomorphic to their double conjugate. Afterreviewing both this construction and Isbell conjugacy itself, we give newexamples and revisit Isbell’s main results from 1960 in a modern cate-gorical context. We establish the sense in which reflexive completion isfunctorial, and find conditions under which two categories have equivalentreflexive completions. We describe the relationship between the reflexiveand Cauchy completions, determine exactly which limits and colimits existin an arbitrary reflexive completion, and make precise the sense in whichthe reflexive completion of a category is the intersection of the categoriesof covariant and contravariant functors on it.
Contents
Isbell conjugacy inhabits the same basic level of category theory as the Yonedalemma, springing from the most primitive concepts of the subject: category,functor and natural transformation. It can be understood as follows.Let A be a small category. Any functor X : A op → Set gives rise to a newfunctor X (cid:48) : A op → Set defined by X (cid:48) ( a ) = [ A op , Set ]( A ( − , a ) , X ) , ∗ School of Mathematics, University of Edinburgh, Edinburgh EH9 3FD, Scotland;[email protected]. Supported by a Leverhulme Trust Research Fellowship. a r X i v : . [ m a t h . C T ] F e b nd so, in principle, an infinite sequence X, X (cid:48) , X (cid:48)(cid:48) , . . . of functors A op → Set .Of course, they are all canonically isomorphic, by the Yoneda lemma. But X also gives rise to a functor X ∨ : A →
Set , its
Isbell conjugate , defined by X ∨ ( a ) = [ A op , Set ]( X, A ( − , a )) . (1)The same construction with A in place of A op produces from X ∨ a furtherfunctor X ∨∨ : A op → Set , and so on, giving an infinite sequence
X, X ∨ , X ∨∨ , . . . of functors on A with alternating variances. Although it makes no sense to askwhether X ∨ is isomorphic to X (their types being different), one can ask whether X ∨∨ ∼ = X . This is false in general. Thus, there is nontrivial structure.The conjugacy operations define an adjunction between [ A op , Set ] and[ A , Set ] op , so that [ A op , Set ]( X, Y ∨ ) ∼ = [ A , Set ]( Y, X ∨ )naturally in X : A op → Set and Y : A →
Set . The unit and counit of theadjunction are canonical maps X → X ∨∨ and Y → Y ∨∨ , and a covariant orcontravariant functor on A is said to be reflexive if the canonical map to itsdouble conjugate is an isomorphism.The reflexive completion R ( A ) of A is the category of reflexive functorson A (covariant or contravariant; it makes no difference). Put another way, R ( A ) is the invariant part of the conjugacy adjunction. It contains A , sincerepresentables are reflexive. Its properties are the main subject of this work.The reflexive completion is very natural category-theoretically, but cate-gories of reflexive objects also appear in other parts of mathematics. That is,there are many notions of duality in mathematics, in most instances there is acanonical map η X : X → X ∗∗ from each object X to its double dual, and specialattention is paid to those X for which η X is an isomorphism. For example, inlinear algebra, the vector spaces X with this property are the finite-dimensionalones, and in functional analysis, there is a highly developed theory of reflexivityfor Banach spaces and topological vector spaces. Content of the paper
We begin with the definition of conjugacy on small categories, giving several characterizations of the conjugacy operations andmany examples (Sections 2 and 3). Defining conjugacy on an arbitrary categoryis more delicate, and we review and use the notion of small functor (Section 4).This allows us to state the definition of the reflexive completion of an arbitrarycategory, and again, we give many examples (Sections 5 and 6).Up to here, there are no substantial theorems, but the examples provide somesurprises. For instance, the reflexive completion of a nontrivial group is simplythe group with initial and terminal objects adjoined—except when the group isof order 2, in which case it is something more complicated (for reasons relatedto the fact that 2 + 2 = 2 ×
2; see Example 6.5). There is also a finite monoidwhose reflexive completion is not even small, a fact due to Isbell (Examples 6.8and 8.7). Other examples involve the Dedekind–MacNeille completion of anordered set (Examples 6.10 and 6.11) and the tight span of a metric space (atthe end of Section 6).The second half of the paper develops the theory, as follows.Section 7 collects necessary results on dense and adequate functors. (SeeDefinition 7.10 and Remark 7.11 for this terminology.) Many of them are stan-dard, but we address points about set-theoretic size that do not seem to have2 ( A ) (cid:98) A A (cid:98) R ( A ) AA complete cocompletereflexively completeCauchy complete(a) (b)Figure 1: (a) Completions of a category A : the Cauchy completion A , reflexivecompletion R ( A ), free completions (cid:98) A and A (cid:98) with respect to small colimits andsmall limits, and Isbell envelope I ( A ); (b) classes of complete categories.previously been considered. Using the results of Section 7, we give a uniquecharacterization of the reflexive completion that sharpens a result of Isbell’s(Theorem 8.4).Reflexive completion is functorial (Section 9), but only with respect to a verylimited class of functors: the small-adequate ones. It is often the case that thefunctor R ( F ) : R ( B ) → R ( A ) induced by a functor F : A → B is an equivalence.For example, using work of Day and Lack on small functors together with theresults on size just mentioned, we show that R ( F ) is always an equivalence if B is either small or both complete and cocomplete (Corollary 9.8).Our study of functoriality naturally recovers Isbell’s result that reflexivecompletion is idempotent: R ( R ( A )) (cid:39) R ( A ). A category is reflexively complete if it is the reflexive completion of some category, or equivalently if every reflexivefunctor on it is representable.Reflexive completion has certain formal resemblances to Cauchy completion,but the reflexive completion is typically bigger (Figure 1). The relationship isanalysed in Section 10.A reflexively complete category has absolute (co)limits, and if it is the re-flexive completion of a small category then it has initial and terminal objectstoo, but these are all the limits and colimits that it generally has (Section 11).The case of ordered sets, where the reflexive (Dedekind–MacNeille) completionhas all (co)limits, is atypical. On the other hand, it is true that a complete orcocomplete category is reflexively complete (Figure 1).Informally, one can understand R ( A ) as the intersection (cid:98) A∩A (cid:98) , where (cid:98) A and A (cid:98) are the free completions of A under small colimits and small limits. (If A issmall then (cid:98) A = [ A op , Set ] and A (cid:98) = [ A , Set ] op .) Section 12 formalizes this idea,reviewing the definition of the Isbell envelope I ( A ) of a category and provingthat the square in Figure 1(a) is a pullback in the bicategorical sense.We work with categories enriched over a suitable monoidal category V inSections 2–6, then restrict to V = Set from Section 7. While some of the laterresults are particular to V = Set (such as Theorem 11.6 on limits), others canbe generalized to any V . To avoid complicating the presentation, we have notspecified exactly which results generalize, but we have tried to choose proofsthat make any generalization transparent.3 elationship to Isbell’s paper Although there are many new results inthis work, some parts are accounts of results first proved in Isbell’s remarkablepaper [9], and the reader may ask what we bring that Isbell did not. There areseveral answers.First, Isbell’s paper was extraordinarily early. He submitted it in mid-1959,only the year after the publication of Kan’s paper introducing adjoint functors.What we now know about category theory can be used to give shape to Isbell’soriginal arguments. In Grothendieck’s metaphor [19], the rising sea of generalcategory theory has made the hammer and chisel unnecessary.Second, Isbell worked only with full subcategories, where we use arbitraryfunctors. It is true that we will often need to assume our functors to be fulland faithful, so that up to equivalence, they are indeed inclusions of full sub-categories. Nevertheless, the functor-based approach has the benefits of beingequivalence-invariant and of revealing exactly where the full and faithful hypoth-esis is needed. Ulmer emphasized that many naturally occurring dense functorsare not full and faithful (Example 7.3), and the theory of dense and adequatefunctors should be developed as far as possible without that assumption.Third, we modernize some aspects, including the treatment of set-theoreticsize. Isbell used a size constraint on
Set -valued functors that he called proper-ness and Freyd later called pettiness (Remark 4.5). It now seems clear that themost natural such notion is that of small functor, which extends smoothly tothe enriched context and is what we use here.Finally, Isbell simply omitted several proofs; we provide them.
Terminology
Isbell conjugacy has sometimes been called Isbell duality (as inDi Liberti [4]), but that term has also been used for a different purpose entirely(as in Barr, Kennison and Raphael [1]). Reflexive completion has also beenstudied under the name of Isbell completion (as in Willerton [21]).
Conventions
Usually, and always in declarations such as ‘let A be a category’,the word ‘category’ means locally small category. However, we will sometimesform categories such as [ A op , Set ] that are not locally small. For us, the words small and large refer to sets and proper classes.The symbol × denotes both product and copower, so that when S is a setand a is an object of some category, S × a = (cid:96) s ∈ S a .A category is (co)complete when it admits small (co)limits. Certain aspects of conjugacy are simpler for small categories. In this section,we review several descriptions and characterizations of conjugacy on small cate-gories, all previously known. Our categories will be enriched in a complete andcocomplete symmetric monoidal closed category V . Henceforth, we will usuallyabbreviate ‘ V -category’ to ‘category’, and similarly for functors, adjunctions,etc.; all are understood to be V -enriched.Let A be a small category. The (Isbell) conjugate of a functor X : A op →V is the functor X ∨ : A → V defined by X ∨ ( a ) = [ A op , V ]( X, A ( − , a ))4 a ∈ A ). With A op in place of A , this means that the conjugate of a functor Y : A → V is the functor Y ∨ : A op → V defined by Y ∨ ( a ) = [ A , V ]( Y, A ( a, − )) . Remark 2.1
Every functor Z from a small category to V has a single, unam-biguous, conjugate Z ∨ , which is the same whether Z is regarded as a covariantfunctor on its domain or a contravariant functor on the opposite of its domain.Conjugacy defines a pair of functors[ A op , V ] ∨ (cid:47) (cid:47) [ A , V ] op ∨ (cid:111) (cid:111) . (2)Given X : A op → V and Y : A → V , define X (cid:2) Y : A op ⊗ A → V ( a, b ) (cid:55)→ X ( a ) ⊗ Y ( b ) . (3)One verifies that[ A op , V ]( X, Y ∨ ) ∼ = [ A op ⊗ A , V ]( X (cid:2) Y, Hom A ) ∼ = [ A , V ]( Y, X ∨ ) (4)naturally in X and Y . In particular, the conjugacy functors (2) define a con-travariant adjunction on the right.Evidently A ( − , a ) ∨ ∼ = A ( a, − ) , A ( a, − ) ∨ ∼ = A ( − , a )naturally in a ∈ A . Thus, both triangles in the diagram[ A op , V ] ∨ (cid:45) (cid:45) [ A , V ] op ∨⊥ (cid:109) (cid:109) A H • (cid:98) (cid:98) H • (cid:60) (cid:60) (5)commute, where H • and H • are the two Yoneda embeddings. This propertycharacterizes conjugacy: Lemma 2.2
Isbell conjugacy is the unique adjunction such that both trianglesin (5) commute up to isomorphism.
Proof
Let P : [ A op , V ] → [ A , V ] op be a left adjoint satisfying P ◦ H • ∼ = H • . Byhypothesis, P ( X ) ∼ = X ∨ when X is representable. But every object of [ A op , V ]is a small colimit of representables (as A is small), and both P and ( ) ∨ preservecolimits (being left adjoints), so P ∼ = ( ) ∨ . (cid:3) Conjugacy can also be described as a nerve-realization adjunction. Anyfunctor F : A → E induces a nerve functor N F : E → [ A op , V ] E (cid:55)→ A ( F − , E ) . E is cocomplete, the nerve functor has a left adjoint, sometimes calledthe realization functor of F (after the case where F is the standard embeddingof the simplex category ∆ into Top ). It is the left Kan extension of F alongthe Yoneda embedding H • .Taking F to be H • : A → [ A , V ] op , we thus obtain a pair of adjoint functorsbetween [ A op , V ] and [ A , V ] op . This is the conjugacy adjunction. For example,in diagram (5), the functor ( ) ∨ : [ A op , V ] → [ A , V ] op is the left Kan extensionof H • along H • .Yet another derivation of conjugacy uses profunctors. Our convention is thatfor small categories A and B , a profunctor B + −→ A is a functor A op ⊗ B → V ,and the composite of profunctors Q : C + −→ B and P : B + −→ A is denoted by P (cid:12) Q : C + −→ A .The operation of composition with a profunctor, on either the left or theright, has a right adjoint. Indeed, given profunctors C (cid:31) R (cid:54) (cid:54) (cid:31) Q (cid:47) (cid:47) B (cid:31) P (cid:47) (cid:47) A , there are profunctors C (cid:31) [ P,R ] > (cid:47) (cid:47) B (cid:31) [ Q,R ] < (cid:47) (cid:47) A defined by [ P, R ] > ( b, c ) = [ A op , V ]( P ( − , b ) , R ( − , c )) , [ Q, R ] < ( a, b ) = [ C , V ]( Q ( b, − ) , R ( a, − )) , which satisfy the adjoint correspondences P → [ Q, R ] < P (cid:12) Q → RQ → [ P, R ] > (6)Now take B to be the unit V -category I , with C = A and R = Hom A . Theprofunctors P and Q are functors X : A op → V and Y : A → V , respectively.Then [
P, R ] > = X ∨ and [ Q, R ] < = Y ∨ , while P (cid:12) Q = X (cid:2) Y , and the generaladjointness relations (6) reduce to the conjugacy relations (4).Finally, conjugates can be described as Kan extensions or lifts in the bicat-egory V - Prof of V -profunctors. Let X : A op → V . There is a canonical naturaltransformation A (cid:31) X ∨ (cid:47) (cid:47) (cid:0) Hom A ⇐ ε X (cid:31) (cid:31) (cid:95) X (cid:15) (cid:15) A (7)whose ( a, b )-component X ( a ) ⊗ [ A op , V ]( X, A ( − , b )) → A ( a, b )6s defined in the case V = Set by ( x, ξ ) (cid:55)→ ξ b ( x ), and by the obvious general-ization for arbitrary V . Equivalently, using the second of the isomorphisms (4), ε X is the map X (cid:2) X ∨ → Hom A corresponding to the identity on X ∨ .The result is that ε X exhibits X ∨ as the right Kan lift of Hom A through X in V - Prof . (That is, the pair ( X ∨ , ε X ) is terminal of its type.) This followsfrom the second adjointness relation in (6) on taking P = X and R = Hom A .Dually, for Y : A → V , a similarly defined transformation ε Y : Y ∨ (cid:12) Y → Hom A exhibits Y ∨ as the right Kan extension of Hom A along Y in V - Prof . We list some examples of conjugacy, beginning with unenriched categories.
Example 3.1
Let A be a small discrete category, Y : A →
Set , and a ∈ A .Then Y ∨ ( a ) = (cid:40) Y ( b ) = ∅ for all b (cid:54) = a ∅ otherwise.Thus, writing supp Y = { b ∈ A : Y ( b ) is nonempty } , (8)we have Y ∨ ∼ = Y ∼ = 0 A ( − , a ) if supp Y = { a } Example 3.2
Let G be a group, seen as a one-object category. A functor X : G op → Set is a right G -set, and the unique representable such functor is G r , the set G acted on by G by right multiplication. Thus, X ∨ is the left G -setof G -equivariant maps G → G r . We now compute X ∨ explicitly.First suppose that the G -set X is nonempty, transitive and free (for g ∈ G ,if xg = x for some x then g = 1). Then X ∼ = G r , so X ∨ is isomorphic to G (cid:96) ,the set G acted on by the group G by left multiplication.Next suppose that X is nonempty and transitive but not free. Choose x ∈ X and 1 (cid:54) = g ∈ G such that xg = x . Any equivariant α : X → G r satisfies α ( x ) = α ( xg ) = α ( x ) g , a contradiction since g (cid:54) = 1. Hence X ∨ = ∅ .Finally, take an arbitrary G -set X . It is a coproduct (cid:80) i ∈ I X i of nonemptytransitive G -sets, so by adjointness, X ∨ = (cid:81) i ∈ I X ∨ i . By the previous paragraph, X ∨ is empty unless every orbit X i is free, or equivalently unless X is free. If X is free then X is the copower I × G r and X ∨ is the power G I(cid:96) . But I ∼ = X/G , so X ∨ ∼ = (cid:40) G X/G(cid:96) if X is free ∅ otherwise. Example 3.3
Let A be a partially ordered set regarded as a category, and let X : A op → Set . The set supp X ⊆ A (equation (8)) is downwards closed, andwhen X = A ( − , a ), it is ↓ a = { b ∈ A : b ≤ a } . Now X ∨ ( a ) ∼ = (cid:40) X ⊆ ↓ a ∅ otherwise .
7f course, the dual result also holds, involving ↑ a = { b ∈ A : b ≥ a } .A Set -valued functor on a category is subterminal if it is a subobject ofthe terminal functor, or equivalently if all of its values are empty or singletons.Subterminal functors A → Set correspond via supp to upwards closed subsets of A . The conjugate of any functor X : A op → Set is subterminal, correspondingto the upwards closed set of upper bounds of supp X in A . Example 3.4
Write = (0 →
1) with min as monoidal structure. A small -category A is a partially ordered set (up to equivalence), and a -functor X : A op → amounts to a downwards closed subset of A , namely, { a ∈ A : X ( a ) = 1 } . Dually, a -functor A → is an upwards closed subset of A .From this perspective, the conjugacy adjunction is as follows: for a down-wards closed set X ⊆ A , the upwards closed set X ∨ is the set of upper boundsof X , and dually. Example 3.5
Write Ab for the category of abelian groups. A one-object Ab -category R is a ring, and an Ab -functor R op → Ab is a right R -module. Theunique representable on R is R r , the abelian group R regarded as a right R -module. Thus, the conjugate of a right module M is M ∨ = Mod R ( M, R r )with the left module structure induced by the left action of R on itself. When R is a field, M ∨ is the dual of the vector space M . Example 3.6
Consider the ordered set ([0 , ∞ ] , ≥ ) with its additive monoidalstructure. This is a monoidal closed category, the internal hom [ x, y ] being thetruncated difference y ·− x = max { y − x, } . Lawvere [16] famously observed that a [0 , ∞ ]-category is a generalized metricspace, ‘generalized’ in that distances need not be symmetric or finite, and dis-tinct points can be distance 0 apart.Let A = ( A, d ) be a generalized metric space. A [0 , ∞ ]-functor A op → [0 , ∞ ]is a function f : A → [0 , ∞ ] such that f ( a ) ·− f ( b ) ≤ d ( a, b )for all a, b ∈ A . Its conjugate f ∨ : A → [0 , ∞ ] is defined by f ∨ ( a ) = sup b ∈ A (cid:0) d ( b, a ) ·− f ( b ) (cid:1) . To define conjugacy on a general category requires more delicacy than on asmall category. The reader who wants to get on to the reflexive completioncan ignore this section for now. However, because of the phenomenon notedin Example 6.8, the theory of the reflexive completion ultimately requires thismore general definition of conjugacy: it is not possible to confine oneself to smallcategories only.The following example shows that the definition of conjugacy for small cat-egories cannot be extended verbatim to large categories.8 xample 4.1
Let C be a proper class. Let A be the category obtained byadjoining to the discrete category C a further object z and maps p c , p c : z → c for each c ∈ C . Let Y : A →
Set be the functor defined by Y ( a ) = (cid:40) a ∈ C ∅ if a = z. A natural transformation Y → A ( z, − ) is a choice of element of { p c , p c } for each c ∈ C . There is a proper class of such transformations, so there is no Set -valuedfunctor Y ∨ : A op → Set defined by Y ∨ ( a ) = [ A , Set ]( Y, A ( a, − )).Since not every functor has a conjugate, we restrict ourselves to a classof functors that do. These are the small functors introduced by Ulmer ([20],Remark 2.29). We briefly review them now, referring to Day and Lack [3] fordetails.Again we work over a complete and cocomplete symmetric monoidal closedcategory V , understanding all categories, functors, etc., to be V -enriched.For a category A , a functor A → V is small if it can expressed as a smallcolimit of representables, or equivalently if it is the left Kan extension of itsrestriction to some small full subcategory of A , or equivalently if it is the leftKan extension of some V -valued functor on some small category B along somefunctor B → A . Example 4.2
When A is small, every functor A → V is small.
Example 4.3
Taking V = Set , the constant functor 1 on a large discrete cat-egory is not small; nor is the functor Y of Example 4.1. Example 4.4
For later purposes, let us consider an ordered class A and asubterminal functor X : A op → Set (as defined in Example 3.3). Then X issmall if and only if there is some small K ⊆ supp X such that for all a ∈ supp X ,the poset K ∩ ↑ a is connected (and in particular, nonempty). This follows fromthe definition of a small functor as one that is the left Kan extension of itsrestriction to some small full subcategory.For arbitrary functors X, X (cid:48) : A op → V , the V -natural transformations X → X (cid:48) do not always define an object of V , as Example 4.1 shows in the case V = Set . But when X is small, they do: it is the (possibly large) end V - Nat ( X, X (cid:48) ) = (cid:90) a [ X ( a ) , X (cid:48) ( a )] ∈ V . (9)To see that this end exists, first note that by smallness of X , we can choosea small full subcategory C of A such that X is the left Kan extension ofits restriction to C . Since C is small and V has small limits, the functor V -category [ C op , V ] exists, and the universal property of Kan extensions impliesthat [ C op , V ] (cid:0) X | C , X (cid:48) | C (cid:1) is the end (9).In particular, the small functors A op → V form a V -category (cid:98) A . We alsowrite A (cid:98) for the opposite of the V -category of small functors A → V . When A is small, (cid:98) A = [ A op , V ] , A (cid:98) = [ A , V ] op . A is large, the right-hand sides are in general undefined as V -categories.In the case V = Set , the right-hand sides can be interpreted as categories thatare not locally small, but typically (cid:98) A (cid:40) [ A op , Set ] , A (cid:98) (cid:40) [ A , Set ] op . A small colimit of small V -valued functors is small, so the V -category (cid:98) A hassmall colimits, computed pointwise. Indeed, it is the free cocompletion of A :the Yoneda embedding A (cid:44) → (cid:98) A is the initial functor (in a 2-categorical sense)from A to a category with small colimits. Dually, A (cid:98) is the free completion of A . Remark 4.5
Isbell used a different size condition, defining a
Set -valued functorto be proper if it admits an epimorphism from a small coproduct of representa-bles ([9], Section 1). (Freyd later called such functors ‘petty’ [6].) Propernessis a weaker condition than smallness, but the universal properties of (cid:98) A and A (cid:98) make smallness a natural choice, and it generalizes smoothly to arbitrary V . Definition 4.6
A functor F : A → B is representably small if for each b ∈ B ,the functor N F ( b ) = B ( F − , b ) : A op → V is small, and corepresentably small if for each b ∈ B , N F ( b ) = B ( b, F − ) : A → V is small. (This is dual to the convention in Section 8 of Day and Lack [3].)Thus, a representably small functor F : A → B induces a nerve functor N F : B → (cid:98) A , and dually. Lemma 4.7
Let A F −→ B G −→ C be functors. If F and G are representablysmall then so is GF , and dually for corepresentably small. Proof
Suppose that F and G are representably small, and let c ∈ C . Wemust show that C ( GF − , c ) is small. By hypothesis, C ( G − , c ) is a small colimitof representables, say C ( G − , c ) = W ∗ B ( − , D ) where I is a small category, W : I op → V and D : I → B . Then C ( GF − , c ) = W ∗ B ( F − , D ), which byhypothesis is a small colimit of small functors, hence small. (cid:3) This completes our review of smallness. Now let A be a category. The conjugate of a small functor X : A op → V is the functor X ∨ : A → V definedby X ∨ ( a ) = (cid:98) A ( X, A ( − , a )) . Since (cid:100) A op = (cid:0) A (cid:98) (cid:1) op , this implies that the conjugate of a small functor Y : A → V is the functor Y ∨ : A op → V defined by Y ∨ ( a ) = A (cid:98) ( Y, A ( a, − )) . The conjugate of a small functor need not be small:
Example 4.8
Let A be a discrete category on a proper class of objects. Thesmall functors Y : A →
Set are precisely those such that supp Y is small. So theinitial (empty) functor 0 : A →
Set is small, but its conjugate is the terminalfunctor 1, which is not small. 10hen V = Set , conjugacy defines functors( ) ∨ : (cid:98) A → [ A , Set ] op , ( ) ∨ : A (cid:98) → [ A op , Set ] , whose codomains are in general not locally small. For a general V and A ,conjugacy is still contravariantly functorial in X and Y , but there are no V -categories [ A , V ] op and [ A op , V ] to act as the codomains of ( ) ∨ . So it no longermakes sense to speak of a conjugacy adjunction . However, we do have thefollowing. Lemma 4.9
Let A be a category. Then V - Nat ( X, Y ∨ ) ∼ = V - Nat ( Y, X ∨ ) naturally in X ∈ (cid:98) A and Y ∈ A (cid:98) . Since X and Y are small, each side of the claimed isomorphism is a well-defined object of V (equation (9)). Proof
It is routine to verify that each side is naturally isomorphic to V - Nat ( X (cid:2) Y, Hom A ), where (cid:2) was defined in (3). (cid:3) The isomorphism of Lemma 4.9 gives rise in the usual way to a canonicalmap η X : X → X ∨∨ whenever X : A op → V is a small functor such that X ∨ isalso small. Dually, for any small functor Y : A → V with small conjugate, thereis a canonical map η Y : Y → Y ∨∨ . Remark 4.10
The reuse of the letter η is not an abuse, in that η X is the samewhether X is regarded as a contravariant functor on A or a covariant functoron A op . (Compare Remark 2.1.)In the case V = Set , the unit transformation η can be described explicitlyas follows. Let X : A op → Set be a small functor with small conjugate. Let a ∈ A and x ∈ X ( a ). Then η X,a ( x ) ∈ X ∨∨ ( a ) is the natural transformation η X,a ( x ) : X ∨ → A ( a, − )that evaluates at x : its component at b ∈ A is the function (cid:98) A ( X, A ( − , b )) → A ( a, b ) ξ (cid:55)→ ξ a ( x ) . Remark 4.11
Define a category A to be gentle if (cid:98) A is complete and A (cid:98) iscocomplete. Small categories are certainly gentle. Day and Lack proved that (cid:98) A is complete if A is (Corollary 3.9 of [3]), so by duality, any complete andcocomplete category is also gentle. On the other hand, a large discrete category A is not gentle, as (cid:98) A has no terminal object.For a gentle category A , the conjugate of a small functor on A is againsmall, so that conjugacy defines a genuine adjunction between (cid:98) A and A (cid:98) . Thiswas shown by Day and Lack in Section 9 of [3].11 The reflexive completion
The reflexive completion of a category was first defined by Isbell (Section 1of [9]), for unenriched categories. We consider it for categories enriched in acomplete and cocomplete symmetric monoidal closed category V , beginning withthe case of small V -categories and then generalizing to arbitrary V -categories.For small categories over V = Set , our definition is precisely Isbell’s. For generalcategories over
Set , there is the set-theoretic difference that our definition usessmall functors where his used proper functors (Remark 4.5).Recall that every adjunction C F (cid:41) (cid:41) D G ⊥ (cid:104) (cid:104) between V -categories restricts canonically to an equivalence between full sub-categories of C and D . The subcategory of C consists of those objects c for whichthe unit map c → GF c (in the underlying category of C ) is an isomorphism,and dually for D . We call either of these equivalent subcategories the invariantpart of the adjunction.The reflexive completion R ( A ) of a small V -category A is the invariantpart of the conjugacy adjunction (cid:98) A ∨ (cid:41) (cid:41) A (cid:98) . ∨⊥ (cid:105) (cid:105) When R ( A ) is seen as a full subcategory of (cid:98) A , it consists of those functors X : A op → V such that the unit map η X : X → X ∨∨ is an isomorphism; suchfunctors X are called reflexive . Dually, R ( A ) can be seen as the full subcate-gory of A (cid:98) consisting of the reflexive functors A → V .Now let A be an any V -category, not necessarily small. To define reflexivityof a functor X on A , we need X ∨∨ to be defined, so we ask that X and X ∨ aresmall. Definition 5.1
A functor X : A op → V is reflexive if X ∈ (cid:98) A , X ∨ ∈ A (cid:98) , andthe canonical natural transformation η X : X → X ∨∨ is an isomorphism.This extends the earlier definition for small A . Although conjugacy for anarbitrary A does not define an adjunction between (cid:98) A and A (cid:98) , it still inducesan equivalence between the full subcategory of (cid:98) A consisting of the reflexivefunctors A op → V and the full subcategory of A (cid:98) consisting of the reflexivefunctors A → V . The reflexive completion R ( A ) of A is either of theseequivalent categories.In the case V = Set , we have the concrete description of η X given afterLemma 4.9. It implies that X ∈ (cid:98) A is reflexive if and only if for each a ∈ A ,every element of X ∨∨ ( a ) is evaluation at a unique element of X ( a ). Remark 5.2
By Remark 4.10, whether a V -valued functor is reflexive does notdepend on whether it is considered as a covariant functor on its domain or acontravariant functor on the opposite of its domain. It follows that R ( A op ) (cid:39)R ( A ) op for all V -categories A . 12 xample 5.3 Let A be a V -category. For each a ∈ A , A ( − , a ) ∨ ∼ = A ( a, − ) , A ( a, − ) ∨ ∼ = A ( − , a ) , and the unit map A ( − , a ) → A ( − , a ) ∨∨ is an isomorphism. Hence representa-bles are reflexive.The image of the Yoneda embedding A (cid:44) → (cid:98) A therefore lies in R ( A ), whenthe latter is seen as a subcategory of (cid:98) A . A dual statement holds for A (cid:98) . Thereis, then, an unambiguous Yoneda embeddding J A : A → R ( A )such that the diagram of full and faithful functors (cid:98) AA (cid:37) (cid:5) (cid:51) (cid:51) (cid:31) (cid:127) J A (cid:47) (cid:47) (cid:25) (cid:121) (cid:43) (cid:43) R ( A ) (cid:42) (cid:10) (cid:55) (cid:55) (cid:20) (cid:116) (cid:39) (cid:39) A (cid:98) (10)commutes. We begin with unenriched examples.
Example 6.1
Let denote the empty category. Then [ op , Set ] and [ , Set ] op are both the terminal category , so R ( ) (cid:39) . In particular, the reflexivecompletion of a category need not be equivalent to its Cauchy completion. Example 6.2
The conjugacy adjunction for the terminal category consistsof the functors Set (cid:29)
Set op with constant value 1, giving R ( ) (cid:39) . Example 6.3
Let A be a small discrete category with at least two objects. ByExample 3.1, for Y : A →
Set , Y ∨∨ ∼ = Y ∼ = 0 A ( a, − ) if supp Y = { a } A →
Set are those that are initial, terminal orrepresentable. (Contrast this with Example 6.2, in which the initial functor → Set is not reflexive.) It follows that R ( A ) is A with initial and terminalobjects adjoined. Example 6.4
Now let A be a large discrete category. As observed in Exam-ple 4.8, the conjugate of the small functor 0 : A →
Set is the non-small functor1 : A op → Set . Hence neither 0 nor 1 is reflexive. The same argument as inExample 6.3 then shows that the only reflexive functors on A are the repre-sentables. Thus, unlike in the small case, the reflexive completion of a largediscrete category is itself. 13 xample 6.5 Let G be a group, regarded as a one-object category. If G istrivial then R ( G ) (cid:39) by Example 6.2. Suppose not.Let X be a right G -set. In Example 3.2, we showed that X ∨ ∼ = (cid:40) G X/G(cid:96) if X is free ∅ otherwise,and of course a similar result holds for left G -sets. If X is not free then X ∨∨ = 1,so the only non-free reflexive G -set is the terminal G -set 1.Now suppose that X is free. If X is empty then X ∨∨ = 1 ∨ = ∅ (using thenontriviality of G in the second equality), so X is reflexive. Assume now that X is nonempty, write S = X/G , and choose s ∈ S . The left G -action on X ∨ ∼ = G S(cid:96) is free, and each orbit contains exactly one element whose s -component is theidentity element of G , so X ∨ has | G | | S \{ s }| orbits. Writing | S | − | S \ { s }| ,we conclude that X ∨ is a free G -set with | X ∨ /G | = | G | | S |− . Repeating the argument in the dual situation then gives | X ∨∨ /G | = | G | | G | | S |− − . Hence if X is reflexive, | S | = | G | | G | | S |− − . By elementary arguments, this implies that | S | = 1 (in which case X is repre-sentable) or | G | = | S | = 2. Hence when | G | >
2, the only reflexive right G -setsare ∅ , 1 and G r .The remaining case is where G is the two-element group and X is the free G -set on two generators. A direct calculation shows that G r + G r ∼ = G r × G r in [ G op , Set ]. Since G is abelian, the same is true for G (cid:96) . Now using the adjointproperty of conjugates, X ∨ ∼ = ( G r + G r ) ∨ ∼ = G (cid:96) × G (cid:96) ∼ = G (cid:96) + G (cid:96) ,X ∨∨ ∼ = ( G (cid:96) + G (cid:96) ) ∨ ∼ = G r × G r ∼ = G r + G r , giving X ∨∨ ∼ = X . We claim that X is reflexive, that is, the unit map η X : X → X ∨∨ is an isomorphism. One of the triangle identities for the con-jugacy adjunction implies that η W ∨ is split monic for any W : G → Set . But X ∼ = ( X ∨ ) ∨ , so η X is an injection between finite sets of the same cardinality,hence bijective, hence an isomorphism.In summary, the reflexive completion of a group G is as follows: • if | G | = 1 then R ( G ) ∼ = G ; • if | G | = 2 then R ( G ) is the full subcategory of the category of G -setsconsisting of the initial G -set, the terminal G -set, the representable G -set G r , and the four-element G -set G r + G r ∼ = G r × G r .14 if | G | > R ( G ) is the full subcategory of the category of right G -setsconsisting of the initial G -set, the terminal G -set, and the representable G -set. It is equivalent to G with initial and terminal objects adjoined. Remark 6.6
For a
Set -valued functor X , the sequence X, X ∨ , X ∨∨ , X ∨∨∨ , . . . need not ever repeat itself. For consider functors on the three-element group C . By Example 6.5, the conjugate of the free C -set n × C on n elements is3 n − × C . Since 3 n − > n for all n ≥
2, no two of the iterated conjugates of2 × C are isomorphic. Example 6.7
Let M be the two-element commutative monoid whose non-identity element e is idempotent. A covariant or contravariant functor from M to Set amounts to a set X together with an idempotent endomorphism f .The representable such functor corresponds to the set M = { id , e } together withthe endomorphism with constant value e .Given a pair ( X, f ), write X = im f and X = X \ X . A routine calculationshows that (cid:12)(cid:12)(cid:0) X ∨∨ (cid:1) (cid:12)(cid:12) = 1 and (cid:12)(cid:12)(cid:0) X ∨∨ (cid:1) (cid:12)(cid:12) = 2 | X | − − . So if X is reflexive then | X | = 1 and | X | = 2 | X | − −
1, and the latter equationimplies that | X | is 0 or 1. Thus, X is either the representable functor or theterminal functor on M . Both are indeed reflexive.The reflexive completion of M is, therefore, the full subcategory of the cate-gory of M -sets consisting of M itself and 1. This is the free category on a splitepimorphism, and is the same as the Cauchy completion of M . Example 6.8
There is a 7-element monoid whose reflexive completion is large.In particular, the reflexive completion of a finite category need not even besmall. This is an example of Isbell to which we return in Example 8.7.
Remark 6.9
The Cauchy completion of a category A has a well-known con-crete description: an object is an object a ∈ A together with an idempotent e on a , a map ( a, e ) → ( a (cid:48) , e (cid:48) ) is a map f : a → a (cid:48) in A such that e (cid:48) f e = f ,composition is as in A , and the identity on ( a, e ) is e . It follows that the Cauchycompletion of a finite or small category is finite or small, respectively. Exam-ple 6.8 implies that the reflexive completion can have no very similar description. Example 6.10
Let A be a poset, regarded as a category. The conjugacy ad-junction of A , when restricted to subterminal functors (Example 3.3), is anadjunction { downwards closed subsets of A } ↑ (cid:47) (cid:47) { upwards closed subsets of A } op . ↓ (cid:111) (cid:111) Here both sets are ordered by inclusion, and ↑ S and ↓ S are the sets of upperand lower bounds of a subset S ⊆ A , respectively. The adjointness states that X ⊆ ↓ Y ⇐⇒ Y ⊆ ↑ X .A reflexive functor A op → Set amounts to a downwards closed set X ⊆ A such that X = ↓ ↑ X . The reflexive completion R ( A ) is the set of such subsets X , ordered by inclusion. This is the Dedekind–MacNeille completion of the15oset A (Definition 7.38 of Davey and Priestley [2]). For example, the reflexivecompletion of ( Q , ≤ ) is the extended real line [ −∞ , ∞ ].The Dedekind–MacNeille completion of a poset is always a complete lattice.(And conversely, any complete lattice is the Dedekind–MacNeille completion ofitself.) However, the reflexive completion of an arbitrary small category is farfrom complete, as Theorem 11.6 shows. Example 6.11
Now let A be a poset, but regarded as a category enriched in . By Example 3.4 and the same argument as in Example 6.10, the reflexivecompletion of A as a -category is again its Dedekind–MacNeille completion. Example 6.12
As in Example 3.5, let V = Ab , let R be a ring, and let M bea right R -module. The double conjugate of M is its double dual M ∨∨ = R Mod ( Mod R ( M, R r ) , R (cid:96) ) , so the notion of reflexive functor on R coincides with the algebraists’ notionof reflexive module (as in Section 5.1.7 of McConnell and Robson [18]). Hencethe reflexive completion of a ring, viewed as a one-object Ab -category, is itscategory of reflexive modules. In particular, the reflexive completion of a field k is the category of finite-dimensional k -vector spaces.The rest of this section concerns the case V = [0 , ∞ ] (Example 3.6), sum-marizing results from Willerton’s analysis [21] of the reflexive completion of ageneralized metric space A . It follows from Example 3.6 that R ( A ) is the set ofdistance-decreasing functions f : A op → [0 , ∞ ] such that f ( c ) = sup b ∈ A inf a ∈ A (cid:16) d ( c, b ) ·− (cid:0) d ( a, b ) ·− f ( a ) (cid:1)(cid:17) for all c ∈ A , with metric d ( f, g ) = sup a (cid:0) g ( a ) ·− f ( a ) (cid:1) . (11)The reflexive completion of A consists of the [0 , ∞ ]-valued functors equal totheir double conjugate. But when A is symmetric, covariant and contravariantfunctors on A can be identified, so we can form the set T ( A ) = { distance-decreasing functions f : A → [0 , ∞ ] such that f ∨ = f } of [0 , ∞ ]-valued functors equal to their single conjugate. Then T ( A ), metrizedas in equation (11), is called the tight span of A . Evidently A ⊆ T ( A ) ⊆ R ( A ).Both inclusions can be strict, as the following example shows. Example 6.13
Take the symmetric metric space { , D } consisting of twopoints distance D apart (Figure 2). It follows from the description above that R ( { , D } ) is the set [0 , D ] with metric d (cid:0) ( t , t ) , ( u , u ) (cid:1) = max { u ·− t , u ·− t } (Example 3.2.1 of [21]). The Yoneda embedding { , D } → [0 , D ] is given by0 (cid:55)→ (0 , D ) and D (cid:55)→ ( D, T ( { , D } ) is the interval [0 , D ] withits usual metric, and embeds in R ( { , D } ) as shown.160 , D ) ( D, T ( { , D } ) R ( { , D } )Figure 2: The symmetric metric space { , D } embedded into its tight span T ( { , D } ) and reflexive completion R ( { , D } ) (Example 6.13).The tight span construction has been discovered independently several times,as recounted in the introduction of [21]. That the form given here is equivalentto other forms of the definition was established by Dress (Section 1 of [5]). Thefirst to discover it was Isbell [10], who called it the ‘injective envelope’, T ( A )being the smallest injective metric space containing A . But Isbell does not seemto have noticed the connection with Isbell conjugacy.The tight span is only defined for symmetric spaces, and is itself symmetric(not quite trivially). On the other hand, the reflexive completion of a symmetricspace need not be symmetric, as the two-point example shows. Theorem 4.1.1of [21] states that the tight span is the symmetric part of the reflexive comple-tion: Theorem 6.14 (Willerton)
Let A be a symmetric metric space. Then thetight span T ( A ) is the largest symmetric subspace of R ( A ) containing A . Here ‘largest’ is with respect to inclusion. A nontrivial corollary is that R ( A ) has a largest symmetric subspace containing A .Finally, reflexive completion of metric spaces has arisen in fields far fromcategory theory. Pursuing a project in combinatorial optimization, Hirai andKoichi [8] defined the ‘directed tight span’ of a generalized metric space. AsWillerton showed (Theorem 4.2.1 of [21]), it is exactly the reflexive completion. Here we gather results on dense and adequate functors that will be used later tocharacterize the reflexive completion. Some can be found in Isbell’s or Ulmer’sfoundational papers [9, 20] or in Chapter 5 of Kelly [13], while some appear tobe new. For the rest of this work, we restrict to unenriched categories, althoughmuch of what we do can be extended to the enriched setting.
Definition 7.1
A functor F : A → B is dense if its nerve functor N F : B → [ A op , Set ] (Definition 4.6) is full and faithful, and codense if N F : B → [ A , Set ] op is full and faithful.A functor is small-dense if dense and representably small, and small-codense if codense and corepresentably small. Remark 7.2
It is a curious fact (not needed here) that while F is dense if andonly if N F is full and faithful, F is full and faithful if and only if N F is dense.17 xample 7.3 Let F : A → B be a functor with a right adjoint G . It is verywell known that G is full and faithful if and only if the counit of the adjunctionis an isomorphism. Less well known, but already pointed out by Ulmer in 1968(Theorem 1.13 of [20]), is that these conditions are also equivalent to F beingdense. Thus, any functor F with a full and faithful right adjoint is dense.Indeed, F is small-dense, since B ( F − , b ) is representable for each b ∈ B .A functor F : A → B is small-dense when N F is full, faithful, and takesvalues in the category (cid:98) A of small functors A op → Set . Then B embeds fullyinto (cid:98) A . When A is small, every dense functor A → B is small-dense.
Example 7.4
The archetypal dense functor is the Yoneda embedding A (cid:44) → [ A op , Set ], and the archetypal small-dense functor is the Yoneda embedding A (cid:44) → (cid:98) A .A standard result is that F : A → B is dense if and only if every object of B is canonically a colimit of objects of the form F a ; that is, for each b ∈ B , thecanonical cocone on the diagram( F ↓ b ) pr −→ A F −→ B with vertex b is a colimit cocone (Section X.6 of Mac Lane [17]). In terms ofcoends, this means that b ∼ = (cid:90) a B ( F a, b ) × F a.
That the Yoneda embedding is dense gives the density formula X ∼ = (cid:90) a X ( a ) × A ( − , a )for functors X : A op → Set . Example 7.5
When B is an ordered set (or class) and A ⊆ B with the inducedorder, the inclusion A (cid:44) → B is dense if and only if it is join-dense : every elementof B is a join of elements of A . Lemma 7.6
Let F : A → B be a small-dense functor. Then for each b ∈ B ,there is a small diagram ( a i ) i ∈I in A such that b ∼ = colim i F a i . Proof
Let b ∈ B . Since F is representably small, we can choose a small diagram( a i ) in A such that B ( F − , b ) ∼ = colim i A ( − , a i ). Then by density of F and thedensity formula, b ∼ = (cid:90) a B ( F a, b ) × F a ∼ = (cid:90) a,i A ( a, a i ) × F a ∼ = (cid:90) i F a i . (cid:3) We will be especially interested in the (co)density of functors that are full andfaithful. Up to equivalence, such functors are inclusions of full subcategories,which are called (co)dense or small-(co)dense subcategories if the inclusionfunctor has the corresponding property.We now state some basic lemmas on full, faithful and dense functors, begin-ning with one whose proof is immediate from the definitions.18 emma 7.7 Let F : A → B be a full and faithful functor. Then the composite A F −→ B N F −→ [ A op , Set ] is isomorphic to the Yoneda embedding. (cid:3) The next lemma follows from the corollary to Proposition 3.2 in Lambek [15],but we include the short proof for completeness.
Lemma 7.8
Every full and faithful dense functor preserves all (not just small)limits.
Proof
We use Lemma 7.7. Since N F is full and faithful, it reflects arbitrarylimits; but the Yoneda embedding preserves them, so F does too. (cid:3) The composite of two full and faithful dense functors need not be dense.Isbell gave one counterexample (paragraph 1.2 of [9]) and Kelly gave another(Section 5.2 of [13]). Nevertheless:
Lemma 7.9
Let A F −→ B G −→ C be dense functors, and suppose that G pre-serves arbitrary colimits. Then GF is dense. Proof
For c ∈ C , we have canonical isomorphisms c ∼ = (cid:90) b C ( Gb, c ) × Gb ( G is dense) ∼ = (cid:90) b C ( Gb, c ) × G (cid:18)(cid:90) a B ( F a, b ) × F a (cid:19) ( F is dense) ∼ = (cid:90) a,b C ( Gb, c ) × B ( F a, b ) × GF a ( G preserves colimits) ∼ = (cid:90) a C ( GF a, c ) × GF a (density formula),so GF is dense. (cid:3) Definition 7.10
A functor is adequate if it is full, faithful, dense and codense,and small-adequate if it is full, faithful, small-dense and small-codense.
Remark 7.11
Isbell’s foundational paper [9] considered adequacy only for fullsubcategories. Up to equivalence, this amounts to working only with functorsthat are full and faithful. For him, fullness and faithfulness were implicit as-sumptions rather than explicit hypotheses. He used ‘left/right adequate’ forwhat is now called dense/codense, and ‘adequate’ for dense and codense. Theword ‘dense’ was introduced later by Ulmer [20], who extended the theory toarbitrary functors.
Example 7.12
Let f : A → B be an order-preserving map between partiallyordered classes. It is full and faithful if and only if it is an order-embedding (thatis, reflects the order relation), and by Example 7.5, it is adequate if and onlyif it is also both join-dense and meet-dense. Small-adequacy means that eachelement of B is a join of some small family of elements of im f , and similarlyfor meets. 19 emma 7.13 The classes of adequate and small-adequate functors are eachclosed under composition.
Isbell proved an analogue of this result for properly adequate functors (state-ment 1.6 of [9]), using a different argument.
Proof
Let A F −→ B G −→ C be adequate functors. Then G preserves arbitrarycolimits by the dual of Lemma 7.8, so GF is dense by Lemma 7.9. Dually, GF is codense. So GF is adequate. If F and G are small-adequate then so is GF ,by Lemma 4.7. (cid:3) Lemma 7.14
For adequate functors A F (cid:47) (cid:47) B G (cid:47) (cid:47) G (cid:48) (cid:47) (cid:47) C , if GF ∼ = G (cid:48) F then G ∼ = G (cid:48) . Proof
We prove the stronger result that if F is codense and G and G (cid:48) are full,faithful and dense then GF ∼ = G (cid:48) F ⇒ G ∼ = G (cid:48) . Indeed, under these assumptions,Lemma 7.8 implies that G preserves all limits, so by codensity of F , Gb ∼ = G (cid:90) a [ B ( b, F a ) , F a ] ∼ = (cid:90) a [ B ( b, F a ) , GF a ]naturally in b ∈ B (where [ − , − ] denotes a power). The same holds for G (cid:48) , so if GF ∼ = G (cid:48) F then G ∼ = G (cid:48) . (cid:3) For full subcategories
A ⊆ B ⊆ C , if A is dense in C then both intermediateinclusions are dense. More generally: Lemma 7.15
Let A F −→ B G −→ C be functors, with G full and faithful.i. If GF is dense then so are F and G .ii. If GF is small-dense then so is F . That G is dense was asserted without proof in statement 1.1 of Isbell [9]. Proof
For (i), to prove that F is dense, note that the composite functor B G −→ C N GF −→ [ A op , Set ] b (cid:55)−→ Gb (cid:55)−→ C ( GF − , Gb ) ∼ = B ( F − , b )is isomorphic to N F . But both G and N GF are full and faithful, so N F is too.To prove that G is dense, let c ∈ C . Then c ∼ = (cid:90) a C ( GF a, c ) × GF a ( GF is dense) ∼ = (cid:90) a,b C ( Gb, c ) × B ( F a, b ) × GF a (density formula) ∼ = (cid:90) b C ( Gb, c ) × (cid:90) a C ( GF a, Gb ) × GF a ( G is full and faithful) ∼ = (cid:90) b C ( Gb, c ) × Gb ( GF is dense)naturally in c , as required.For (ii), suppose that GF is small-dense. We must prove that F is repre-sentably small. Let b ∈ B . Since G is full and faithful, B ( F − , b ) ∼ = C ( GF − , Gb ),which is small since GF is small-dense. (cid:3)
204 3 25 ∞ (3 , ∞ ) P Q C = P × Q Figure 3: The ordered classes of Example 7.19, with A ⊆ C shown in blue. Proposition 7.16
For every category A , the Yoneda embedding J A : A →R ( A ) is small-adequate. Proof
We refer to diagram (10) (p. 13). Certainly J A is full and faithful.Lemma 7.15(ii) applied to A J A −→ R ( A ) (cid:44) → (cid:98) A implies that J A is small-dense.By duality, it is also small-codense. (cid:3) Lemma 7.15 and its dual immediately imply:
Proposition 7.17
Let A F −→ B G −→ C be functors, with G full and faithful. If GF is small-adequate then G is adequate and F is small-adequate. (cid:3) Propositions 7.16 and 7.17 have the following corollary. It is implicit inSection 1 of Isbell [9], modulo the difference in size conditions (Remark 4.5).
Corollary 7.18 (Isbell)
Let A be a category. For any full subcategory B of R ( A ) containing the representables, the inclusion A (cid:44) → B is small-adequate. (cid:3) Proposition 7.17 does not conclude that G must be small -adequate. Indeed,it need not be. This apparently technical point becomes important later, sowe give both a counterexample and a sufficient condition for G to be small-adequate. Example 7.19
We exhibit ordered classes A ⊆ B ⊆ C such that A (cid:44) → C issmall-adequate (hence A (cid:44) → B is too, by Proposition 7.17) but B (cid:44) → C is not.Let P be the 5-element poset shown in Figure 3, and let Q be the orderedclass of ordinals with a greatest element ∞ adjoined. Put C = P × Q, B = C \ { (3 , ∞ ) } , A = C \ ( { } × Q ) , with the product order on C and the induced orders on A, B ⊆ C . We willprove that A (cid:44) → C is small-adequate but B (cid:44) → C is not representably small,and, therefore, not small-adequate.First we show that A (cid:44) → C is dense, that is, A is join-dense in C (Exam-ple 7.5). Let c ∈ C . If c ∈ A then c is trivially a join of elements of A . Otherwise, c = (3 , q ) for some q ∈ Q , and then c = (1 , q ) ∨ (2 , q ) with (1 , q ) , (2 , q ) ∈ A .21econd, A (cid:44) → C is codense by the same argument, because it used nothingabout the ordered class Q and because P ∼ = P op .Third, we show that A (cid:44) → C is representably small; that is, for each c ∈ C ,the subterminal functor A op → Set with support A ∩ ↓ c is small. Let c ∈ C .By Example 4.4, we must find a small K ⊆ A ∩ ↓ c such that for all a ∈ A , theposet K ∩ ↑ a is connected.If c ∈ A , we can take K = { a } . Otherwise, c = (3 , q ) for some q ∈ Q .Put K = { (1 , q ) , (2 , q ) } . Given a ∈ A ∩ ↓ c , we may suppose without loss ofgenerality that a = (1 , q (cid:48) ) for some q (cid:48) ≤ q , and then K ∩ ↑ a is the connectedposet { (1 , q ) } .Fourth, A (cid:44) → C is corepresentably small by the same argument, again be-cause it used nothing about Q and because P ∼ = P op .Finally, we show that B (cid:44) → C is not representably small. In fact, we showthat the subterminal functor B op → Set with support B ∩ ↓ (3 , ∞ ) is not small.Suppose for a contradiction that it is small. Then we can choose a small class K ⊆ B ∩ ↓ (3 , ∞ ) such that for all b ∈ B , the poset K ∩ ↑ b is connected. Inparticular, every element of B is less than or equal to some element of K . Nowfor each ordinal q we have (3 , q ) ∈ B , so (3 , q ) ≤ k for some k ∈ K , and then k = (3 , q (cid:48) ) for some ordinal q (cid:48) with q ≤ q (cid:48) . So if we put Q (cid:48) = { ordinals q (cid:48) :(3 , q (cid:48) ) ∈ K } then Q (cid:48) is small (since K is) and every ordinal is less than or equalto some element of Q (cid:48) . But there is no set of ordinals with this property, acontradiction.The following companion to Proposition 7.17 uses the notion of gentle cate-gory from Remark 4.11. Lemma 7.20
Let A F −→ B G −→ C be functors, with G full and faithful. Supposethat B is gentle. If GF is small-adequate then so is G . Proof If GF is adequate then G is adequate by Proposition 7.17, so it onlyremains to prove that G is representably and corepresentably small. By duality,it is enough to show that G is representably small.Let c ∈ C . Since GF is small-codense, the dual of Lemma 7.6 implies that c = lim i GF a i for some small diagram ( a i ) in A . Then C ( G − , c ) ∼ = lim i C ( G − , GF a i ) ∼ = lim i B ( − , F a i )as G is full and faithful. Hence C ( G − , c ) is a small limit in [ B op , Set ] of repre-sentables.Since B is gentle, the subcategory (cid:98) B of [ B op , Set ] is complete. Limits in (cid:98) B are computed pointwise because it contains the representables (as noted by Dayand Lack in Section 3 of [3]). Hence (cid:98) B is closed under small limits in [ B op , Set ],and in particular, C ( G − , c ) is small. (cid:3) Here we prove a theorem characterizing the reflexive completion of a categoryuniquely up to equivalence. It is a refinement and variant of Theorem 1.8 ofIsbell [9]. Roughly put, the result is that the reflexive completion of a category A is the largest category into which A embeds as a small-adequate subcategory.22his is formally similar to the fact that the completion of a metric space A isthe largest metric space in which A is dense. Lemma 8.1
Let F : A → B be a full and faithful small-dense functor. Then B ( F − , b ) ∨ ∼ = B ( b, F − ) naturally in b ∈ B ; that is, the diagram (cid:98) A ∨ (cid:15) (cid:15) B N F (cid:53) (cid:53) N F (cid:41) (cid:41) [ A , Set ] op commutes up to natural isomorphism. The smallness hypothesis guarantees that B ( F − , b ) ∨ is defined. Proof
By the hypotheses on F , B ( F − , b ) ∨ ( a ) ∼ = (cid:98) A ( B ( F − , b ) , A ( − , a )) ∼ = (cid:98) A ( B ( F − , b ) , B ( F − , F a )) ∼ = B ( b, F a )naturally in a ∈ A and b ∈ B . (cid:3) For a representably small functor F : A → B , the nerve functor N F hasimage in (cid:98) A . When does it have image in R ( A )? The next result provides ananswer (given without proof as statement 1.5 of [9]). Proposition 8.2 (Isbell)
Let F : A → B be a full and faithful small-densefunctor. Then B ( F − , b ) is reflexive for each b ∈ B if and only if F is small-adequate. Proof
Suppose that B ( F − , b ) is reflexive for each b ∈ B . Then we have functors A F −→ B N F −→ R ( A )whose composite is the Yoneda embedding J A : A → R ( A ) (Lemma 7.7). ByProposition 7.16, J A is small-adequate. Hence by Proposition 7.17, F is small-adequate.Conversely, suppose that F is small-adequate. Let b ∈ B . Then byLemma 8.1 and its dual, there are canonical isomorphisms B ( F − , b ) ∨∨ ∼ = B ( b, F − ) ∨ ∼ = B ( F − , b ) , and B ( F − , b ) is reflexive. (cid:3) Corollary 8.3
Let F : A → B be a small-adequate functor. Then there is afunctor N ( F ) : B → R ( A ) , unique up to isomorphism, such that the diagram (cid:98) AB N F (cid:51) (cid:51) N ( F ) (cid:47) (cid:47) N F (cid:43) (cid:43) R ( A ) (cid:42) (cid:10) (cid:55) (cid:55) (cid:20) (cid:116) (cid:39) (cid:39) A (cid:98) ommutes up to isomorphism. Moreover, N ( F ) is full and faithful. (cid:3) Precomposing this whole diagram with the functor F : A → B gives thediagram (10) of Yoneda embeddings, by Lemma 7.7.The main theorem of this section is as follows.
Theorem 8.4
Let F : A → B be a small-adequate functor. Then the functor N ( F ) : B → R ( A ) is adequate, and the triangle B N ( F ) (cid:47) (cid:47) R ( A ) A F (cid:95) (cid:95) (cid:46)(cid:14) J A (cid:60) (cid:60) commutes up to isomorphism. Moreover, N ( F ) is the unique full and faithfulfunctor B → R ( A ) such that the triangle commutes, up to isomorphism. This result is mostly due to Isbell (Theorem 1.8 of [9]). He proved a versionfor properly adequate functors (Remark 4.5), but without the conclusion thatthe functor
B → R ( A ) is adequate or unique. Proof
The triangle commutes by Lemma 7.7, N ( F ) is full and faithful byCorollary 8.3, and then N ( F ) is adequate by Propositions 7.16 and 7.17. Foruniqueness, the same two propositions prove the adequacy of any full and faithfulfunctor making the triangle commute, and the result follows from Lemma 7.14. (cid:3) Theorem 8.4 characterizes the reflexive completion uniquely up to equiv-alence. Indeed, given a category A , form the 2-category whose objects aresmall-adequate functors out of A and whose maps are adequate functors be-tween their codomains making the evident triangle commute. Theorem 8.4states that its terminal object (in a 2-categorical sense) is the Yoneda embed-ding J A : A (cid:44) → R ( A ). Remark 8.5
Corollary 7.18 and Theorem 8.4 together imply that the cate-gories containing A as a small-adequate subcategory are, up to equivalence,precisely the full subcategories of R ( A ) containing A . When B is a full subcate-gory of R ( A ) containing A , writing F : A (cid:44) → B for the inclusion, the uniquenesspart of Theorem 8.4 implies that N ( F ) is the inclusion B (cid:44) → R ( A ). Example 8.6
Let A be a poset, and recall Examples 6.10 and 7.12. Loosely,Theorem 8.4 for A states that its Dedekind–MacNeille completion is the largestordered class containing A and with the property that every element can beexpressed as both a join and a meet of elements of A . For example, any posetcontaining Q as a join- and meet-dense full subposet embeds into [ −∞ , ∞ ]. Example 8.7
The following example is due to Isbell (Example 1 of [12]). Let B be the category of sets and partial bijections, and let A be the full subcategoryconsisting of a single two-element set. Thus, A corresponds to a 7-elementmonoid. Isbell showed that the inclusion A (cid:44) → B is adequate. It is small-adequate since A is small. Hence by Theorem 8.4, there is an adequate functor B → R ( A ). In particular, there is a full and faithful functor from a largecategory into R ( A ), so R ( A ) is large (in the strong sense that it is not equivalentto any small category). This proves the statement in Example 6.8: the reflexivecompletion of a finite category can be large.24heorem 8.4 shows that when F is small-adequate, N ( F ) is adequate. But N ( F ) need not be small- adequate, as the following lemma and example show. Lemma 8.8
Let F : A → B be a small-adequate functor such that N ( F ) issmall-adequate. Then for every full and faithful functor G : B → C such that GF is small-adequate, G is also small-adequate. For a general small-adequate F , without the hypothesis that N ( F ) is small-adequate, Proposition 7.17 implies that any such functor G is adequate. So theforce of the conclusion is that G is small -adequate. Proof
The proof will use the functors in the following diagram. C N ( GF ) (cid:47) (cid:47) R ( A ) B G (cid:94) (cid:94) N ( F ) (cid:52) (cid:52) A F (cid:95) (cid:95) J A (cid:63) (cid:63) Let G be a full and faithful functor such that GF is small-adequate. Then thereis an induced adequate functor N ( GF ) as shown. Also, since GF is adequate,Proposition 7.17 implies that G is adequate. Hence by Lemma 7.13, N ( GF ) ◦ G isadequate. Now by Theorem 8.4, N ( F ) is the unique adequate functor satisfying N ( F ) ◦ F = J A . Since also N ( GF ) ◦ G ◦ F = J A by definition of N ( GF ), wehave N ( GF ) ◦ G = N ( F ). The hypothesis that N ( F ) is small-adequate andProposition 7.17 then imply that G is small-adequate. (cid:3) Example 8.9
Let F be the inclusion A (cid:44) → B of Example 7.19. As shown there,the conclusion of Lemma 8.8 is false for F . Hence N ( F ) is not small-adequate.So, for full subcategories A ⊆ B ⊆ R ( A ), it is true that A (cid:44) → B is small-adequate and B (cid:44) → R ( A ) is adequate, but B (cid:44) → R ( A ) need not be small -adequate. The reflexive completion differs from many other completions in that it is onlyfunctorial in a very restricted sense. First, there is no way to make it act on all functors:
Proposition 9.1
There is no covariant or contravariant pseudofunctor Q from CAT to CAT such that Q ( A ) (cid:39) R ( A ) for all A ∈
CAT . Proof
Suppose that there is. Write C for the two-element group, viewed asa one-object category. Then C is a retract of C × C , so Q ( C ) is a retract(up to natural isomorphism) of Q ( C × C ). Hence Q ( C ) has at most as manyisomorphism classes of objects as Q ( C × C ). But by Example 6.5, Q ( C ) hasfour isomorphism classes and Q ( C × C ) has three, a contradiction. (cid:3) F : A → B , the composite J B ◦ F is small-adequateby Lemma 7.13, giving a functor R ( F ) = N ( J B ◦ F ) : R ( B ) → R ( A ) . By Theorem 8.4, R ( F ) is adequate, and up to isomorphism, it is the unique fulland faithful functor such that the diagram R ( A ) R ( B ) R ( F ) (cid:111) (cid:111) A F (cid:47) (cid:47) (cid:63)(cid:31) J A (cid:79) (cid:79) B (cid:63)(cid:31) J B (cid:79) (cid:79) (12)commutes. The uniqueness implies that R defines a pseudofunctor R : (categories and small-adequate functors) op → (categories and adequate functors) . When the reflexive completion of a category is regarded as a subcategory ofthe
Set -valued functors on it, R ( F ) is simply composition with F : Lemma 9.2
Let F : A → B be a small-adequate functor. Then the squares [ A op , Set ] [ B op , Set ] −◦ F (cid:111) (cid:111) R ( A ) (cid:63)(cid:31) (cid:79) (cid:79) R ( B ) (cid:63)(cid:31) (cid:79) (cid:79) R ( F ) (cid:111) (cid:111) [ A , Set ] op [ B , Set ] op −◦ F (cid:111) (cid:111) R ( A ) (cid:63)(cid:31) (cid:79) (cid:79) R ( B ) (cid:63)(cid:31) (cid:79) (cid:79) R ( F ) (cid:111) (cid:111) commute up to isomorphism. Proof
Identify R ( A ) with the category of reflexive functors A op → Set , andsimilarly for B . For Z ∈ R ( B ),( R ( F ))( Z ) = ( N ( J B ◦ F ))( Z ) = R ( B ) (cid:0) ( J B ◦ F )( − ) , Z (cid:1) , which at a ∈ A gives (cid:0) ( R ( F ))( Z ) (cid:1) ( a ) = [ B op , Set ] (cid:0) B ( − , F a ) , Z (cid:1) ∼ = Z ( F a ) . Hence R ( F ) ∼ = − ◦ F , proving the commutativity of the first square. The secondfollows by duality. (cid:3) For example, when F is the inclusion of a small-adequate subcategory andreflexive completions are viewed as categories of functors, R ( F ) is restriction.The pseudofunctor R applied to a small-adequate functor F produces afunctor R ( F ) that is adequate but not always small-adequate:26 heorem 9.3 Let F : A → B be a small-adequate functor. The following areequivalent:i. N ( F ) is small-adequate;ii. R ( F ) is small-adequate;iii. R ( F ) is an equivalence.If these conditions hold then N N ( F ) is defined and pseudo-inverse to R ( F ) . These equivalent conditions do not always hold, by Example 8.9.
Proof
First, since R ( F ) ◦ J B is an adequate functor satisfying R ( F ) ◦ J B ◦ F ∼ = J A (diagram (12)), Theorem 8.4 gives R ( F ) ◦ J B ∼ = N ( F ) , (13)Trivially, (iii) implies (ii). Now assuming (ii), equation (13) gives (i), sincethe composite of small-adequate functors is small-adequate.Finally, assume (i). Then N N ( F ) is defined, and we show that it is pseudo-inverse to R ( F ), proving (iii) and the final assertion. Consider the diagram R ( A ) NN ( F ) (cid:50) (cid:50) R ( B ) R ( F ) (cid:114) (cid:114) A (cid:63)(cid:31) J A (cid:79) (cid:79) F (cid:47) (cid:47) B . (cid:63)(cid:31) J B (cid:79) (cid:79) N ( F ) (cid:99) (cid:99) The bottom-left triangle and the top-right triangle involving
N N ( F ) commuteby Theorem 8.4, the top-right triangle involving R ( F ) commutes by equa-tion (13), and the outer square commutes by definition of R ( F ). Simple diagramchases then show that R ( F ) ◦ N N ( F ) ◦ J A ∼ = J A , N N ( F ) ◦ R ( F ) ◦ J B ∼ = J B . Hence by Lemma 7.14, R ( F ) and N N ( F ) are mutually pseudo-inverse. (cid:3) It follows that reflexive completion is idempotent:
Corollary 9.4 (Isbell)
For every category A , the functors R ( A ) J R ( A ) (cid:45) (cid:45) RR ( A ) R ( J A ) (cid:108) (cid:108) define an equivalence R ( A ) (cid:39) RR ( A ) . A version of this result appeared as part of Theorem 1.8 of Isbell [9], with apartial proof.
Proof
Take F = J A : A → R ( A ) in Theorem 9.3. We have N ( J A ) = 1 R ( A ) ,which is certainly small-adequate, so R ( J A ) and N N ( J A ) are pseudo-inverse.But N N ( J A ) = N (1 R ( A ) ) = J R ( A ) . (cid:3) orollary 9.5 The following conditions on a category A are equivalent:i. J A : A (cid:44) → R ( A ) is an equivalence;ii. every reflexive functor A op → Set is representable;iii. every reflexive functor
A →
Set is representable;iv.
A (cid:39) R ( B ) for some category B . Proof J A is always full and faithful, so it is an equivalence just when it isessentially surjective on objects. Hence (i) ⇐⇒ (ii) ⇐⇒ (iii) by diagram (10).That (iv) is equivalent to (i) follows from Corollary 9.4. (cid:3) A category satisfying the equivalent conditions of Corollary 9.5 is reflex-ively complete . It is a self-dual condition: A is reflexively complete if andonly if A op is.In the introduction to Section 8, we compared reflexive completion to metriccompletion, drawing an analogy between Theorem 8.4 and the characterizationof the completion of a metric space A as the largest metric space in which A is dense. The completion of a metric space A can also be characterized as thesmallest complete metric space containing A . However, the reflexive analogueof that characterization is false: Example 9.6
Let F : A → B be a small-adequate functor such that N ( F ) isnot small-adequate, as in Example 8.9. Then by Theorem 9.3, the full andfaithful functor R ( F ) : R ( B ) → R ( A ) is not an equivalence. Its image is a fullsubcategory of R ( A ) that is reflexively complete and contains A , but is strictlysmaller than R ( A ).Such examples can be excluded by restricting to categories that are gentle.There, the pseudofunctor R acts somewhat trivially, in the sense of part (i) ofCorollary 9.8 below. Proposition 9.7
Let F : A → B be a small-adequate functor. If B is gentlethen N ( F ) is small-adequate. Proof
Apply Lemma 7.20 with G = N ( F ), recalling that N ( F ) ◦ F is thesmall-adequate functor J A . (cid:3) Corollary 9.8
Let B be a gentle category. Then:i. for every category A and small-adequate functor F : A → B , the functor R ( F ) : R ( B ) → R ( A ) is an equivalence;ii. R ( A ) (cid:39) R ( B ) for every small-adequate subcategory A ⊆ B . Proof
Part (i) follows from Proposition 9.7 and Theorem 9.3, and part (ii) isthen immediate. (cid:3)
For example, every small-adequate subcategory of a complete and cocom-plete category B has the same reflexive completion as B , which by Proposi-tion 11.11 below is B itself. 28 Some formal resemblances are apparent between the reflexive and Cauchy com-pletions. Both are idempotent completions; both commute with the operationof taking opposites; and to the analogy between reflexive and metric completionmentioned in the introduction to Section 8, one can add the fact that metriccompletion is Cauchy completion in the [0 , ∞ ]-enriched setting. On the otherhand, the reflexive and Cauchy completions are different, as even the exampleof the empty category shows (Example 6.1). In this section, we describe therelationship between them. Proposition 10.1
Let A be a category.i. In [ A op , Set ] , the class of reflexive functors is closed under small absolutecolimits.ii. R ( A ) is Cauchy complete.iii. A ⊆ R ( A ) , when the Cauchy completion A and reflexive completion R ( A ) are viewed as subcategories of [ A op , Set ] .iv. R ( A ) (cid:39) R ( A ) . Proof
For (i), let X = colim i X i be a small absolute colimit of reflexive functors X i in [ A op , Set ]. Each X i is small, so X is small, and X is the absolute colimitof the X i in (cid:98) A . Since ∨ : (cid:98) A → [ A , Set ] op preserves this colimit, X ∨ ∼ = colim i X ∨ i in [ A , Set ] op , again an absolute colimit. Since each X i is reflexive, each X ∨ i issmall. Now A (cid:98) ⊆ [ A , Set ] op is complete, hence Cauchy complete, hence closedunder small absolute colimits in [ A , Set ] op . So X ∨ is small and is the absolutecolimit of the X ∨ i in A (cid:98) . Since ∨ : A (cid:98) → [ A op , Set ] preserves this colimit, X ∨∨ ∼ =colim i X ∨∨ i . Since each X i is reflexive, so is X .This proves (i). Colimits in R ( A ) ⊆ [ A op , Set ] are computed pointwise, so R ( A ) has absolute colimits, proving (ii). And A is the closure of A ⊆ [ A op , Set ]under absolute colimits, giving (iii).For (iv), Corollary 7.18 and (iii) imply that the inclusion F : A (cid:44) → A is small-adequate. We will prove that the induced adequate functor N ( F ) : A → R ( A )is representably small. It will follow by duality that N ( F ) is small-adequate,and so by Theorem 9.3 that R ( F ) : R ( A ) → R ( A ) is an equivalence.First recall that by the 2-universal property of Cauchy completion, restric-tion along F is an equivalence[ A op , Set ] (cid:39) −→ [ A op , Set ] . (14)Its pseudo-inverse is left Kan extension along F , and left Kan extension alongany functor preserves smallness, so every Z : A op → Set such that Z | A op issmall is itself small.To prove that N ( F ) : A → R ( A ) is representably small, we regard A and R ( A ) as subcategories of [ A op , Set ] and recall that N ( F ) is then the inclusion A (cid:44) → R ( A ) (Remark 8.5). For each X ∈ R ( A ), the Yoneda lemma gives R ( A )( − , X ) | A op ∼ = X, so R ( A )( − , X ) | A op is small, so R ( A )( − , X ) | A op is small, as required. (cid:3) emark 10.2 There is another, more elementary, proof of (iv). One showsthat the equivalence (14) restricts to an equivalence (cid:98) A (cid:39) −→ (cid:98) A , and dually. Thenone shows that the conjugacy operations on A and A commute with theseequivalences. It follows that the equivalence (14) also restricts to an equivalence R ( A ) (cid:39) −→ R ( A ).Proposition 10.1(iv) implies: Corollary 10.3
Morita equivalent categories have equivalent reflexive comple-tions. (cid:3)
Hence the category R ( A ) is determined by the category [ A op , Set ], withoutknowledge of A itself.Conjugacy (as opposed to reflexivity) also plays a role in the theory ofCauchy completions. For vector spaces X and Z , there is a canonical linearmap X ∨ ⊗ Z → Vect ( X, Z ) ξ ⊗ z (cid:55)→ ξ ( − ) · z, (15)where X ∨ is the linear dual of X . Analogously, for a category A and X, Z ∈ (cid:98) A ,there is a canonical map of sets κ X,Z : X ∨ (cid:12) Z → (cid:98) A ( X, Z ) , to be defined. Here X ∨ (cid:12) Z = (cid:90) a X ∨ ( a ) × Z ( a ) , and the coend exists since Z is small. The map κ X,Z can be defined concretelyby specifying a natural family of functions X ∨ ( a ) × Z ( a ) → [ X ( b ) , Z ( b )]( a, b ∈ A ), which are taken to be( ξ, z ) (cid:55)→ (cid:16) x (cid:55)→ (cid:0) Z ( ξ b ( x )) (cid:1) ( z ) (cid:17) . Equivalently, X ∨ (cid:12) Z is the composite profunctor (cid:31) Z (cid:47) (cid:47) A (cid:31) X ∨ (cid:47) (cid:47) , and the map κ X,Z corresponds under the adjunctions (6) to X (cid:12) X ∨ (cid:12) Z ε X (cid:12) Z −−−−→ Hom A (cid:12) Z ∼ = Z, where ε X : X (cid:12) X ∨ → Hom A is the natural transformation of (7). Proposition 10.4
Let A be a small category. The following conditions on afunctor X : A op → Set are equivalent:i. X ∈ A , when A is regarded as a subcategory of (cid:98) A ; i. (cid:98) A ( X, − ) : (cid:98) A →
Set preserves small colimits;iii. X : + −→ A has a right adjoint in the bicategory Prof ;iv. X : + −→ A has right adjoint X ∨ in Prof , with counit ε X ;v. κ X,Z : X ∨ (cid:12) Z → (cid:98) A ( X, Z ) is a bijection for all Z ∈ (cid:98) A . Proof
The equivalence of (i)–(iii) is standard, and (iv) ⇒ (iii) is trivial.To prove (iii) ⇒ (iv), suppose that X has a right adjoint Y in Prof , withcounit β . As in any bicategory, this implies that β exhibits Y as the rightKan lift of Hom A through X . (Here we are using a result more often given inthe dual form, involving Kan extensions. See, for instance, Theorem X.7.2 ofMac Lane [17], which is stated for CAT but the proof is valid in any bicategory.)But as shown at the end of Section 2, ε X exhibits X ∨ as the right Kan lift ofHom A through X . Hence ( X ∨ , ε X ) ∼ = ( Y, β ) and (iv) follows.Now assume (v). The maps κ X,Z are natural in Z ∈ (cid:98) A , so( X ∨ (cid:12) − ) ∼ = (cid:98) A ( X, − ) : (cid:98) A →
Set . But X ∨ (cid:12) − preserves small colimits by the adjointness relations (6), giving (ii).Finally, assuming (ii), we prove (v). When Z is representable, the function κ X,Z : X ∨ (cid:12) Z → (cid:98) A ( X, Z )is bijective. But every Z ∈ (cid:98) A is a small colimit of representables, and both X ∨ (cid:12) − and (cid:98) A ( X, − ) preserve small colimits, so κ X,Z is bijective for all Z . (cid:3) This result can be generalized to enriched categories. When A is a one-object Ab -category corresponding to a field, A is the category of finite-dimensionalvector spaces, the maps κ X,Z are as defined in equation (15), and we recoverthe following fact: a vector space X is finite-dimensional if and only if κ X,Z isan isomorphism for all vector spaces Z .Further results on Isbell conjugacy and Cauchy completion can be found inSections 6 and 7 of Kelly and Schmitt [14].
11 Limits in reflexive completions
A partially ordered set is complete if and only if it is reflexively complete (Ex-ample 6.10). In one direction, we show that, more generally, every complete orcocomplete category is reflexively complete (Proposition 11.11). But the con-verse is another matter entirely: in general, reflexively complete categories havevery few limits. We identify exactly which ones.
Lemma 11.1
Let A be a category.i. The inclusion J A : A (cid:44) → R ( A ) preserves and reflects both limits and col-imits.ii. The inclusion R ( A ) (cid:44) → [ A op , Set ] preserves limits and reflects both limitsand colimits. R ( A ) (cid:44) → [ A op , Set ] does not preserve co limits, since J A : A (cid:44) →R ( A ) preserves colimits but the Yoneda embedding A (cid:44) → [ A op , Set ] does not.
Proof
The statements on reflection are immediate, since both functors are fulland faithful.For (i), the embedding J A : A (cid:44) → R ( A ) is adequate by Proposition 7.16, sopreserves limits and colimits by Lemma 7.8 and its dual.For (ii), the composite of R ( A ) (cid:44) → [ A op , Set ] with J A : A (cid:44) → R ( A ) isthe Yoneda embedding, which is dense, so R ( A ) (cid:44) → [ A op , Set ] is dense byLemma 7.15. Since it is also full and faithful, it preserves limits by Lemma 7.8. (cid:3)
We have already shown that reflexively complete categories are Cauchy com-plete, that is, have absolute limits and colimits (Proposition 10.1(ii)). The nextfew results show that the reflexive completion of a small category also has initialand terminal objects, but that the (co)limits just mentioned are the only onesthat generally exist.For a small category A and b ∈ A , write Cone(id , b ) for the set of cones fromthe identity to b (natural transformations from id A to the constant endofunctor b ). This defines a functor Cone(id , − ) : A →
Set . Lemma 11.2
Let A be a small category. Then Cone(id , − ) ∨ is the terminalfunctor A op → Set . Proof
Fixing a ∈ A , we must show that there is exactly one natural transfor-mation α : Cone(id , − ) → A ( a, − ). There is at least one, since we can define α b ( p ) = p a for each b ∈ A and p ∈ Cone(id , b ).To prove uniqueness, let β : Cone(id , − ) → A ( a, − ). Let b ∈ B and p ∈ Cone(id , b ). We must prove that β b ( p ) = p a .Since p is a cone, p b ◦ p c = p c for all c ∈ A . So we have an equality of cones p b ◦ p = p , and then naturality of β gives p b ◦ β b ( p ) = β b ( p ). On the other hand, p b ◦ β b ( p ) = p a since p is a cone. Hence β b ( p ) = p a , as required. (cid:3) Proposition 11.3
The reflexive completion of a small category has initial andterminal objects.
Proof
Let A be a small category. Write 1 for the terminal functor A op → Set .Then 1 ∨ ∼ = Cone(id , − ), so 1 ∨∨ ∼ = 1 by Lemma 11.2. Hence 1 is reflexive. It fol-lows that 1 is a terminal object of R ( A ) ⊆ [ A op , Set ]. By duality (Remark 5.2), R ( A ) also has an initial object. (cid:3) Remark 11.4
When R ( A ) is viewed as a subcategory of [ A op , Set ], its initialobject is not in general the initial (empty) functor A op → Set . The case A = already shows this (Example 6.2).Proposition 11.3 is false for large categories. The proof fails because theterminal functor need not be small. Any large discrete category A is a coun-terexample to the statement, since then R ( A ) (cid:39) A (Example 6.4).To show that reflexive completions have no other (co)limits in general, we usethe following lemma. A category I is an absolute limit shape if all I -limitsare absolute. Lemma 11.5
A small category is an absolute limit shape if and only if it admitsa cone on the identity.
Proof
Let I be a small category admitting a cone (cid:0) k u i −→ i (cid:1) i ∈I on the identity. Take functors I D −→ A F −→ B and a limit cone (cid:0) L p i −→ Di (cid:1) i ∈I (16)on D . The cone ( Dk Du i −→ Di ) induces a map Dk f −→ L in A . Then f ◦ p k = 1 L .To show that ( F L
F p i −→ F Di ) is a limit cone on
F D , let (cid:0) B r i −→ F Di (cid:1) i ∈I be any other cone on F D . We have a map B r k −→ F Dk
F f −→ F L , and it isroutine to check that this is the unique map of cones from ( r i ) to ( F p i ). Hencethe limit cone (16) is absolute.We sketch the proof of the converse, which will not be needed here. Sup-pose that I -limits are absolute. The limit of the Yoneda embedding I (cid:44) → (cid:98) I isCone( − , id I ). By absoluteness, this limit is preserved by colim I : (cid:98) I →
Set . Itfollows that colim I (Cone( − , id I )) = 1, so there exists a cone on id I . (cid:3) Theorem 11.6
Let I be a small category. The following are equivalent:i. I -limits exist in the reflexive completion of every small category;ii. I -limits exist in every Cauchy complete category with a terminal object;iii. I is empty or an absolute limit shape. By Remark 5.2, dual results hold for colimits.
Proof
Every Cauchy complete category has small absolute limits, so (iii) ⇒ (ii).Every reflexive completion of a small category is Cauchy complete with a ter-minal object (Propositions 10.1(ii) and 11.3), so (ii) ⇒ (i). It remains to prove(i) ⇒ (iii), which we do by contradiction.Assume (i), and that I is neither empty nor an absolute limit shape. ByLemma 11.5, I admits no cone on the identity.Let J be the category obtained from I by adjoining a new object z andmaps p i , p i : z → i for each i ∈ I , subject to u ◦ p εi = p εj for each map i u −→ j in I and ε ∈ { , } . By assumption, the composite I (cid:44) → J (cid:44) → R ( J )has a limit, L . Since the inclusion R ( J ) (cid:44) → [ J op , Set ] preserves limits(Lemma 11.1), L is the limit of the composite I (cid:44) → J (cid:44) → [ J op , Set ] , and is reflexive. Now for i ∈ I , L ( i ) = lim i (cid:48) ∈I J ( i, i (cid:48) ) = Cone( i, id I ) = ∅ , L ( z ) = lim i (cid:48) ∈I J ( z, i (cid:48) ) ∼ = { , } π I , where π I is the set of connected-components of I . Write S = L ( z ). Since I isnonempty, | S | ≥
2. Then L ∼ = S × J ( − , z ) , so L ∨ ∼ = J ( z, − ) S . Since L is reflexive, the unit map η L,z : L ( z ) → L ∨∨ ( z ) is surjective. That is,every natural transformation α : J ( z, − ) S → J ( z, − ) (17)is the s -projection for some s ∈ S .We will derive a contradiction by constructing a transformation (17) not ofthis form. For i ∈ I , let α i be the function α i : J ( z, i ) S → J ( z, i ) (cid:0) p ε s i (cid:1) s ∈ S (cid:55)→ p min s ε s i , and let α z be the unique function J ( z, z ) S → J ( z, z ). It is routine to checkthat α defines a natural transformation (17). Hence α is t -projection for some t ∈ S .Choose some i ∈ I , as we may since I is nonempty. Writing δ for theKronecker delta and recalling that | S | ≥ α i (cid:0)(cid:0) p δ st i (cid:1) s ∈ S (cid:1) = p min s δ st i = p i . But the t -projection of (cid:0) p δ st i (cid:1) s ∈ S is p δ tt i = p i , a contradiction. (cid:3) Remark 11.7
Theorem 11.6 might suggest the idea that reflexive completionis Cauchy completion followed by the adjoining of initial and terminal objects,and there are examples in Section 6 where this is indeed the case. But the groupof order 2 (Example 6.5) shows that this is false in general.Theorem 11.6 concerns limits in reflexive completions of small categories.Every reflexively complete category is the reflexive completion of some category(Corollary 9.5), but not always of a small category. For example, any largediscrete category is reflexively complete (Example 6.4), but does not have aterminal object and so cannot be the reflexive completion of a small category.The following corollary is the analogue of Theorem 11.6 for the larger class ofreflexively complete categories.
Corollary 11.8
Let I be a small category. The following are equivalent:i. I -limits exist in every reflexively complete category;ii. I -limits exist in every Cauchy complete category;iii. I is an absolute limit shape. roof Certainly (iii) ⇒ (ii), and (ii) ⇒ (i) because reflexively complete categoriesare Cauchy complete. Assuming (i), I is empty or an absolute limit shape byTheorem 11.6. But any large discrete category is a reflexively complete categorywith no terminal object, so I is not empty, proving (iii). (cid:3) Every
Set -valued functor on a small category can be expressed as a smallcolimit of representables. Not every such functor can be expressed as a small limit of representables, since then it would preserve small limits.
Lemma 11.9
Let A be a category. A functor A op → Set is a small limit ofrepresentables if and only if it is the conjugate of some small functor
A →
Set . Proof
Let X : A op → Set . If X ∼ = lim i A ( − , a i ) for some small diagram ( a i ) in A then X is the conjugate of the small functor colim i A ( a i , − ). Conversely, everysmall functor A →
Set can be expressed as a small colimit of representables,and its conjugate is the corresponding small limit of representables. (cid:3)
Proposition 11.10
Every reflexive
Set -valued functor preserves small limits.
Proof
By Lemma 11.9, every reflexive functor X is a small limit of representa-bles. But representables preserve small limits, so X does too. (cid:3) A reflexive functor need not preserve co limits. For example, the uniquereflexive functor → Set is the terminal functor, which does not preserveinitial objects.Moreover, although a reflexive functor
A →
Set preserves all limits thatexist in A , it need not be flat. Indeed, many of the examples in Section 6 are ofcategories A where the initial functor 0 : A →
Set is reflexive; but 0 is not flat.
Proposition 11.11
Every complete or cocomplete category is reflexively com-plete.
Proof
Let A be a complete category. By Lemma 11.9, every reflexive functor X : A op → Set is a small limit of representables. But A is complete, so X isrepresentable. This proves that every complete category if reflexively complete.Since reflexive completeness is a self-dual condition, the dual result follows. (cid:3)
12 The Isbell envelope
This short section describes the relationship between two constructions due toIsbell. As well as introducing the reflexive completion of a category A in 1960 [9],he also defined what is now called the Isbell envelope I ( A ) in 1966 (naming itthe ‘couple category’: [11], p. 622). See also Garner [7] for a thorough moderntreatment of the Isbell envelope.We begin with an arbitrary adjunction C F ⊥ (cid:42) (cid:42) D , G (cid:105) (cid:105) η and counit ε . We already defined the invariant part Inv ( F, G )(Section 5), which comes with full and faithful inclusion functors
Inv ( F, G ) (cid:47) (cid:47) (cid:15) (cid:15) CD The envelope Env ( F, G ) of the adjunction is the category of quadruples( c ∈ C , d ∈ D , f : c → Gd, g : F c → d )such that f and g are each other’s transposes. A map ( c, d, f, g ) → ( c (cid:48) , d (cid:48) , f (cid:48) , g (cid:48) )in Env ( F, G ) is a pair of maps( p : c → c (cid:48) , q : d → d (cid:48) )such that ( Gq ) ◦ f = f (cid:48) ◦ p , or equivalently, q ◦ g = g (cid:48) ◦ ( F p ). There are canonicalfunctors C (cid:15) (cid:15) D (cid:47) (cid:47) Env ( F, G ) (18)defined by c (cid:55)→ ( c, F c, η c , F c ) , d (cid:55)→ ( Gd, d, Gd , ε d )( c ∈ C , d ∈ D ), which are full and faithful.The invariant part and the envelope are related as follows. Lemma 12.1
Let F (cid:97) G : D → C be an adjunction.i. The full and faithful functors
Inv ( F, G ) (cid:47) (cid:47) (cid:15) (cid:15) C (cid:15) (cid:15) D (cid:47) (cid:47) Env ( F, G ) defined above form a 2-pullback square (in the up-to-isomorphism sense).ii. The composite functor Inv ( F, G ) → Env ( F, G ) defines an equivalence be-tween Inv ( F, G ) and the full subcategory of objects ( c, d, f, g ) of Env ( F, G ) such that f and g are isomorphisms. Proof
The proof is a series of elementary checks, omitted here. In outline, anobject of the 2-pullback of the two functors into
Env ( F, G ) consists of objects c ∈ C and d ∈ D together with an isomorphism( c, F c, η c , F c ) ∼ = −→ ( Gd, d, Gd , ε d )in Env ( F, G ), and one verifies that this amounts to an object of
Inv ( F, G ). (cid:3) emark 12.2 The
Inv and
Env constructions can be understood as ad-joints. Let
ADJ be the 2-category defined as follows. Objects are adjunc-tions ( C , D , F, G ). A map ( C , D , F, G ) → ( C (cid:48) , D (cid:48) , F (cid:48) , G (cid:48) ) consists of functors andnatural transformations (cid:0) C P −→ C (cid:48) , D Q −→ , D (cid:48) , F (cid:48) P α −→ QF, P G β −→ G (cid:48) Q (cid:1) such that α and β are each other’s mates. The 2-cells are the evident ones. Let ADJ ∼ = be the sub-2-category of ADJ consisting of all objects, just those maps(
P, Q, α, β ) for which α and β are isomorphisms, and all 2-cells between them.There are 2-functors and 2-adjunctions ADJCAT (cid:124) (cid:124) E n v (cid:62) (cid:39) (cid:7) (cid:52) (cid:52) (cid:97) (cid:97) I n v ⊥ (cid:23) (cid:119) (cid:42) (cid:42) ADJ ∼ = (cid:63)(cid:31) (cid:79) (cid:79) Here, the embedding of
CAT into
ADJ ∼ = associates to a category the identityadjunction on it, and Inv is its right 2-adjoint. Similarly,
Env is the right2-adjoint of
CAT (cid:44) → ADJ .Now consider the conjugacy adjunction (cid:98) A (cid:29) A (cid:98) of a small category A . Itsinvariant part is R ( A ). Its envelope is the Isbell envelope I ( A ). An objectof I ( A ) is a quadruple( X : A op → Set , Y : A →
Set , φ : X → Y ∨ , ψ : Y → X ∨ )such that φ and ψ correspond to one another under the conjugacy adjunction.By the isomorphisms (4), I ( A ) is equivalently the category of triples( X : A op → Set , Y : A →
Set , χ : X (cid:2) Y → Hom A ) , with the obvious maps between them (as Isbell observed in Section 1.1 of [11]).With this formulation of I ( A ), the canonical functor (cid:98) A → I ( A ) (as in dia-gram (18)) maps X ∈ (cid:98) A to ( X, X ∨ , ε X ), where ε X is the natural transformationof (7). A dual statement holds for A (cid:98) .Lemma 12.1 immediately implies: Proposition 12.3
For a small category A , the canonical full and faithful func-tors R ( A ) (cid:47) (cid:47) (cid:15) (cid:15) (cid:98) A (cid:15) (cid:15) A (cid:98) (cid:47) (cid:47) I ( A ) form a 2-pullback square. (cid:3) Thus, informally, R ( A ) = (cid:98) A ∩ A (cid:98) . 37 eferences [1] M. Barr, J. F. Kennison, and R. Raphael. Isbell duality.
Theory and Applicationsof Categories , 20(15):504–542, 2008.[2] B. A. Davey and H. A. Priestley.
Introduction to Lattices and Order . CambridgeUniversity Press, Cambridge, 2nd edition, 2002.[3] B. J. Day and S. Lack. Limits of small categories.
Journal of Pure and AppliedAlgebra , 210:651–663, 2007.[4] I. Di Liberti. Codensity: Isbell duality, pro-objects, compactness and accessibility.
Journal of Pure and Applied Algebra , 224(10):106379, 2020.[5] A. W. M. Dress. Trees, tight extensions of metric spaces, and the cohomologicaldimension of certain groups: a note on combinatorial properties of metric spaces.
Advances in Mathematics , 53(3):321–402, 1984.[6] P. Freyd. Several new concepts: lucid and concordant functors, pre-limits, pre-completeness, the continuous and concordant completions of categories. In
Cat-egory Theory, Homology Theory and Their Applications (Proceedings of the Bat-telle Institute Conference, Seattle, Washington, 1968, Volume 3) , volume 99 of
Lecture Notes in Mathematics , pages 196–241. Springer, Berlin, 1969.[7] R. Garner. The Isbell monad.
Advances in Mathematics , 274:516–537, 2015.[8] H. Hirai and S. Koichi. On tight spans for directed distances.
Annals of Combi-natorics , 16(3):543–569, 2012.[9] J. R. Isbell. Adequate subcategories.
Illinois Journal of Mathematics , 4:541–552,1960.[10] J. R. Isbell. Six theorems about injective metric spaces.
Commentarii Mathe-matici Helvetici , 39:65–74, 1964.[11] J. R. Isbell. Structure of categories.
Bulletin of the American MathematicalSociety , 72:619–655, 1966.[12] J. R. Isbell. Small adequate subcategories.
Journal of the London MathematicalSociety , 43:242–246, 1968.[13] G. M. Kelly.
Basic Concepts of Enriched Category Theory , volume 64 of
LondonMathematical Society Lecture Note Series . Cambridge University Press, Cam-bridge, 1982. Also
Reprints in Theory and Applications of Categories
Theory and Applications of Categories , 14(7):399–423, 2005.[15] J. Lambek.
Completions of Categories , volume 24 of
Lecture Notes in Mathemat-ics . Springer, Berlin, 1966.[16] F. W. Lawvere. Metric spaces, generalized logic and closed categories.
Rendicontidel Seminario Matematico e Fisico di Milano , XLIII:135–166, 1973. Also
Reprintsin Theory and Applications of Categories
Categories for the Working Mathematician . Graduate Texts inMathematics 5. Springer, New York, 1971.[18] J. C. McConnell and J. C. Robson.
Noncommutative Noetherian Rings . Gradu-ate Studies in Mathematics. American Mathematical Society, Providence, RhodeIsland, 2001.[19] C. McLarty. The rising sea: Grothendieck on simplicity and generality. In
Episodes in the History of Modern Algebra (1800–1950) , volume 32 of
Historyof Mathematics , pages 301–325. American Mathematical Society, Providence, RI,2007.[20] F. Ulmer. Properties of dense and relative adjoint functors.
Journal of Algebra ,8:77–95, 1968.[21] S. Willerton. Tight spans, Isbell completions and semi-tropical modules.
Theoryand Applications of Categories , 28(22):696–732, 2013., 28(22):696–732, 2013.