Theory and Applications of Categories, Vol. 10, No. 20, 2002, pp. 486–497.

A DUALITY RELATIVE TO A LIMIT DOCTRINE

C. CENTAZZO AND E.M. VITALE

ABSTRACT. We give a unified proof of Gabriel-Ulmer duality for locally finitely presentable categories, Adámek-Lawvere-Rosický duality for varieties and Morita duality for presheaf categories. As an application, we compare presheaf categories and varieties.

1. Introduction

The celebrated Gabriel-Ulmer duality states that there is a contravariant biequivalence

Lex^{op}  \overline{}LFP

where Lex is the 2-category of small finitely complete categories, finite limit preserving functors and natural transformations, and LFP is the 2-categories of locally finitely presentable categories, finitary right adjoint functors and natural transformations (see [7]). More recently, Adámek, Lawvere and Rosický established a similar biequivalence

Th^{op}  \overline{}VAR

where Th is the 2-category of small Cauchy complete categories with finite products, finite product preserving functors and natural transformations, and VAR is the 2-category of multisorted finitary varieties, algebraically exact functors and natural transformations (see [2]).

The first one of these dualities is based on the interplay between finite limits and filtered colimits, which are those colimits commuting in Set with finite limits and which are used to define locally finitely presentable categories. Similarly, the second duality is based on finite products and sifted colimits, which can be defined as those colimits commuting in Set with finite products and which can be used to give an abstract characterization of finitary multisorted varieties.

It seems natural to look for a common generalization of these dualities, starting from a good class 𝔻 of limits and using 𝔻-filtered colimits, i.e. those colimits commuting in Set with 𝔻-limits. The notion of locally finitely presentable category has been recently generalized by Adámek, Borceux, Lack and Rosický in [1], where the basic theory of locally 𝔻-presentable categories is developed. In particular, Adámek, Borceux, Lack and Rosický have the expected representation theorem of locally 𝔻-presentable categories as categories of 𝔻-continuous set-valued functors. The aim of this note is just to put this theorem in its natural context, which is a biequivalence

Received by the editors 2002 July 18 and, in revised form, 2002 November 29.

Transmitted by Jiri Rosicky. Published on 2002 December 7.

2000 Mathematics Subject Classification: 18C10, 18C35, 18A25, 18A30, 18A35.

Key words and phrases: limit doctrines, locally 𝔻-presentable categories, duality.

\begin{matrix}c \\ ◯ \end{matrix} C. Centazzo and E.M. Vitale, 2002. Permission to copy for private use granted.

𝔻-Th^{op}  \overline{}𝔻-LP

where 𝔻-Th is the 2-category of essentially small Cauchy complete and 𝔻-complete categories, 𝔻-continuous functors and natural transformations, and 𝔻-LP is the 2-category of locally 𝔻-presentable categories, 𝔻-accessible right adjoint functors and natural transformations.

In Section 2, we recall from [1] the basic definitions and results we need. In section 3 we state and prove the duality between 𝔻-Th and 𝔻-LP. The proof is achieved thanks to several simple lemmas. These lemmas are inserted when needed in the proof, but their own proofs are postponed to Section 4, so to make the reading of the main result easier. We follow the lines of the remarkable exposition of Gabriel-Ulmer duality given in [3]. In the last section, we give a simple application of the duality theorem comparing presheaf categories and varieties.

We are grateful to the referee for his useful comments.

2. Preliminaries

All definitions and results contained in this section can be found in [1].

2.1. Definition. A collection 𝔻 of small categories is called a limit doctrine, or just doctrine, if 𝔻, regarded as full subcategory of Cat, is essentially small. A 𝔻-limit is the limit of a functor with domain in 𝔻. A category which has all 𝔻-limits is said 𝔻-complete, and a functor between 𝔻-complete categories is called 𝔻-continuous if it preserves all 𝔻limits. Dually, there are the notions of 𝔻-cocompleteness and 𝔻-cocontinuity. 𝔻^{op} stands for the doctrine consisting of all categories D^{op} for D ∈ 𝔻.

As general consideration, let us point out that a category A is 𝔻-complete if and only if its dual category A^{op} is 𝔻^{op}-cocomplete.

2.2. Definition. We say that a small category C is 𝔻-filtered if C-colimits commute in Set with 𝔻-limits, i.e. for any D ∈ 𝔻 and any functor F : C × D ⟶ Set the canonical map

colim_{C∈C} lim_{D∈D} F (C, D) ⟶ lim_{D∈D} colim_{C∈C} F (C, D)

is an isomorphism.

2.3. Definition. Let K and K′ be categories with 𝔻-filtered colimits. A functor F: K ⟶ K′ is 𝔻-accessible if it preserves 𝔻-filtered colimits. An object K of K is 𝔻-presentable if the representable functor K(K, −) : K ⟶ Set is 𝔻-accessible.

2.4. Lemma. A 𝔻^{op}-colimit of 𝔻-presentable objects is 𝔻-presentable itself.

2.5. Definition. A doctrine 𝔻 is sound if for a functor F : A ⟶ Set, with A a small category, the left Kan extension Lan_{Y}F : [A^{op}, Set] ⟶ Set of F along the Yoneda embedding preserves 𝔻-limits if and only if it preserves 𝔻-limits of representables.

From now on, we assume the doctrine 𝔻 to be sound.

2.6. Lemma. Let A be a small and 𝔻-complete category. A functor F: A ⟶ Set is 𝔻-continuous if and only if it is a 𝔻-filtered colimit of representable functors.

2.7. Definition. To say that a locally small category K is 𝔻-accessible is to say that it has 𝔻-filtered colimits and admits a small set A of 𝔻-presentable objects such that any object of K is a 𝔻-filtered colimit of objects of A.

2.8. Lemma. Let K be a 𝔻-accessible category. The full subcategory K_{𝔻} of 𝔻-presentable objects is essentially small.

2.9. Definition. A locally small category K is locally 𝔻-presentable if it is 𝔻-accessible and cocomplete (equivalently, if it is 𝔻-accessible and complete).

Observe that, by Lemma 2.4, if K is locally 𝔻-presentable, then K_{𝔻} is 𝔻^{op}-cocomplete.

If A is a small and 𝔻-complete category, we write 𝔻-cont[A, Set] for the full subcategory of the functor category [A, Set] given by 𝔻-continuous functors. By general reasons (see for example [8]), the full inclusion

i_{A}: 𝔻-cont[A, Set] ⟶ [A, Set]

has a left adjoint, which we denote by r_{A}.

2.10. Theorem.

1. Let A be a small and 𝔻-complete category. Then the category 𝔻-cont[A, Set] is locally 𝔻-presentable.

2. Let K be a locally 𝔻-presentable category. Then K is equivalent to 𝔻-cont[K_{𝔻}^{op}, Set].

3. The duality theorem

Let 𝔻 be a sound doctrine, we denote by 𝔻-LP the 2-category of locally 𝔻-presentable categories, 𝔻-accessible right adjoint functors and natural transformations. Whereas, write 𝔻-Th for the 2-category of essentially small 𝔻-complete and Cauchy complete categories, 𝔻-continuous functors and natural transformations. Recall that a category C is Cauchy complete if all idempotents of C split (see [4]).

3.1. Theorem. There is a biequivalence of 2-categories

ε : 𝔻-Th^{op}  \overline{}𝔻-LP

Proof. Let A be in 𝔻-Th. We put ε(A) = 𝔻-cont[A, Set], which is locally 𝔻-presentable by part 1 of Theorem 2.10. Let F: A ⟶ B be a morphism in 𝔻-Th. To define ε(F), consider the following diagram

[figure: 𝔻-cont[A, Set]; [A, Set]; 𝔻-cont[B, Set]; [B, Set]; r_{B}; i_{B}; i_{A}; ε(F); T; − · F]  (1)

where T is the left Kan extension of Y_{B} · F^{op}: A^{op} ⟶ B^{op} ⟶ [B, Set] along the Yoneda embedding Y_{A}: A^{op} ⟶ [A, Set]. Since F is 𝔻-continuous, composing with F on the right actually restricts to a functor ε(F): 𝔻-cont[B, Set] ⟶ 𝔻-cont[A, Set]. We wish to show that ε(F) is in fact a morphism of 𝔻-LP. For this, we need two lemmas. The second one is part of Proposition 3.3 in [1], which states that 𝔻-cont[A, Set] is the free completion of A^{op} under 𝔻-filtered colimits.

3.2. Lemma. Consider the following diagram of categories and functors:

[figure: C; D; A; B; r; i; j; h; f; g]

Suppose: j full and faithful, g · i ≃ j · h, r ⊣ i and f ⊣ g. Then, the composite r · f · j satisfies r · f · j ⊣ h.

3.3. Lemma. Let A be an essentially small and 𝔻-complete category. The full embedding i_{A} : 𝔻-cont[A, Set] ⟶ [A, Set] preserves and reflects 𝔻-filtered colimits.

Since T is left adjoint to − · F: [B, Set] ⟶ [A, Set], we can apply Lemma 3.2 to diagram (1) and we get a left adjoint to ε(F): 𝔻-cont[B, Set] ⟶ 𝔻-cont[A, Set]. Now, observe that −·F: [B, Set] ⟶ [A, Set] preserves all colimits (because thay are computed pointwise in Set) and then, by Lemma 3.3, i_{A} · ε(F) ≃ (− · F) · i_{B} preserves 𝔻-filtered colimits. Finally, since i_{A} reflects 𝔻-filtered colimits (Lemma 3.3 again), ε(F) preserves them. This complete the definition of

ε : 𝔻-Th^{op}  \overline{}𝔻-LP

on objects and morphisms. It is defined in the obvious (covariant) way on 2-cells and it is clearly a 2-functor. Let K be a locally 𝔻-presentable category and consider its full subcategory K_{𝔻} of 𝔻presentable objects. To prove that ε is surjective on objects up to equivalence, we apply part 2 of Theorem 2.10 and we have only to check that K_{𝔻}^{op} is Cauchy complete. For later use, we state a slightly more general fact.

3.4. Lemma. Consider two functors F, G: K ⟶ Set, with K a locally small category with 𝔻-filtered colimits. If F is 𝔻-accessible and G is a retract of F, then G is 𝔻-accessible.

This lemma implies that K_{𝔻} is closed in K under retracts, so that it is Cauchy complete (and then K_{𝔻}^{op} is Cauchy complete too). It remains to prove that

ε : 𝔻-Th^{op}  \overline{}𝔻-LP

is locally an equivalence, that is, for A and B in 𝔻-Th, the functor

ε_{A,B}: 𝔻-Th(A, B) ⟶ 𝔻-LP(ε(B), ε(A))

is an equivalence of categories. Proving that ε_{A,B} is full and faithful is routine calculation based on Yoneda Lemma, which can be used because the Yoneda embedding A^{op} ⟶ [A, Set] clearly factors through 𝔻-cont[A, Set]. We prove now that ε_{A,B} is essentially surjective. For this, consider a morphism R: ε(B) ⟶ ε(A) in 𝔻-LP, together with its left adjoint L and the corestricted Yoneda embeddings as in the following diagram

[figure: A^{op}; 𝔻-cont[A, Set]; Y_{A}; B^{op}; 𝔻-cont[B, Set]; Y_{B}; L_{|}; L; R; ⊣]

We claim that L restricts to a functor A^{op} ⟶ B^{op}. For this, we use the following lemmas. The first one can be deduced from Lemma 3.9 in [1]. We state it explicitly for sake of clearness.

3.5. Lemma. Let A be a locally small and 𝔻-complete category.

1. Any representable functor is a 𝔻-presentable object in 𝔻-cont[A, Set].

2. Any 𝔻-presentable object of 𝔻-cont[A, Set] is a retract of a representable functor.

3.6. Lemma. Let K and K′ be locally small categories with 𝔻-filtered colimits, and consider two functors R: K ⟶ K′, L: K′ ⟶ K with L ⊣ R. If R is 𝔻-accessible, then L preserves 𝔻-presentable objects.

Consider now an object A in A. By the first part of Lemma 3.5 and by Lemma 3.6, L(A(A, −)) is 𝔻-presentable. Then, by the second part of Lemma 3.5, L(A(A, −)) is a retract of a representable functor. Since B is a Cauchy complete category, L(A(A, −)) is itself representable. It follows then that for every A ∈ A we can choose an object B := F A in B such that B(B, −) ≃ L(A(A, −)). Let us fix a natural isomorphism l_{A} : L(A(A, −)) ⟶ B(F A, −) for all A ∈ A. For a morphism f : A ⟶ A′ in A, the Yoneda Lemma guarantees that there exists a unique map F f : F A ⟶ F A′ for which the natural transformation

l_{A} · L(A(f, −)) · l_{A′}^{−1} : B(F A′, −) ⟶ B(F A, −)

is given by composition with F f. In this way, we get a functor F : A ⟶ B, which will show to be 𝔻-continuous by making use of the following lemma.

3.7. Lemma. Let A be an essentially small and 𝔻-complete category. The restricted Yoneda embedding

Y_{A}: A^{op} ⟶ 𝔻-cont[A, Set]

preserves and reflects 𝔻^{op}-colimits.

Since L is a left adjoint and, by Lemma 3.7, Y_{A} preserves 𝔻^{op}-colimits, the composite LY_{A} preserves 𝔻^{op}-colimits. Thus Y_{B}F^{op} preserves 𝔻^{op}-colimits and, by Lemma 3.7 again, Y_{B} reflects them. So F^{op} also preserves them: that is, F preserves 𝔻-limits. Finally, it only remains to prove that the functor ε(F): 𝔻-cont[B, Set] ⟶ 𝔻-cont[A, Set] is isomorphic to R. Since L ⊣ R, it is sufficient to prove that ε(F) is right adjoint to L, i.e., that there is an isomorphism Nat(L(M), G) ≃ Nat(M, G · F) natural in the variables M ∈ 𝔻-cont[A, Set] and G ∈ 𝔻-cont[B, Set]. First, remark that for a representable functor M = A(A, −), where A ∈ A, this is just the Yoneda Lemma. For an arbitrary 𝔻-continuous functor, one just use this fact together Lemma 2.6 (or the fact that 𝔻-cont[A, Set] is the free completion of A^{op} under 𝔻-filtered colimits). The proof of Theorem 3.1 is now complete.

3.8. Example.

1. If we take as 𝔻 the doctrine of finite limits, then Theorem 3.1 is exactly Gabriel-Ulmer duality.

2. If we take as 𝔻 the doctrine of finite products, then Theorem 3.1 is exactly Adamek-Lawvere-Rosicky duality between algebraic theories and multisorted finitary varieties.

3. If we take as 𝔻 the empty doctrine, we have a duality between the 2-category of small Cauchy complete categories, functors and natural transformations, and the 2-category of categories of the form [A, Set] for A small, cocontinuous right adjoint functors and natural transformations. This duality refines the well-known fact that two small categories A and B are Morita equivalent, that is [A, Set] ≃ [B, Set], if and only if they are Cauchy equivalent, that is they have equivalent Cauchy completions. (A Cauchy completion of a small category A is given by the full subcategory of [A, Set] of retracts of representable functors, see [4].)

4. Proof of some lemmas

Proof of Lemma 3.2. It follows from the following bijections natural in C ∈ C and A ∈ A.

A(r · f · j(C), A) ≃ B(f · j(C), i(A)) ≃ D(j(C), g · i(A)) ≃ D(j(C), j · h(A)) ≃ C(C, h(A))

Proof of Lemma 3.4. Consider

[figure: G; F; β; α]

where β · α = 1_{G}. We wish to show that G : K ⟶ Set preserves 𝔻-filtered colimits. For it, take J a 𝔻-filtered category and a functor H : J ⟶ K. For any J ∈ J applying G to the colimit injection σ_{J} : H(J) ⟶ colim_{J∈J} H(J) gives an arrow G(σ_{J}) : G(H(J)) ⟶ G(colim_{J∈J} H(J)). In this way, we get a unique comparison φ : colim_{J∈J} G(H(J)) ⟶ G(colim_{J∈J} H(J)) such that φ · δ_{J} = G(σ_{J}), where δ_{J} are the colimit injections corresponding to the diagram G · H. We will show that φ is an isomorphism. The natural transformations α and β not only induce arrows, for any J ∈ J, as in the following diagram

[figure: G(H(J)); F (H(J)); colim_{J∈J} G(H(J)); colim_{J∈J} F (H(J)); δ_{J}; γ_{J}; β̂; β_{H(J)}; α̂; α_{H(J)}]

which gives, by the universal property of the colimits involved, two morphisms α̂ and β̂ making the correspondent squares commute, but also, for any X ∈ K they provide G(X) as retract of F (X), since β_{X} · α_{X} = (β · α)_{X} = (1_{G})_{X} = 1_{G(X)}. Consider then the diagram

[figure: F (colim_{J∈J} H(J)); colim_{J∈J} F (H(J)); ≃; G(colim_{J∈J} H(J)); colim_{J∈J} G(H(J)); φ; β_{colim_{J∈J} H(J)}; β̂; G(colim_{J∈J} H(J)); colim_{J∈J} G(H(J)); φ; α_{colim_{J∈J} H(J)}; α̂]

where we know that F (colim_{J∈J} H(J)) ≃ colim_{J∈J} F (H(J)), since F is 𝔻-accessible. To exhibit β̂ · α_{colim_{J∈J} H(J)} as the inverse of φ, it is sufficient to show that both squares commute. Commutativity of the bottom square is evident by naturality of α once we compose with the colimit injections δ_{J}, as well as commutativity of the upper square derives from naturality of β by composing with the colimit injections γ_{J}.

Proof of Lemma 3.6. Let M ∈ K′ be 𝔻-presentable. To prove that K(L(M), −) : K ⟶ Set preserves 𝔻-filtered colimits, consider a 𝔻-filtered category J and a functor H : J ⟶ K. We get:

K(L(M), colim_{J∈J} H(J)) ≃ K′(M, R(colim_{J∈J} H(J)))

≃ K′(M, colim_{J∈J} R · H(J))

≃ colim_{J∈J} K′(M, R · H(J))

≃ colim_{J∈J} K(L(M), H(J))

(where the second isomorphism follows from R being 𝔻-accessible and the third one from M being 𝔻-presentable).

Proof of Lemma 3.7. (Note that, when A is Cauchy complete, this is Lemma 2.4.) For any D ∈ 𝔻 and any functor S : D ⟶ A, consider its 𝔻-limit cone (lim_{D∈D} S(D); λ_{D} : lim_{D∈D} S(D) ⟶ S(D)) in A. We wish to show that applying Yoneda to this diagram produces a 𝔻^{op}-colimit cone in 𝔻-cont[A, Set], namely that

(A( lim_{D∈D} S(D), −); A(λ_{D}, −) : A(S(D), −) ⟶ A( lim_{D∈D} S(D), −))

is the 𝔻^{op}-colimit of the diagram Y · S. This is obviously a cocone in 𝔻-cont[A, Set]. To show that it actually coincides with colim_{D∈D^{op}} A(S(D), −) (which exists, 𝔻-cont[A, Set] being cocomplete) we take any object E ∈ 𝔻-cont[A, Set] and just use the Yoneda Lemma, once we have remarked that the functor E is in fact 𝔻-continuous:

Nat(A( lim_{D∈D} S(D), −), E) ≃ E( lim_{D∈D} S(D))

≃ lim_{D∈D} E(S(D))

≃ lim_{D∈D} Nat(A(S(D), −), E)

≃ Nat(colim_{D∈D^{op}} A(S(D), −), E)

Since this is valid for any E ∈ 𝔻-cont[A, Set], again from Yoneda Lemma we can derive that A(lim_{D∈D} S(D), −) ≃ colim_{D∈D^{op}} A(S(D), −), as desired.

4.1. Remark. To end this section, let us point out a simple consequence of Lemmas 3.3 and 3.6. Let K be a locally 𝔻-presentable category and consider a reflective subcategory i: K′ ⟶ K. If the functor i is 𝔻-accessible, then K′ is locally 𝔻-presentable. When 𝔻 is such that [A, Set] is locally 𝔻-presentable for any A in 𝔻-Th, then locally 𝔻-presentable categories are precisely the 𝔻-accessible reflections of presheaf categories. This is the case when 𝔻 is as in Example 3.8. (B. Mesablishvili pointed out to us that, when 𝔻 is the empty doctrine, the previous remark can be generalized to the enriched context, replacing Set by any complete and cocomplete symmetric monoidal closed category. One has just to observe that K′ is monadic over K and then use Theorem 3.12 in [6].)

5. A simple application

To end this note, we sketch a simple application of the duality theorem. In [9, 10], Pedicchio and Wood establish the following precise comparison between the 2-category VAR of varieties and the 2-category LFP of locally finitely presentable categories.

1. The (not full) inclusion of VAR into LFP has a left biadjoint;

2. A small category with finite limits C is such that Lex[C, Set] is equivalent to a variety if and only if every object of C is a regular subobject of an effective injective (an object E ∈ C is effective injective if C(−, E): C^{op} ⟶ Set preserves coequalizers of reflexive graphs).

Here is the general setting: consider two sound doctrines 𝔻_{1} ↪ 𝔻_{2}, so that there are a not full inclusion 𝔻_{2}-Th ⟶ 𝔻_{1}-Th and a corresponding 2-functor j: 𝔻_{2}-LP ⟶ 𝔻_{1}-LP defined by j(𝔻_{2}-cont[C, Set]) = 𝔻_{1}-cont[C, Set]. If every A ∈ 𝔻_{1}-Th has a 𝔻_{1}- conservative 𝔻_{2}-completion, that is if there is a 𝔻_{1}-Th morphism A ⟶ \overline{A} with \overline{A} ∈ 𝔻_{2}- Th universal w.r.t. 𝔻_{2}-complete categories (see [11]), then 𝔻_{2}-Th ⟶ 𝔻_{1}-Th has a left biadjoint. Via the duality, we have:

1. j: 𝔻_{2}-LP ⟶ 𝔻_{1}-LP has a right biadjoint j^{∗} which is an “inclusion” in the sense that j^{∗}(K) ≃ K for any K ∈ 𝔻_{1}-LP;

2. C ∈ 𝔻_{2}-Th is such that 𝔻_{2}-cont[C, Set] ∈ 𝔻_{1}-LP if and only if C is of the form \overline{A} for some A ∈ 𝔻_{1}-Th.

The case studied in [9, 10] corresponds to choosing as 𝔻_{1} the doctrine of finite products and as 𝔻_{2} the doctrine of finite limits. The completion is then the completion under equalizers of coreflexive graphs and the interesting result of [9] is the characterization of those finitely complete categories which are completion of categories with finite products.

The other relevant (even if much more easy) example is when 𝔻_{1} is the empty doctrine and 𝔻_{2} is the doctrine of finite products, so that 𝔻_{1}-LP is the 2-category PS of presheaf categories and 𝔻_{2}-LP is VAR. The next lemma provides the context for this example. We state it in terms of the more popular dual completion: we write Fam_{f}A for the category of finite families of objects of A, which is the completion of A under finite coproducts. Part 2 of the lemma is simply the finitary version of Proposition 6.1.5 in [5]. We call an object finitely connected if the associated representable functor preserves finite coproducts.

5.1. Lemma.

1. If A is small and Cauchy complete, then Fam_{f}A is small and Cauchy complete. Conversely, if C has finite coproducts and is Cauchy complete, then the full subcategory of finitely connected objects is Cauchy complete.

2. A category C with finite coproducts is of the form Fam_{f}A if and only if each object is a finite coproduct of finitely connected objects; when this is the case, one can choose as A the full subcategory of finitely connected objects.

Proof of 1. One implication can be proved using the same arguments as in the proof of Lemma 3.4. Conversely, assume that A is small and Cauchy complete. An object in Fam_{f}A is a pair (I, f) with I a finite set and f: I ⟶ A a functor; an arrow (a, α): (I, f) ⟶ (J, g) is given by a functor a: I ⟶ J and a natural transformation α: f ⇒ g · a. Clearly, Fam_{f}A is small because its objects are in bijection with ∐_{I}(ObA)^{I} for I varying in the category of finite sets, which is (essentially) small. Consider now an idempotent (a, α): (I, f) ⟶ (I, f). The condition (a, α) · (a, α) = (a, α) means

\{ \begin{matrix}a · a = a & that is a is an idempotent in Set \\ α_{a(i)} · α_{i} = α_{i} & ∀i ∈ I \end{matrix}  (2)

We can consider the splitting of the idempotent a in Set, which is given by the following equalizer

\xymatrix{J \ar[rr]^{c} &  & I \ar[r]^{a} \ar[r]_{1_{I}} & I \\  & I \ar[ul]^{b} \ar@{-->}[ur]_{a}}

so that J = {j ∈ I | a(j) = j}, c · b = a, b · c = 1_{J}. Now, for each j ∈ J, the second equation in (2) says that α_{j}: f(j) ⟶ f(j) is an idempotent in A, and then it splits. Let us consider its splitting

\xymatrix{X_{j} \ar[rr]^{s_{j}} &  & f(j) \ar[r]^{α_{j}} \ar[r]_{1_{f(j)}} & f(j) \\  & f(j) \ar[ul]^{t_{j}} \ar@{-->}[ur]_{α_{j}}}

that is s_{j} · t_{j} = α_{j} and t_{j} · s_{j} = 1_{X_{j}}. In this way we get an object (J, g: J ⟶ A: j ↦ X_{j}) in Fam_{f}A and two arrows (b, β): (I, f) ⟶ (J, g) and (c, γ): (J, g) ⟶ (I, f) defined by

- b: I ⟶ J b(i) = a(i) , β_{i} = t_{a(i)} · α_{i}
- c: J ⟶ I c(j) = j , γ_{j} = s_{j}

and one can check that (c, γ) · (b, β) = (a, α), (b, β) · (c, γ) = 1_{(J,g)}.

Accordingly with the general setting, we have:

1. the inclusion PS ⟶ VAR is right biadjoint to j: VAR ⟶ PS;

2. An algebraic theory C is such that the corresponding variety is equivalent to a presheaf category if and only if each object of C is a finite product of “finitely coconnected” objects (that is, objects X such that C(−, X): C^{op} ⟶ Set preserves finite coproducts).

Putting together the PS - VAR comparison and the VAR - LFP comparison, we have:

5.2. Corollary. Let C be a finitely complete small category. The following conditions are equivalent:

1. Lex[C, Set] is equivalent to a presheaf category;

2. C is equivalent to the free completion under finite limits of a small category;

3. Any object of C is a regular subobject of a finite product ∏_{I} X_{i} such that, for all i ∈ I, the functor C(−, X_{i}): C^{op} ⟶ Set preserves finite colimits.

Proof. Once again we work with the dual of the theory. We write B ⟶ B_{rc} for the completion under coequalizers of reflexive graphs of a category B with finite coproducts. Because of the general setting stated above, we have to prove that:

1′. C ≃ (Fam_{f}A)_{rc} for some small Cauchy complete category A

if and only if (the dual of) condition 3 holds. (Indeed, the implication 2 ⇒ 1 is obvious and the implication 1′ ⇒ 2 is proved in [9].) 1′ ⇒ 3 : Note that a functor defined on a finitely cocomplete category preserves finite colimits if it preserves coequalizers of reflexive graphs and finite coproducts. Each object of C is a regular quotient of a finite coproduct of objects in A. Each object of A is finitely connected in B = Fam_{f}A and it is effective projective in C, so that we have only to prove that B ⟶ B_{rc} preserves finitely connected objects. But this follows, via the commutativity of finite coproducts with coequalizers of reflexive graphs, from the fact that each object of B_{rc} is the coequalizer of a reflexive graph in B. 3 ⇒ 1′ : Let us write A for the full subcategory of C of the objects A such that C(A, −): C ⟶ Set preserves finite colimits, B for the full subcategory spanned by finite coproducts of objects of A, and E for the full subcategory of effective projectives. Since each A ∈ A is effective projective and effective projectives are closed under finite coproducts, we have that B is contained in E and C has enough effective projectives, so that C ≃ E_{rc}. Moreover, since C has finite colimits, each object of E is regular projective and then it is a retract of an object of B. By Lemma 5.1, B is equivalent to Fam_{f}A and then it is Cauchy complete because A is Cauchy complete (same argument as in the proof of Lemma 3.4). Finally, E ≃ B and C ≃ (Fam_{f}A)_{rc}.

References

[1] J. Adámek, F. Borceux, S. Lack and J. Rosický: A classification of accessible categories, Journal of Pure and Applied Algebra 175 (2002) 7–30.

[2] J. Adámek, F.W. Lawvere and J. Rosický: On the duality between varieties and algebraic categories, Algebra Universalis (to appear).

[3] J. Adámek and H. Porst: Algebraic theories of quasivarieties, Journal of Algebra 208 (1998) 379–398.

[4] F. Borceux: Handbook of categorical algebra I, Cambridge University Press (1994).

[5] F. Borceux and G. Janelidze: Galois theories, Cambridge University Press (2001).

[6] J. Fisher-Palmquist and D.C. Newell: Triples on functor categories, Journal of Algebra 25 (1973) 226-258.

[7] P. Gabriel and F. Ulmer: Lokal präsentierbare Kategorien, Springer LNM 221 (1971).

[8] G.M. Kelly: Basic concepts of enriched category theory, Cambridge University Press (1982).

[9] M.C. Pedicchio and R.J. Wood: A simple characterization of theories of varieties, Journal of Algebra 233 (2000) 483-501.

[10] M.C. Pedicchio and R.J. Wood: A note on effectively projective objects, Journal of Pure and Applied Algebra 158 (2001) 83-87.

[11] J. Velebil and J. Adámek: A remark on conservative cocompletions of categories, Journal of Pure and Applied Algebra 168 (2002) 107-124.

Institut de mathématique pure et appliquée Université catholique de Louvain Chemin du Cyclotron 2 B 1348 Louvain-la-Neuve, Belgique

Email: centazzo@math.ucl.ac.be and vitale@math.ucl.ac.be

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