Theory and Applications of Categories, Vol. 30, No. 48, 2015, pp. 1647–1658.

LIMITS OF ABSTRACT ELEMENTARY CLASSES

M. LIEBERMAN, J. ROSICKÝ

Abstract. We show that the category of abstract elementary classes (AECs) and concrete functors is closed under constructions of “limit type,” which generalizes the approach of Mariano, Zambrano and Villaveces away from the syntactically oriented framework of institutions. Moreover, we provide a broader view of this closure phenomenon, considering a variety of categories of accessible categories with additional structure, and relaxing the assumption that the morphisms be concrete functors.

1. Introduction

One of the main virtues of accessible categories is that they are closed under constructions of limit type ([9]). This should be made precise by considering accessible functors between accessible categories and showing that the resulting 2-category is closed under appropriate limits. These limits can be reduced to products, inserters and equifiers and are called PIE-limits. Proofs of this result (see [9], or [1]) also show that the category of accessible categories with directed colimits and functors preserving directed colimits is closed under PIE-limits. The needed 2-categorical limits are explained both in [9] and [1] and we recommend [4] for a more systematic introduction.

Recent papers [3], [6] and [7] have shown that abstract elementary classes ([2]) can be understood as special accessible categories with directed colimits. In [7], in particular, the authors develop a hierarchy of such categories, extending from accessible categories with directed colimits to AECs themselves. Here we show that each stage in this hierarchy is closed under PIE-limits as well, provided we take the morphisms to be directed colimit preserving functors. This closure becomes more problematic if we insist that the morphisms be concrete functors: here we see that the iso-fullness axiom for AECs (heretofore unneeded in the category-theoretic analysis thereof) is essential to guarantee the existence of desired limits.

Schematically, our results encompass the categories in the figure below, where the downward-accumulating properties of the objects are described in the left margin, and the properties of the morphisms are listed at the top.

Supported by the Grant agency of the Czech republic under the grant P201/12/G028.

Received by the editors 2015-05-22 and, in revised form, 2015-11-17.

Transmitted by Tom Leinster. Published on 2015-11-30.

2010 Mathematics Subject Classification: 18C35, 03C48, 03C95.

Key words and phrases: accessible category, abstract elementary class, PIE limit.

\begin{matrix}c \\ ◯ \end{matrix} M. Lieberman, J. Rosický, 2015. Permission to copy for private use granted.

Preserve directed colimits Subconcrete Concrete Accessible Acc Directed colimits _{Acc_{0}} Concrete directed colimits _{Acc_{1}}  _{Acc{† \atop 1}}  _{Acc{∗ \atop 1}} Coherent, concrete monos _{Acc_{2}}  _{Acc{† \atop 2}}  _{Acc{∗ \atop 2}} Iso-full  _{Acc_{3}}  _{Acc{† \atop 3}}  _{Acc{∗ \atop 3}}

Subconcrete functors, introduced in Definition 3.1 below, are a natural generalization of the concrete case. We show that all the pictured categories are closed under PIE-limits in Acc, with the exception of Acc_{3}, Acc_{1}^{∗} and Acc_{3}^{∗}. The last category has PIE-limits but it is closed in Acc only under inserters and equifiers while products are calculated in Acc ↓ Set.

We note that the objects in categories along the bottom row are (equivalent to) AECs, but equipped with three different notions of morphism, ranging from the most general— functors preserving directed colimits—to a very close generalization of the syntacticallyderived functors in [10], namely directed-colimit preserving functors that are concrete, i.e. respect underlying sets. We note also that while AECs are replete (their individual isomorphism classes are of maximal size, in a sense discussed immediately following Remark 2.6 below), this property need not be shared by categories to which they are equivalent and, in particular, by the objects of Acc_{3}, Acc_{3}^{†}, and Acc_{3}^{∗}. Provided that we are not concerned with the sizes of isomorphism classes, however, this is of no consequence. Modulo this technical detail, the closure result corresponding to the Acc_{3}^{∗} is the promised generalization of [10], shifting it out of the framework of institutions and into a more intrinsic, purely syntax-free characterization. We consider the precise relationship between our result and that of [10] in Remark 3.4.

In fact, our ambitions are broader: inspired by the example of metric AECs, in which directed colimits need not be concrete but ℵ_{1}-directed colimits always are, we consider a second version of this diagram in which we require only that the categories from the third row down have concrete κ-directed colimits for a given κ—such categories will be distinguished by the superscript κ. In particular, the category Acc_{3}^{†κ} will consist of κ-CAECs as defined in [8], with subconcrete functors as morphisms. We obtain a closure result there as well.

2. Accessible categories with directed colimits

Recall that a category K is λ-accessible, λ a regular cardinal, if it has λ-directed colimits (i.e. colimits indexed by a λ-directed poset) and contains, up to isomorphism, a set A of λ-presentable objects such that each object of K is a λ-directed colimit of objects from A. Here, an object K is λ-presentable if its hom-functor K(K, −) : K → Set preserves λ-directed colimits. A category is accessible if it is λ-accessible for some λ. A functor F : K → L between λ-accessible categories is called λ-accessible if it preserves λ-directed colimits. F is called accessible if it is λ-accessible for some λ. In this way, we get the category Acc whose objects are accessible categories and morphisms are accessible functors.

2.1. Remark. We work in the Gödel-Bernays set theory. Thus a category K is a class of objects together with a class K(A, B) of morphisms A → B for each object A and B. It is called locally small if all K(A, B) are sets. Any accessible category is locally small. It is important to observe that Acc is a category which is not locally small. The reason is that a λ-accessible functor F : K → L is determined by its restriction on the full subcategory A of λ-presentable objects.

We may regard Acc as a 2-category where the 2-cells are natural transformations. As noted above, Acc is closed under appropriate 2-limits, namely PIE-limits, where “PIE” abbreviates “products,” “inserters” and “equifiers.” This means that these 2-limits exist in Acc and are calculated in the non-legitimate category CAT of categories, functors and natural transformations. It follows that Acc is closed under lax limits and under pseudolimits (see [9] or [1]).

Recall that, given functors F, G : K → L, the inserter category Ins(F, G) is the subcategory of the comma category F ↓ G consisting of all objects f : F K → GK and all morphisms

\xymatrix{F K \ar[r]^{f} \ar[d]_{F k} & GK \ar[d]^{Gk} \\ F K′ \ar[r]_{f′} & GK′}

The projection functor P : Ins(F, G) → K sends f : F K → GK to K. The universal property of Ins(F, G) is the existence of a natural transformation φ : F P → GP (given as φ_{P f} = f) in the sense that for any H : H → K with ψ : F H → GH there is a unique H̄ : H → Ins(F, G) such that P H̄ = H and φH = ψ (see [4]). Since Acc is full in CAT with respect to 2-cells, we can ignore the 2-dimensional aspect of universality.

Given functors F, G : K → L and natural transformations φ, ψ : F → G, the equifier Eq(φ, ψ) is the full subcategory of K consisting of all objects K such that φ_{K} = ψ_{K}. Let P : Eq(φ, ψ) → K be the inclusion. The universal property of Eq(φ, ψ) is that φP = ψP and for any H : H → K with φH = ψH there is a unique H̄ : H → K such that P H̄ = H (see [4]); the 2-dimensional aspect of universality can be ignored again.

We now consider accessible categories having all directed colimits. Let Acc_{0} be the 2-category whose objects are accessible categories with directed colimits, morphisms are functors preserving directed colimits and 2-cells are natural transformations.

2.2. Theorem. Acc_{0} is closed under PIE-limits in Acc.

Proof. Let K_{i}, i ∈ I be accessible categories with directed colimits. Following [1] 2.67, the product ∏_{i∈I} K_{i} is an accessible category. Clearly, it has all directed colimits and the projections P_{i} : ∏ K_{i} → K_{i} preserve them. Let L be an accessible category with directed colimits and Q_{i} : L → K_{i} functors preserving directed colimits. Then the induced functor L → ∏ K_{i} preserves directed colimits. Hence ∏ K_{i} is the product in Acc_{0}.

Let K, L be accessible categories with directed colimits and F, G : K → L be functors preserving directed colimits. Following [1] 2.72, Ins(F, G) is an accessible category which clearly has directed colimits. Let H be an accessible category with directed colimits, H : H → K preserve directed colimits and ψ : F H → GH a natural transformation. Then the induced functor H̄ : H → Ins(F, G) preserves directed colimits. Hence Ins(F, G) is an inserter in Acc_{0}.

Finally, let K, L be accessible categories with directed colimits, F, G : K → L functors preserving directed colimits and φ, ψ : F → G natural transformations. Following [1] 2.76, Eq(φ, ψ) is an accessible category. Again, it is clear that Eq(φ, ψ) has all directed colimits. Let H be an accessible category with directed colimits and H : H → K a functor preserving directed colimits with φH = ψH. Then the induced functor H̄ : H → Eq(φ, ψ) preserves directed colimits. Hence Eq(φ, ψ) is an equifier in Acc_{0}.

We say that (K, U) is an accessible category with concrete directed colimits if K is an accessible category with directed colimits and U : K → Set is a faithful functor to the category of sets that preserves directed colimits. Let Acc_{1} be the full sub-2-category of Acc_{0} consisting of accessible categories with concrete directed colimits. In particular, morphisms in Acc_{1} are functors preserving directed colimits.

2.3. Theorem. Acc_{1} is closed under PIE-limits in Acc.

Proof. We must show that PIE-limits of accessible categories with concrete directed colimits have concrete directed colimits (since Acc_{1} is a full subcategory of Acc_{0}, we do not need to bother about universal properties). This is evident for inserters and equifiers because, in the first case, the projection functor P : Ins(F, G) → K is faithful and, in the second case, Eq(φ, ψ) is a full subcategory of K. Consider accessible categories with concrete directed colimits (K_{i}, U_{i}), i ∈ I. Then the functor U : ∏_{i∈I} K_{i} → Set sending (A_{i})_{i∈I} to ∐_{i∈I} U_{i}A_{i} is faithful. Since

colim ∐_{i∈I} U_{i}A_{i} ≅ ∐_{i∈I} colim U_{i}A_{i},

∏_{i∈I} K_{i} is an accessible category with concrete directed colimits.

2.4. Remark. (1) We could also consider the subcategory Acc_{1}^{∗} having the same objects as Acc_{1} but whose morphisms are concrete functors F : K_{1} → K_{2} preserving directed colimits. By “concrete,” we mean that F commutes with the relevant underlying set functors, i.e. U_{2}F = U_{1}. The category Acc_{1}^{∗} is closed in Acc under inserters and equifiers but not under products. In fact, we are in the comma category Acc ↓ Set where ∏_{i∈I}(K_{i}, U_{i}) is the multiple pullback of U_{i} over Set. While Acc has multiple pseudopullbacks, it does not have multiple pullbacks. For multiple pullbacks, we would need all of the functors U_{i} to be transportable in the sense that for any isomorphism f : U_{i}A → X there is a unique isomorphism \overline{f} : A → B such that U_{i}(\overline{f}) = f (this also implies U_{i}\overline{B} = X). Then a multiple pullback of U_{i} is equivalent to their multiple pseudopullback and thus it belongs to Acc. This is done for a pullback in [9] 5.1.1 and the multiple case is analogous.

(2) Theorem 2.3 is also valid for the full sub-2-category Acc_{1}^{κ} of Acc_{0} consisting of accessible categories with directed colimits where κ-directed colimits are concrete. These categories appear in [8] in connection with metric abstract elementary classes.

An accessible category (K, U) with concrete directed colimits is coherent if for each commutative triangle

\xymatrix{UA \ar[rr]^{U(h)} \ar[dr]_{f} &  & UC \\  & UB \ar[ur]_{U(g)}}

there is \overline{f} : A → B in K such that U(\overline{f}) = f.

We say that morphisms of K are concrete monomorphisms if any morphism of K is a monomorphism which is preserved by U. Let Acc_{2} be the full sub-2-category of Acc_{1} consisting of coherent accessible categories with concrete monomorphisms.

2.5. Theorem. Acc_{2} is closed under PIE-limits in Acc.

Proof. Since there is no problem with concrete monomorphisms, we have to show that PIE-limits of coherent accessible categories are coherent (universal properties are again clear). This is evident for equifiers because Eq(φ, ψ) is a full subcategory of K. Consider coherent accessible categories (K_{i}, U_{i}), i ∈ I. We have to show that U : ∏_{i∈I} K_{i} → Set sending (A_{i})_{i∈I} to ∐_{i∈I} U_{i}A_{i} is coherent. Consider a commutative triangle

\xymatrix{U(A_{i}) \ar[rr]^{U(h)} \ar[dr]_{f} &  & U(C_{i}) \\  & U(B_{i}) \ar[ur]_{U(g)}}

and a ∈ U_{i}A_{i}. Assume that fa_{i} ∈ U_{j}B_{j} for j ≠ i. Then (Ug)fa_{i} ∈ U_{j}C_{j} and (Uh)a_{i} ∈ U_{i}C_{i}, which is impossible. Thus f = ∐_{i∈I} f_{i}. Since each U_{i} is coherent, there are morphisms \overline{f}_{i} : A_{i} → B_{i} such that U(\overline{f}_{i}) = f. Hence ∏_{i∈I} K_{i} is coherent.

Consider morphisms F, G : K → L in Acc_{2}. We have to show that the composition

Ins(F, G) ⟶^{P} K ⟶^{U} Set

is coherent. Consider a commutative triangle

\xymatrix{UP f_{1} \ar[rr]^{UP (h)} \ar[dr]_{f} &  & UP f_{3} \\  & UP f_{2} \ar[ur]_{UP (g)}}

where f_{i} : F K_{i} → GK_{i}, i = 1, 2, 3. Thus we have a commutative triangle

\xymatrix{UK_{1} \ar[rr]^{U(h)} \ar[dr]_{f} &  & UK_{3} \\  & UK_{2} \ar[ur]_{U(g)}}

and, since U is coherent, we have f = U\overline{f}. Thus we get the diagram

\xymatrix{F K_{1} \ar[r]^{f_{1}} \ar[d]_{F \overline{f}} & GK_{1} \ar[d]^{G\overline{f}} \\ F K_{2} \ar[r]^{f_{2}} \ar[d]_{F g} & GK_{2} \ar[d]^{Gg} \\ F K_{3} \ar[r]_{f_{3}} & GK_{3}}

where the outer rectangle and the bottom square commute. Since Gg is a monomorphism, the upper square commutes as well. Hence \overline{f} : f_{1} → f_{2} is a morphism in Ins(F, G) and f = UP \overline{f}. Therefore P U is coherent.

2.6. Remark. (1) The assumption that objects of Acc_{2} have concrete monomorphisms was needed in the proof of closure under inserters.

(2) Theorem 2.5 is also valid for the full sub-2-category Acc_{2}^{κ} of Acc_{1}^{κ} consisting of coherent accessible categories with directed colimits and concrete monomorphisms.

Abstract elementary classes can be characterized as coherent accessible categories K with directed colimits and with concrete monomorphisms satisfying two additional conditions dealing with finitary function and relation symbols interpretable in K (see [7]). Here, finitary relation symbols interpretable in K are subfunctors R of U^{n} = Set(n, U−) where n is a finite cardinal. Finitary function symbols interpretable in K are natural transformations h : U^{n} → U. Since n-ary function symbols can be replaced by (n + 1)- ary relation symbols, we can confine ourselves to finitary relation symbols interpretable in K. Let Σ_{K} consist of those finitary relation symbols R interpretable in K for which K-morphisms f : A → B behave as embeddings. This means that if (Uf)^{n}(a) ∈ R_{B} then a ∈ R_{A}. We get the functor E : K → Emb(Σ_{K}) where Emb(Σ_{K}) is the category of Σ_{K}-structures whose morphisms are substructure embeddings. Now, K is an abstract elementary class if and only if the functor E is full with respect to isomorphisms and replete. The first condition means that if f : EA → EB is an isomorphism then there is an isomorphism \overline{f} : A → B with E\overline{f} = f. In this case, we might also say, equivalently, that the relations R ∈ Σ_{K} detect isomorphisms. The second condition means that if EA is isomorphic to X then there is B ∈ K such that A is isomorphic to B and EB = X.

We note that abstract elementary classes are commonly presented via an embedding K → Emb(Σ). In this case, Σ ⊆ Σ_{K} and, in fact, Σ_{K} is the largest relational signature in which K can be presented.

Let Acc_{3} be the full sub-2-category of Acc_{2} consisting of categories equivalent to abstract elementary classes. Following the discussion in the introduction, such categories need not be replete in the sense described above—their isomorphism classes may be smaller, in principle, than one would expect in an abstract elementary class. To be precise, then, Acc_{2} and Acc_{3} differ only in that we assume objects in the latter are full with respect to isomorphisms.

2.7. Proposition. Acc_{3} is closed under products and equifiers in Acc.

Proof. The closedness under equifiers immediately follows from the fact that Eq(φ, ψ) → K is a replete, full embedding. Consider (K_{i}, U_{i}), i ∈ I, in Acc_{3}. Given n-ary relation symbols R_{i} ∈ Σ_{K_{i}} where i ∈ I, we get the n-ary relation symbol R = ∐_{i} R_{i} belonging to Σ_{∏_{i} K_{i}}. It includes unary interpretable relation symbols given by R_{j} = U_{j} and R_{i} = ∅ for i ≠ j. It is easy to see that these R detect isomorphisms. Thus E : ∏_{i} K_{i} → Emb(Σ_{∏ K_{i}}) is full with respect to isomorphisms.

2.8. Remark. In the case of inserters, any finitary relation symbol R interpretable in K yields the finitary relation symbol RP interpretable in Ins(F, G). Let f_{i} : F K_{i} → GK_{i} for i = 1, 2 and

f : UK_{1} = UP f_{1} → UP f_{2} = UK_{2}

be a bijection such that f^{n} induces a bijection between Sf_{1} and Sf_{2} for each n-ary relation symbol S interpretable in Ins(F, G). By taking S = RP we get that f^{n} induces a bijection between RK_{1} and RK_{2} for each n-ary relation symbol R interpretable in K. Since E : K → Emb(Σ_{K}) is full with respect to isomorphisms, there is an isomorphism \overline{f} : K_{1} → K_{2} with U\overline{f} = f. But we do not know whether \overline{f} : f_{1} → f_{2} is a morphism, i.e., whether the square

\xymatrix{F K_{1} \ar[r]^{f_{1}} \ar[d]_{F \overline{f}} & GK_{1} \ar[d]^{G\overline{f}} \\ F K_{2} \ar[r]_{f_{2}} & GK_{2}}

commutes.

2.9. Problem. Is Acc_{3} closed under inserters in Acc?

3. Abstract elementary classes

3.1. Definition. Let (K_{1}, U_{1}) and (K_{2}, U_{2}) be concrete categories. We say that a functor H : K_{1} → K_{2} is subconcrete if there is a natural monotransformation α : U_{2}H → U_{1} such that if (U_{1}f)a ∈ U_{2}HB then a ∈ U_{2}HA for each f : A → B in K_{1}.

This means that U_{2}H is a unary relation symbol belonging to Σ_{K_{1}}. Any concrete functor is subconcrete. Since a composition of subconcrete functors is subconcrete, we get the subcategory Acc_{1}^{†} of Acc_{1} consisting of accessible categories with concrete directed colimits and subconcrete functors preserving directed colimits. Analogously, we get the full subcategory Acc_{2}^{†} of Acc_{1}^{†} consisting of coherent accessible categories and concrete monomorphisms whose morphisms are subconcrete functors preserving directed colimits. Finally, we have the category

Acc_{3}^{†} = Acc_{3} ∩ Acc_{2}^{†}

of categories equivalent to abstract elementary classes and subconcrete functors preserving directed colimits.

3.2. Theorem. Acc_{1}^{†}, Acc_{2}^{†} and Acc_{3}^{†} are closed under PIE-limits in Acc.

Proof. In Acc_{1}^{†} and Acc_{2}^{†}, equifiers and inserters are the same as in Acc_{1} and Acc_{2}. The functors P : Eq(φ, ψ) → K and P : Ins(F, G) → K are concrete. Thus it remains only to check that the induced maps arising from the universal property are subconcrete. We will do it for inserters (the case of equifiers is the same), and present the argument only for Acc_{1}: the proof for Acc_{2} is identical. Let (K, U), (L, U′) be accessible categories with concrete directed colimits and let F, G : K → L be subconcrete functors preserving directed colimits. Let (H, V) be an accessible category with concrete directed colimits, H : H → K a subconcrete functor preserving directed colimits and ψ : F H → F G a natural transformation. We have to show that the induced functor H̄ : H → Ins(F, G) is subconcrete. We have a natural transformation α : UH → V witnessing the subconcreteness of H. Then α : (UP)H̄ = UH → V witnesses subconcreteness of H̄.

In the case of products, the projections P_{i} : ∏_{i} K_{i} → K_{i} are subconcrete — take the coproduct injections U_{i}P_{i} → U where U : ∏_{i} K_{i} → Set is from the proof of 2.3. Again, we have to check the universal property. Let (H, V) be a an accessible category with concrete directed colimits and H_{i} : H → K_{i} be subconcrete functors preserving directed colimits with witnesses α_{i} : U_{i}H_{i} → V. Then, for A ∈ H, the natural transformation α with components

α_{A} : UHA = ∐_{i} U_{i}H_{i}A → V A

induced by (α_{i})_{A} : U_{i}H_{i}A → V A witnesses the sub concreteness of H̄ : H → ∏_{i} K_{i}.

It remains to prove that Acc_{3}^{†} is closed under inserters. Let (K_{1}, U_{1}) and (K_{2}, U_{2}) be abstract elementary classes and F, G : K_{1} → K_{2} subconcrete functors. First, in the notation of 2.8, we show that the square

\xymatrix{F K_{1} \ar[r]^{f_{1}} \ar[d]_{F \overline{f}} & GK_{1} \ar[d]^{G\overline{f}} \\ F K_{2} \ar[r]_{f_{2}} & GK_{2}}

commutes. Since F and G are subconcrete, we get unary relation symbols U_{2}F, U_{2}G ∈ Σ_{K}. Hence we have unary relation symbols

U_{2}F P, U_{2}GP ∈ Σ_{Ins(F,G)}.

Thus we have a binary relation symbol R ∈ Σ_{Ins(F,G)} such that (a, b) ∈ R_{g}, g : F K → GK, if a ∈ U_{2}F P g, b ∈ U_{2}GP g and b = (U_{2}g)a. To see that the above square commutes, notice that (a, (U_{2}f_{1})a) ∈ R_{f_{1}} for each a ∈ U_{2}F K_{1}. It follows that ((U_{2}F \overline{f})a, U_{2}(G(\overline{f})f_{1})a) ∈ R_{f_{2}}, and therefore that U_{2}(G(\overline{f})f_{1})a = U_{2}(f_{2}F \overline{f})a for each a ∈ U_{2}F K_{1}. Hence

U_{2}(G(\overline{f})f_{1}) = U_{2}(f_{2}F \overline{f})

and, since U_{2} is faithful,

G(\overline{f})f_{1} = f_{2}F \overline{f}.

Finally, we have the category

Acc_{3}^{∗} = Acc_{3} ∩ Acc_{1}^{∗}

of categories equivalent to abstract elementary classes whose morphisms are concrete functors preserving directed colimits.

3.3. Theorem. Acc_{3}^{∗} has PIE-limits.

Proof. Since concrete functors are subconcrete, Acc_{3}^{∗} is closed in Acc under inserters and equifiers. Concerning their universal property, if H is concrete in the notation of the proof of 3.2 then H̄ is concrete as well. Products ∏_{i∈I}(K_{i}, U_{i}) are calculated in Acc ↓ Set, i.e., they are multiple pullbacks. Since any abstract elementary class (K, U) has U transportable, multiple pullbacks are equivalent to multiple pseudopullbacks (see 2.4). Following 3.2, Acc_{3}^{†} is closed in Acc under PIE-limits and, consequently, under pseudolimits. Thus ∏_{i∈i}(K_{i}, U_{i}) belongs to Acc_{3}^{∗} and is the product of (K_{i}, U_{i}) there.

3.4. Remark. (1) Let H : K_{1} → K_{2} be a morphism in Acc_{3}^{†}. Since (U_{2}H)^{n} is an n-ary relation symbol belonging to Σ_{K_{1}}, we get an embedding of signatures \overline{H} : Σ_{K_{2}} → Σ_{K_{1}} sending R to RH. In particular, it sends U_{2} to U_{2}H. This induces the subconcrete functor Emb(\overline{H}) : Emb(Σ_{K_{1}}) → Emb(Σ_{K_{2}}) given by taking reducts. The square

\xymatrix{K_{1} \ar[r]^{H} \ar[d]_{E_{1}} & K_{2} \ar[d]^{E_{2}} \\ Emb(Σ_{K_{1}}) \ar[r]_{Emb(\overline{H})} & Emb(Σ_{K_{2}})}

clearly commutes.

If H is concrete then Emb(\overline{H}) is concrete as well. This relates our morphisms of abstract elementary classes to the syntactically-derived morphisms considered in [10].

(2) On the other hand, let G : Σ_{2} → Σ_{1} be an embedding of signatures. Let K_{1} → Emb(Σ_{1}) and K_{2} → Emb(Σ_{2}) be abstract elementary classes—in the classical sense— presented in signatures Σ_{1} and Σ_{2}: note that, when paired with their natural underlying set functors U_{i} : K_{i} → Set, they satisfy the purely category-theoretic characterization of AECs following Remark 2.6. Moreover, let H : K_{1} → K_{2} be a functor such that the square

\xymatrix{K_{1} \ar[r]^{H} \ar[d] & K_{2} \ar[d] \\ Emb(Σ_{1}) \ar[r]_{Emb(G)} & Emb(Σ_{2})}

commutes. Since Emb(G) is concrete, H is a morphism in Acc_{3}^{∗}. These are precisely the morphisms of abstract elementary classes considered in [10].

More generally, consider relational signatures Σ_{1}, Σ_{2} and let L(Σ_{1}), L(Σ_{2}) be the corresponding languages, i.e., sets of all formulas of Σ_{1}, Σ_{2}. Consider a mapping −^{∗} : Σ_{2} → Σ_{1} of signatures preserving the arity of symbols and let P be a unary relation symbol in Σ_{1}. Let G : L(Σ_{2}) → Ł(Σ_{1}) be a morphism of languages sending each (n-ary) relation symbol R in Σ_{1} to P^{n} ∧ R^{∗}. This defines G on the atomic formulas of L(Σ_{1}); we extend it recursively to all of L(Σ_{1}). In particular, G sends = to the equality =_{P} on P.

Then Emb(G) : Emb(Σ_{1}) → Emb(Σ_{2}) is a subconcrete functor. Let K_{1} → Emb(Σ_{1}) and K_{2} → Emb(Σ_{2}) be abstract elementary classes and H : K_{1} → K_{2} be a functor such that the square

\xymatrix{K_{1} \ar[r]^{H} \ar[d] & K_{2} \ar[d] \\ Emb(Σ_{1}) \ar[r]_{Emb(G)} & Emb(Σ_{2})}

commutes. Then H is a morphism in Acc_{3}^{†}.

(3) Let K be the category of infinite sets and monomorphisms. Then K is an abstract elementary class in the empty signature Σ_{2}. Let Σ_{1} contain just P and =_{P} from (2) and G : Σ_{2} → Σ_{1} be the corresponding morphism of languages. We also have H : Σ_{1} → Σ_{2} sending =_{P} to = and P to the formula x = x. In this way, we make K isomorphic to an abstract elementary class in the signature Σ_{1}, where we interpret objects of K as Σ_{1}-structures K with P_{K} infinite and (¬P)_{K} countable (see [3], 5.8(3) motivated by [5] 2.10).

(4) Theorem 3.2 is also valid for categories Acc_{1}^{†κ}, Acc_{2}^{†κ} and Acc_{3}^{†κ} where Acc_{1} is replaced by Acc_{1}^{κ}.

Analogously, Theorem 3.3 is valid for Acc_{3}^{∗κ}.

3.5. Lemma. Any morphism in Acc_{3}^{†} is coherent and transportable.

Proof. Consider the square from 3.4(1). Since the functor Emb(\overline{H}) is coherent, the composition Emb(\overline{H})E_{1} is coherent as well. Since E_{2} is faithful, H is coherent.

Since Σ_{K_{1}} and Σ_{K_{2}} contain only relation symbols, the functor Emb(\overline{H}) is surjective on objects and full (by interpreting the missing relations as empty). Consider an isomorphism f : HA → B. We get the isomorphism

E_{2}f : Emb(\overline{H})E_{1}A = E_{2}HA → E_{2}B = Emb(\overline{H})B̃.

and thus the isomorphism f̃ : E_{1}A → B̃ such that Emb(\overline{H})f̃ = E_{2}f. Since E_{1} is transportable, there is an isomorphism \overline{f} : A → \overline{B} such that E_{1}\overline{B} = B̃ and E_{1}\overline{f} = f̃. Clearly, H\overline{f} = f. Thus H is transportable.

3.6. Remark. Following 3.5 and 2.4, pullbacks in Acc_{3}^{†} are equivalent to pseudopullbacks. Thus 3.2 implies that Acc_{3}^{†} is closed in CAT under pullback. Consequently, the same holds for Acc_{3}^{∗}, which was proved in [10].

References

[1] J. Adámek and J. Rosický, Locally Presentable and Accessible Categories, volume 189 of London Math. Soc. Lecture Notes, Cambridge UP, 1994.

[2] J. Baldwin, Categoricity, vol. 50 of Univ. Lecture Series, Amer. Math. Soc., 2009.

1658  M. LIEBERMAN, J. ROSICKÝ

[3] T. Beke and J. Rosický, Abstract elementary classes and accessible categories, Ann. Pure Appl. Logic 163(12):2008-2017, 2012.

[4] G. M. Kelly, Elementary observations on 2-categorical limits, Bull. Austral. Math. Soc., 39:301-317, (1989).

[5] D. W. Kueker, Abstract elementary classes and infinitary logic, Ann. Pure Appl. Logic 156(2):274-286, (2008).

[6] M. Lieberman, Category theoretic aspects of abstract elementary classes, Annals Pure Appl. Logic 162(11):903-915, (2011).

[7] M. Lieberman and J. Rosický, Classification theory for accessible categories, to appear in Jour. Symb. Logic, arXiv:1404.2528.

[8] M. Lieberman and J. Rosický, Metric abstract elementary classes as accessible categories, submitted, arXiv:1504.02660.

[9] M. Makkai and R. Paré, Accessible Categories: The Foundations of Categorical Model Theory, volume 104 of Contemp. Math., Amer. Math. Soc., 1989.

[10] H. L. Mariano, P. H. Zambrano and A. Villaveces, A global approach to AECs, arXiv:1405.4488.

Department of Mathematics and Statistics Masaryk University, Faculty of Sciences Kotlářská 2, 611 37 Brno, Czech Republic

Email: lieberman@math.muni.cz
    rosicky@math.muni.cz
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