Theory and Applications of Categories, Vol. 10, No. 7, 2002, pp. 148–161.

MORE ON INJECTIVITY IN LOCALLY PRESENTABLE CATEGORIES

Dedicated to Horst Herrlich on the occasion of his 60th birthday

J. ROSICKÝ^{1) 2)}, J. ADÁMEK^{1)} AND F. BORCEUX

ABSTRACT. Injectivity with respect to morphisms having λ-presentable domains and codomains is characterized: such injectivity classes are precisely those closed under products, λ-directed colimits, and λ-pure subobjects. This sharpens the result of the first two authors (Trans. Amer. Math. Soc. 336 (1993), 785-804). In contrast, for geometric logic an example is found of a class closed under directed colimits and pure subobjects, but not axiomatizable by a geometric theory. A more technical characterization of axiomatizable classes in geometric logic is presented.

1. Introduction

In [2], classes of objects injective with respect to a set M of morphisms of a locally presentable category K were characterized: they are precisely the classes closed under products, λ-directed colimits and λ-pure subobjects for some cardinal λ (see Part 2 below for the concept of λ-pure subobject). In fact, the formulation in [2] did not use λ-pure subobjects, but accessibility of the class in question. However, a full subcategory of K, closed under λ-directed colimits, is accessible iff it is closed under λ′-pure subobjects for some λ′ (see [3], Corollary 2.36). The main result of our paper is a “sharpening” of the previous result to a given regular cardinal λ: by a λ-injectivity class we call a class of all objects injective with respect to M, where M is a set of morphisms with λ-presentable domains and codomains. If K is a locally λ-presentable category, we prove that λ-injectivity classes in K are precisely the full subcategories closed under products, λ-directed colimits, and λ-pure subobjects.

In contrast, the generalization from injectivity with respect to morphisms to injectivity with respect to cones, which was presented by H. Hu and M. Makkai [7], does not allow a corresponding “sharpening” – which is unfortunate, since this is closely connected to geometric logic. Hu and Makkai proved that classes given by injectivity with respect to a set of cones are precisely those closed under λ-directed colimits and λ-pure subobjects for some λ (again, we use Corollary 2.36 of [3] to put the original result into our terminology). We present an example of a class of Σ-structures, for a finitary signature Σ, which is closed under directed colimits and pure subobjects, but which is not a cone-injectivity class for any set of cones with finitely presentable domains and codomains. In particular, the class is not axiomatizable by geometric logic in StrΣ. We introduce a rather technical concept of being strongly closed under pure subobjects. This, then, is a full characterization of classes axiomatizable by geometric logic.

1) Partially supported by the Grant Agency of the Czech Republic under the grant No. 201/96/0119.

2) Partially supported by the Université Catholique Louvain-La-Neuve.

Received by the editors 1999 November 17 and, in revised form, 2002 March 1.

Transmitted by R.J. Wood. Published on 2002 March 15.

2000 Mathematics Subject Classification: 18C35, 03C99.

Key words and phrases: locally presentable category, injectivity class, geometric logic.

\begin{matrix}c \\ ◯ \end{matrix} J. Rosický, J. Adámek and F. Borceux, 2002. Permission to copy for private use granted.

Let us recall that, given an arrow h : A → A′ in a category K, an object X is called h-injective, provided that every morphism from A to X factors through h, i.e. the map

hom (h, X) : hom (A′, X) → hom (A, X), f ↦ fh

is surjective. This is an important concept in algebra, where h is usually required to be a monomorphism, and in model theory (where general h’s are considered). A class A of objects is called an injectivity class provided that there is a collection M of morphisms such that A consists of precisely all objects that are h-injective for all h ∈ M (we write A = M-Inj); if M is small, we speak about a small-injectivity class, and if all domains and all codomains of M-maps are λ-presentable objects of K, we speak about a λ-injectivity class. In case that the base category K is locally presentable (in the sense of Gabriel and Ulmer [5]), then every small-injectivity class is a λ-injectivity class for some λ, and conversely, every λ-injectivity class is a small-injectivity class. But small-injectivity classes are interesting also in such categories as Top - see e.g. [6], where they are called implicational subcategories.

A full characterization of small-injectivity classes in a locally presentable category K has been presented in [2]: they are precisely the classes A of objects, for which there exists a regular cardinal λ such that A is closed under

(i) products,

(ii) λ-directed colimits,

and

(iii) λ-pure subobjects.

(Let us recall that λ-pure monomorphisms are precisely the λ-directed colimits of retractions (as objects of K^{→}), and if K is the category of all Σ-structures for some λ-ary signature Σ, then this categorical concept coincides with that of a λ-pure submodel, used in model theory.) In the present paper, we prove that for each locally λ-presentable category K, the conditions (i) – (iii) above precisely characterize λ-injectivity classes in K. The proof is substantially more difficult than that in [2].

More generally, given a cone H = (h_{i} : A → A′_{i})_{i∈I}, an object X is H-injective iff every map from A to X factors through h_{i} for some i ∈ I. A class of objects is called a (small-)cone-injectivity class if it can be axiomatized by injectivity with respect to a class (or set) of cones. If all domains and codomains of all those cones are λ-presentable, we speak about λ-cone-injectivity classes.

Injectivity in locally presentable categories is treated in [3] where the reader also finds standard categorical concepts used in our paper. We use the notation Str Σ for the category of Σ-structures and homomorphisms, where Σ is a finitary many-sorted signature. This is a locally finitely presentable category.

2. Characterization of Injectivity Classes

2.1. Remark. Recall that a morphism m : A → B of a locally λ-presentable category K is called λ-pure provided that in each commutative square

\xymatrix{X \ar[r]^{h} \ar[d]_{f} & Y \ar[d]^{g} \\ A \ar[r]_{m} & B}

with X and Y λ-presentable the morphism f factors through h. As an object of A ↓ K, m is λ-pure iff m is a λ-directed colimit of retractions, see [3]. Every λ-pure morphism is a regular monomorphism.

2.2. Theorem. Let K be a locally λ-presentable category. A full subcategory of K is a λ-injectivity class iff it is closed under

(i) products

(ii) λ-directed colimits

and

(iii) λ-pure subobjects.

Proof. The necessity is easy: closedness under products and λ-directed colimits is trivial. To prove closedness under λ-pure subobjects m : P → Q, we are to show that if Q is h-injective then P is h-injective for all maps h : A → B with A, B both λ-presentable. Let u : A → P be a morphism. There exists v : B → Q making the following square

\xymatrix{A \ar[r]^{h} \ar[d]_{u} & B \ar[d]^{v} \\ P \ar[r]_{m} & Q}

commutative (h-injectivity of Q) and since u factors through h (λ-purity of m), the hinjectivity of P follows.

For the sufficiency, we can assume that K is the category Str Σ of all Σ-structures for some finitary relational signature Σ. (The general case follows from the fact that K can be considered as a full reflective subcategory of Str Σ closed under λ-directed colimits – see 1.47 of [3]. This implies that K is closed under (i) – (iii), see Remark 2.31 in [3]. Consequently, given a full subcategory L of K closed under (i) – (iii) in K, then L = M-Inj for some set M of morphisms of Str Σ having λ-presentable domains and codomains. The reflector F : Str Σ → K obviously preserves λ-presentability of objects and satisfies L = F (M)-Inj.)

Thus, let L be a full subcategory of Str Σ, where Σ is a finitary relational signature, closed under (i) – (iii). Put

M = {f : A → B in Str Σ ; A and B are λ-presentable

and every object of L is injective with respect to f}.

We will prove that L = M-Inj. We first observe that L is weakly reflective, i.e., every object K of Str Σ has a morphism r_{K} : K → K^{∗} (weak reflection) with K^{∗} in L such that every object of L is injective with respect to r_{K}. This follows from 2.36 and 4.8 in [3]. Now L ⊆ M-Inj by the choice of M, and to prove M-Inj ⊆ L, we verify the following implication:

K ∈ M-Inj ⇒ any weak reflection of K in L is λ-pure. It then follows from (iii) that K ∈ L, and this will prove

M-Inj = L .

Thus, given K ∈ M-Inj and a weak reflection r : K → K^{∗} in L, we are to prove that in any commutative square

\xymatrix{A \ar[r]^{h} \ar[d]_{u} & B \ar[d]^{v} \\ K \ar[r]_{r} & K^{∗}}

with A and B λ-presentable the map u factors through h. We will work with the arrowcategory (Str Σ)^{→} and consider the morphism (u, v) : h → r.

Claim: There is a factorization u = u_{2}·u_{1} and a morphism (u_{1}, v_{1}) : h → r̄ for some object r̄ of (Str Σ)^{→}, where r̄: K̄ → K̄^{∗} is a weak reflection of K̄ in L and K̄ is λ-presentable in Str Σ.

Proof of claim. Consider all morphisms (u_{1}, v_{1}) : h → r̄ where r̄ : K̄ → K̄^{∗} is a weak reflection of K̄ in L and u = u_{2} · u_{1} for some u_{2}. Since (u, v) : h → r is such a morphism, we can take the smallest α such that K̄ is α-presentable in Str Σ. We are to prove α ≤ λ. Assuming α > λ we derive a contradiction. Let us first remark that an object A of Str Σ is α-presentable iff it has (a) less than α elements (i.e., the union of all underlying sets of all sorts has cardinality

< α)

and

(b) less than α relational symbols σ ∈ Σ with σ_{A} nonempty.

(See 1.14 (2) in [3].) Consequently, for every regular cardinal α ≥ ℵ_{0}, any α-presentable object in Str Σ is a colimit of a smooth α-chain (i.e., a chain whose limit steps form a colimit of the preceding part) of objects of presentability smaller than α.

Let us express the above object K̄ as a colimit of a smooth α-chain k_{ij} : K_{i} → K_{j} (i ≤ j < α) of objects K_{i} of presentability less than α, let k_{iα} : K_{i} → K̄ (i < α) denote the colimit cocone. We define an α-chain k_{ij}^{∗} : K_{i}^{∗} → K_{j}^{∗} (i ≤ j < α) in L and a natural transformation

r_{i} : K_{i} → K_{i}^{∗} (i < α)

by transfinite induction. (This idea has been used in [2] already, see the proof of IV. 3.)

(i) r_{0} : K_{0} → K_{0}^{∗} is a weak reflection of K_{0} in L.

(ii) i ↦ i + 1: Form a pushout of r_{i} and k_{i,i+1}

\xymatrix{K_{i} \ar[r]^{r_{i}} \ar[d]_{k_{i,i+1}} & K_{i}^{∗} \ar[d]^{k̂_{i,i+1}} \\ K_{i+1} \ar[r]_{r̂_{i}} & K̂_{i+1}}

Choose a weak reflection r_{i+1}^{∗} : K̂_{i+1} → K_{i+1}^{∗} in L and put r_{i+1} = r_{i+1}^{∗}r̂_{i} and k_{i,i+1}^{∗} = r_{i+1}^{∗} · k̂_{i,i+1}.

(iii) Given a limit ordinal j, form a colimit

K̂_{j} = colim_{i<j} K_{i}^{∗}

with colimit maps k̂_{ij} : K_{i}^{∗} → K̂_{j} (i < j) and choose a weak reflection r_{j}^{∗} : K̂_{j} →

K_{j}^{∗} in L, then r_{j} = r_{j}^{∗} · (colim_{i<j} r_{i}) and k_{ij}^{∗} = r_{j}^{∗}k̂_{ij}.

Observe that every object L of L is r_{i}-injective for all i < α (i.e., each r_{i} : K_{i} → K_{i}^{∗} is a weak reflection of K_{i}).

Proof. This is obvious for i = 0. If this holds for all smaller ordinals than i (> 0), then it holds for i as well: given f : K_{i} → L, we can define a compatible cocone f_{ij}^{∗} : K_{j}^{∗} → L (j ≤ i) with f_{j}^{∗}r_{j} = f k_{ji} by transfinite induction as follows:

(i) f_{0}^{∗} is any map with f_{0}^{∗}r_{0} = f k_{0i} (this exists by the choice of r_{0});

(ii) given f_{j}^{∗}, the pushout property yields a unique f̂_{j+1} : K̂_{j+1} → L and we choose any map f_{j+1}^{∗} : K_{j+1}^{∗} → L with f̂_{j+1} = f_{j+1}^{∗}r_{j+1}^{∗};

(iii) given a limit ordinal j and f_{k}^{∗} for k < j, the j-chain-colimit property yields a unique f̂_{j} : K̂_{j} → L and we choose any map f_{j}^{∗} : K_{j}^{∗} → L with f̂_{j} = f_{j}^{∗} · r_{j}^{∗}.

The object K_{α}^{∗} = colim_{i<α} K_{i}^{∗} lies in L because α-chains are λ-directed (recall that α > λ is

a regular cardinal). Thus, for the map

r_{α} = colim_{i<α} r_{i} : K̄ → K_{α}^{∗}

there exists s : K̄^{∗} → K_{α}^{∗} with r_{α} = sr̄. (Recall that r̄ is a weak reflection of K̄.) In (Str Σ)^{→} we thus obtain a morphism

(u_{1}, sv_{1}) : h → r_{α} .

Now h is, obviously, λ-presentable in (Str Σ)^{→}, thus that last map factors through some of the objects r_{i}, i < α. This is the desired contradiction with the minimality of α: each r_{i} : K_{i} → K_{i}^{∗} is a weak reflection of K_{i}, and K_{i} has smaller presentation rank than α. This proves the claim.

We are ready to prove that u factors through h. Let us consider a factorization u = u_{2} · u_{1} and a morphism (u_{1}, v_{1}) : h → r̄ as in the above claim. Let us express K̄^{∗} as a λ-directed colimit of λ-presentable objects Q_{t}, t ∈ T, in Str Σ with a colimit cocone q_{t} : Q_{t} → K̄^{∗}. Since both K̄ and B are λ-presentable, the maps r̄ and v_{1} both factor through q_{t_{0}} for some t_{0} ∈ T and then there exists (since A is λ-presentable) t_{1} ⩾ t_{0} in T with a commutative diagram as follows:

\xymatrix{A \ar[dd]_{u_{1}} \ar[rr]^{h} &  & B \ar[dd]^{v_{1}} \ar[dl]_{ṽ_{1}} \\  & Q_{t_{1}} \ar[dr]^{q_{t_{1}}} \\ K̄ \ar[ur]^{r̃} \ar[rr]_{r̄} &  & K̄^{∗}}

Since all objects of L are r̄-injective, they are also r̃-injective; moreover, K̄ and Q_{t_{1}} are both λ-presentable, thus,

r̃ ∈ M .

This implies that K is r̃-injective. Choose d : Q_{t_{1}} → K with u_{2} = dr̃ to obtain

u = u_{2}u_{1} = dr̃u_{1} = dṽ_{1}h ,

the desired factorization. Thus, r is λ-pure, and K ∈ L.

2.3. Remark. The reader may well wonder whether the condition of closedness under λ-pure subobjects is not superfluous: after all

(i) closedness under λ-directed colimits implies closedness under split subobjects

and

(ii) every λ-pure subobject is a λ-directed colimit of split subobjects, see [3] 2.30.

However, (ii) holds in the arrow-category of K, not in K itself. The following Example demonstrates that, indeed, λ-pure subobjects are essential:

2.4. Example. A class of graphs which is closed under products and directed colimits, but is not an ω-injectivity class: We denote by Gra the category of graphs, i.e., pairs (X, R) of sets with R ⊆ X × X, and graph homomorphisms. Let A be the class of all graphs containing nodes x_{n} ∈ X (n ∈ N) such that x_{n}Rx_{n+1} for each n ∈ N. It is clear that A is closed under products and nonempty colimits in Gra. However, A is not closed under ω-pure subobjects. In fact, let A denote the graph obtained from an infinite path x_{0}, x_{1}, x_{2}, … (starting in the node x_{0}) by gluing a path of length n to x_{0} for each n ∈ N. Then

A ∈ A and B ∉ A

where B is the strong subgraph of A over all nodes distinct from x_{k} for k ≥ 1. However, B is an ω-pure subgraph of A because given any finite (= finitely presentable) subgraph V of A there exists a morphisms h : V → B with h(x) = x for all nodes x of B. In fact: let n be the largest index with x_{n} ∈ V and for each k ≤ n denote by x′_{k} the k-th node on the path of length n in A, then put h(x) = x if x = B and h(x_{k}) = x′_{k}.

3. Characterization of Classes Axiomatizable by Geometric Logic

Recall that geometric logic is the first-order logic using formulas of the following form

(1)  (∀ x)(φ(x) → ⋁_{i∈I}(∃ y_{i}) ψ(x, y_{i}))

where x and each y_{i} is a finite string of variables, and φ and each ψ_{i} is a finite conjunction of atomic formulas. For example, given a cone H with finitely presentable domain and codomains in Str Σ, injectivity with respect to H can be axiomatized by a formula (1), where φ(x) expresses the presentation of the domain of H, and ψ(x, y_{i}) expresses the presentation of the i-th codomain, and the connecting morphism. Consequently, every ω-cone-injectivity class in Str Σ is axiomatizable in geometric logic. Conversely, every formula (1) can be translated to ω-cone-injectivity in Str Σ, see [3].

Observation. Every class of structures axiomatizable by geometric logic is closed in Str Σ under directed colimits and pure subobjects.

The converse does not hold:

3.1. Example. A class of Σ-structures (Σ finitary) closed under directed colimits and strong subobjects, which is not an ω-cone-injectivity class: Let Σ be the (one-sorted) signature of countably many unary relation symbols σ_{n} (n ∈ ω). For every set A ⊆ ω denote by Ā the Σ-structure over the set {0} with σ_{n} ≠ ∅ iff n ∈ A. Further denote by I the initial (empty) object of Str Σ.

Choose a disjoint decomposition

ω = ⋃_{n∈ω} B_{n} , B_{n} infinite

and denote by B the full subcategory of Str Σ consisting precisely of

(i) I

and

(ii) all B̄ where B ⊆ ω is a set such that there exists n ∈ ω and a set M ⊆ ω of cardinality ≤ n with B = B_{n} ∪ M.

B is obviously closed under substructures (strong subobjects), since B-objects have no nontrivial substructures. And it is closed under directed colimits: if B̄_{t} (t ∈ T) is a directed collection in B, then there exists n ∈ ω such that each B̄_{t} has the form B̄_{t} = B_{n} ∪ M_{t} for M_{t} of at most n elements. Since T is directed, the set M = ⋃_{t∈T} M_{t} also has at most n

elements, and colim B̄_{t} = B_{n} ∪ M ∈ B.

However, B is not an ω-cone-injectivity class. In fact, B does not contain the terminal Σ-structure ω̄, although any ω-cone-injectivity class containing B contains ω̄ too. Indeed: ω̄ is injective with respect to all nonempty cones, and we observe that there exists no empty ω-cone H to which all B-objects are injective. (In fact, let M be the domain of H. Since M is finitely presentable, there exists n ∈ ω such that (σ_{k})_{M} = ∅ for all k ≥ n. Put B = B_{n} ∪ {0, 1, …, n − 1}, then the constant map is a Σ-homomorphism from M to B̄, consequently, B̄ is not injective with respect to the empty cone with domain H.)

3.2. Definition. A span A ⟵^{n} C ⟶^{m} B is called λ-pure provided that for each commutative square

\xymatrix{ & X \ar[r]^{h} \ar[d]_{f} & Y \ar[d]^{g} \\ A & C \ar@{-->}[l]^{n} \ar[r]_{m} & B}

with X and Y λ-presentable the morphism nf factors through h (i.e., there is t : Y → A with nf = th). If λ = ω we say just pure.

3.3. Examples. (1) For each morphism r : B → A the inverse relation of the graph of r, i.e., A ⟵^{r} B ⟶^{1} B, is pure. Also any subspan of the inverse of a graph is pure. These are precisely the relations A ⟵^{n} C ⟶^{m} B where n factors through m; we call them split spans.

(2) A λ-directed colimit of split spans is λ-pure. More precisely, let Sp (K) denote the category whose objects are spans A ⟵^{n} C ⟶^{m} B in K and whose morphisms are commutative diagrams

\xymatrix{A \ar[d]_{u} & C \ar[l]_{n} \ar[d]^{v} \ar[r]^{m} & B \ar[d]^{w} \\ A′ \ar[r]_{n′} & C′ \ar[r]_{m′} & B}

If K has λ-directed colimits, then so does Sp(K), and a λ-directed colimit of split spans is λ-pure. This follows easily from the fact that (λ-directed) colimits are formed object-wise in Sp(K). 3.4. Remark. In a locally λ-presentable category the latter example of λ-pure spans is canonical: every λ-pure span is a λ-directed colimit of split spans. The proof is analogous to that of 2.30 in [3]. 3.5. Definition. A subcategory L of a category K is strongly closed under λ-pure subobjects provided that L contains every object A of K with the following property:

given a morphism n : C → A in K with C λ-presentable, there exists a λ-pure span

A ⟵^{n} C ⟶^{m} B with B ∈ L.

3.6. Remark. (1) Given a morphism m : C → D, for each cardinal λ let us denote by H_{λ}(m) the set of all spans (h, f) where h : X → Y is a morphism with X and Y λ-presentable, and f : X → C is a morphism with a commutative square

\xymatrix{X \ar[r]^{h} \ar[d]_{f} & Y \ar@{-->}[d] \\ C \ar[r]_{m} & D}

A span A ⟵^{n} C ⟶^{m} B is λ-pure iff H_{λ}(m) ⊆ H_{λ}(n). Thus, L is strongly closed under λ-pure subobjects iff L contains every object A ∈ K with the following property: given n : C → A, C λ-presentable, there exists m : B → A with B ∈ L such that H_{λ}(m) ⊆ H_{λ}(n).

(2) This is, indeed, stronger than closedness under λ-pure subobjects. For, given L ∈ L and a λ-pure subobject i : A → L, we prove that A ∈ L as follows: Let n : C → A be a morphism with C λ-presentable, then A ⟵^{n} C ⟶^{in} L is a λ-pure span.

(3) Strong closedness under λ-pure subobjects implies closedness under λ-directed colimits (and thus is, indeed, much stronger than closedness under λ-pure subobjects). In fact, let A be a λ-directed colimit of A_{i} ∈ L with a colimit cocone (A_{i} ⟶^{h_{i}} A)_{i∈I}. For each morphism n : C → A with C λ-presentable there exists i ∈ I such that n factors as n = a_{i}n′ for some n′ : C → A_{i}, and we have a pure span A ⟵^{n} C ⟶^{n′} A_{i}, thus, A ∈ L.

3.7. Theorem. Let K be a locally λ-presentable category. Then λ-cone-injectivity classes of K are precisely the full subcategories strongly closed under λ-pure subobjects.

Proof. I. Suppose that L is strongly closed in K under λ-pure subobjects. For every object A in K − L we construct a sink γ_{A} as follows: choose a morphism n : C → A, C λ-presentable, such that no λ-pure span A ⟵^{n} C ⟶^{m} B with B ∈ L exists. Given any morphism m : C → B with B ∈ L we thus have a commutative square

\xymatrix{ & X_{m} \ar[d]_{f_{m}} \ar[r]^{h_{m}} & Y_{m} \ar[d]^{g_{m}} \\ A & C \ar@{-->}[l]_{n} \ar[r]_{m} & B}

on K with X_{m} and Y_{m} λ-presentable such that n · f_{m} does not factor through h_{m}. Form a pushout of f_{m} and h_{m}:

\xymatrix{X_{m} \ar[d]_{f_{m}} \ar[r]^{h_{m}} & Y_{m} \ar[d]^{f_{m}^{∗}} \ar@{-->}[ddr]^{g_{m}} \\ C \ar[r]_{m^{∗}} \ar@{-->}[drr]_{m} & B^{∗} \ar@{-->}[dr] \\  &  & B}

Since C, X_{m} and Y_{m} are λ-presentable, so is B^{∗}. We denote by γ_{A} the cone of all m^{∗} : C → B^{∗} (indexed by all m ∈ ⋃_{B∈L} hom (C, B) ), and we first observe that since γ_{A}

has a λ-presentable domain and all codomains, it is essentially small. We claim that

L = {γ_{A}}_{A∈K−L}-Inj ,

i.e., that an object of K lies in L iff it is cone-injective with respect to each γ_{A}. In fact, given B ∈ L then for every morphism m : C → B we have a factorization of m through m^{∗}, a member of γ_{A}, thus, B is M-injective.

Conversely, let A be an object of K injective to all of the above cones, then we prove that A ∈ L. In fact, assuming A ∈ K − L we have the above cone γ_{A} and since A is γ_{A}-injective, the morphism n : C → A factors through m^{∗} for some m : C → B with B ∈ L. But then n · f_{m} factors through h_{m} (since if t · m^{∗} = n then t · f_{m}^{∗} · h_{m} = n · f_{m}), in contradiction to the above choice of h_{m}, f_{m}, and g_{m}.

II. Suppose that L is a λ-cone-injectivity class in K. Let M be a set of cones with λpresentable domains and codomains such that L = M-Inj. We will prove that L contains any object A satisfying the condition of the above definition of strong closedness under λ-pure subobjects. Thus, given a cone γ = (C ⟶^{c_{i}} C_{i})_{i∈I} in M, we are to show that A is γ-injective. In fact, let n : C → A be a morphism. There exists a λ-pure span A ⟵^{n} C ⟶^{m} B with B ∈ L. Since B ∈ M-Inj, the morphism m factors through some c_{i}, say, m = m′ · c_{i}.

\xymatrix{ & C \ar[d]^{1} \ar[r]^{c_{i}} & C_{i} \ar[d]^{m′} \\ A & C \ar@{-->}[l]_{n} \ar[r]_{m} & B}

From the λ-purity we conclude that n · 1 factors through c_{i}, thus, A is γ-injective.

3.8. Corollary. A class of Σ-structures can be axiomatized by geometric logic iff it is strongly closed under pure subobjects in Str Σ.

3.9. Remark. (1) We can speak, more generally, about infinitary geometric logic: given a regular cardinal λ, a formula (1) is λ-geometric provided that x and y_{i} are strings of less than λ variables and φ and ψ_{i} are conjunctions of less than λ atomic formulas. The above corollary generalizes as expected: classes axiomatized by λ-geometric theories are precisely those strongly closed under λ-pure subobjects.

(2) λ-geometric logic precisely describes λ-cone-injectivity classes in Str Σ. Analogously, λ-injectivity classes are described by λ-regular logic, i.e., logic using formulas

(2)  (∀ x)(φ(x) → ∃ y ψ(x, y))

where, again, x and y are strings of less than λ variables and φ and ψ are conjunctions of less than λ atomic formulas. Thus, Theorem 2.2 characterizes classes of structures axiomatizable by λ-regular logic: they are the classes closed under products, λ-directed colimits and λ-pure subobjects.

This seems to be a new result for λ > ℵ_{0}; let us remark that, however, for λ = ℵ_{0} a more elementary proof follows from Compactness Theorem. A class closed under products and directed colimits is namely closed under ultraproducts; being closed under pure (in particular, under elementary) subobjects, the class is axiomatizable in the first-order logic, and our result then easily follows.

(3) We now turn to a special case of cone-injectivity classes for which a good characterization can be formulated. If M is a set of cones and each cone consists of strong epimorphisms only, we call M-Inj a strong cone-injectivity class.

If the domains and codomains in all those cones are λ-generated objects (i.e., objects A such that hom (A, −) preserves λ-directed unions), then we speak about strong λ-cone-injectivity classes, and we prove that those are precisely the classes closed under λ-directed unions and subobjects. This depends on (strong epi, mono)-factorizations of morphisms. We work, more generally, with an abstract factorization system. Following [1] we say that K is an (E, M)-structured category provided that E is a class of epimorphisms, M is a class of monomorphisms, both closed under composition, with E ∩ M = Iso and every morphism has an essentially unique (E, M)-factorization.

3.10. Definition. Let K be an (E, M)-structured category. A λ-directed colimit whose colimit cocone is formed by M-morphisms is called a λ-directed union. An object K is called λ-generated provided that hom (K, −) preserves λ-directed unions.

3.11. Lemma. In a locally λ-presentable (E, M)-structured category every object is a λ-directed union of λ-generated objects.

Proof. Every object K is a λ-directed colimit of λ-presentable objects K_{i} (i ∈ I). If k_{i} : K_{i} → K (i ∈ I) denotes the colimit cocone and k_{i} = m_{i}e_{i} is an (E, M)-factorization of k_{i}, with m_{i} : K′_{i} → K, then the objects K′_{i} form a λ-directed diagram with a colimit cocone m_{i} : K′_{i} → K (i ∈ I), due to the diagonal fill-in. Since K_{i} is λ-presentable and e_{i} : K_{i} → K′_{i} lies in E, it follows easily from the diagonal fill-in that K′_{i} is λ-generated.

3.12. Remark. In a locally λ-presentable (E, M)-structured category, λ-generated objects are precisely the E-quotients of all λ-presentable objects – this has been proved for E = strong epis in [3], 1.61, and the general case is analogous.

3.13. Definition. A class of objects is called a strong λ-cone-injectivity class if it has the form M-Inj for a set of cones formed by E-morphisms whose domains and codomains are λ-generated.

3.14. Theorem. Let K be a locally λ-presentable (E, M)-structured category. A full subcategory of K is a strong λ-cone-injectivity class iff it is closed in K under λ-directed unions and M-subobjects.

Proof. Necessity is clear. For the sufficiency, let L be a full subcategory of K closed under λ-directed unions and M-subobjects.

(a) L is an accessible category. In fact, L is closed under λ-pure subobjects in K, since by 2.31 in [3],

λ-pure ⇒ regular mono ⇒ member of M .

By 2.36 of [3] it is sufficient to prove that L is closed under μ-directed colimits for some regular cardinal μ. By the above remark, K has only a set of λ-generated objects, thus, there exists a regular cardinal μ such that

λ-generated ⇒ μ-presentable .

Let L_{i} (I ∈ I) form a μ-directed diagram in L with a colimit l_{i} : L_{i} → K (i ∈ I) in K. We are to prove that K ∈ L. By the above Lemma, K is a λ-directed union of λ-generated objects, say, k_{j} : K_{j} → K (j ∈ J). Since K_{j} is μ-presentable, k_{j} factors through some l_{i}, i.e., there exists i ∈ I and d : K_{j} → L_{i} with k_{j} = l_{i} · d. Now k_{j} ∈ M implies d ∈ M, thus, K_{j} ∈ L (since L is closed under M-subobjects). Thus, K is a λ-directed union of L-objects, which proves K ∈ L.

(b) L is a cone-reflective subcategory of K, i.e., for every object K ∈ K there exists a cone with domain K to which all L-objects are injective. This is proved in 2.53 of [3]. Moreover, every object K ∈ K has a cone-reflection in L all members of which are E-epis: this follows immediately from the closedness of K under M-subobjects.

(c) Let H be the following collection of cones in K: for each λ-generated object K of K we choose a cone-reflection (r_{i}^{K} : K → K_{i})_{i∈I(K)} of K in L with r_{i}^{K} ∈ E for each i ∈ I and let H be the set of all these cones. It is sufficient to prove that an object K ∈ K lies in L iff K is injective with respect to all cones of H.

Thus, let K be injective with respect to H-cones. By the above Lemma, we can express K as a λ-directed union of λ-generated objects k_{t} : K_{t} → K (t ∈ T). Since K is initial with respect to a cone-reflection (r_{i}^{K_{t}}), k_{t} factors through some r_{i}^{K_{t}} which, since k_{t} ∈ M and r_{i}^{K_{t}} ∈ E, implies that r_{i}^{K_{t}} is an isomorphism. In other words, K_{t} ∈ L (for each t ∈ T). Since L is closed under λ-directed unions, this implies K ∈ L.

3.15. Remark. The factorization of Σ-homomorphisms to surjective Σ-homomorphisms followed by inclusions of substructures yields Str Σ as an (epi, regular mono)-structured category. Then λ-generated Σ-structures are Σ-structures generated by less than λ elements in a usual sense. If a cone (h_{i} : A → B_{i})_{i∈I} has all h_{i} surjective and all A and B_{i} λ-generated then (2) is (equivalent to) a universal sentence of the logic L_{∞,λ}. We obtain, from Theorem 3.7,

3.16. Corollary. A class of Σ-structures is axiomatizable by a universal theory of the logic L_{∞,λ} iff it is closed under submodels and λ-directed unions.

3.17. Remark. Corollary 3.16 has an easy direct proof: any class L of Σ-structures which is closed under substructures and λ-directed unions can be axiomatized by universal sentences of L_{∞,λ}

(∀x⃗) ⋁_{i∈I} π_{A_{i}}(x⃗)

where {A_{i}}_{i∈I} is a representative set of Σ-structures A_{i} ∈ L generated by less then λelements and π_{A_{i}} is the diagram of A_{i} reduced to generators.

Corollary 3.16 documents the well-known fact that a class of Σ-structures with at most countably many elements, which is axiomatizable in L_{ω_{1},ω_{1}}, cannot be axiomatized by a universal theory in L_{ω_{1}ω_{1}} (see [4] 2.4.11). An analogous example cannot be found in L_{ω,ω}: any class of Σ-structures axiomatizable in L_{ω,ω} and closed under substructures is axiomatizable by a universal theory in L_{ω,ω}.

References

[1] J. Adámek, H. Herrlich and G. Strecker, Abstract and Concrete Categories, John Wiley, New York 1990.

[2] J. Adámek and J. Rosický, On injectivity in locally presentable categories, Trans. Amer. Math. Soc. 336 (1993), 785–804.

[3] J. Adámek and J. Rosický, Locally Presentable and Accessible Categories, Cambridge Univ. Press 1994.

[4] M. A. Dickmann, Large Infinitary Languages, North-Holland, Amsterdam 1975.

[5] P. Gabriel and F. Ulmer, Lokal Präsentierbare Kategorien, Lect. Notes in Math. 221, Springer-Verlag, Berlin 1971.

[6] H. Herrlich, Almost reflective subcategories of Top, Topology and Appl. 49 (1993), 251–261.

[7] H. Hu and M. Makkai, Accessible embeddings and the solution set conditions, Cah. Top. Géom. Diff. Cat. XXXV (1994), 99–108.

[8] M. Makkai and R. Paré, Accessible categories: the foundations of categorical model theory, Contemporary Math. 104, Amer. Math. Soc., Providence 1989.

Department of Algebra and Geometry Masaryk University Janáčkovo nám. 2a 662 95 Brno, Czech Republic

Institut für Theoretische Informatik Technical University Braunschweig Postfach 3329 38023 Braunschweig, Germany

Departement des Mathématiques Université Catholique Louvain-La-Neuve 2, Chemin du Cyclotron 1348 Louvain-La-Neuve, Belgium

Email: rosicky@math.muni.cz
      adamek@iti.cs.tu-bs.de
      borceux@agel.ucl.ac.be

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