Theory and Applications of Categories, Vol. 12, No. 9, 2004, pp. 270–298.

ALGEBRAICALLY CLOSED AND EXISTENTIALLY CLOSED SUBSTRUCTURES IN CATEGORICAL CONTEXT

MICHEL HÉBERT

Abstract. We investigate categorical versions of algebraically closed (= pure) embeddings, existentially closed embeddings, and the like, in the context of locally presentable categories. The definitions of S. Fakir [Fa, 75], as well as some of his results, are revisited and extended. Related preservation theorems are obtained, and a new proof of the main result of Rosický, Adámek and Borceux ([RAB, 02]), characterizing λ-injectivity classes in locally λ-presentable categories, is given.

Introduction

Algebraically closed embeddings are used in module theory and category theory (where they are called pure morphisms), as well as in model theory. The model theoretic definition allows seeing this concept as one of several related types of morphisms (like elementary embeddings and existentially closed embeddings) defined in terms of preservation of certain families of formulas. Adapting this line of thought, S. Fakir ([Fa, 75]) proposed a categorical version of the concepts of algebraically closed and existentially closed morphisms in the context of locally presentable categories.

In this paper, we first revisit Fakir’s definitions, extending and simplifying them using two ideas: the first one is to use λ-presentable morphisms instead of λ-presentable (and λ-generated) objects, and the second one involves what we will call locally presentable factorization systems.

The former idea was already exploited for the algebraically closed case in [H_{1}, 98]. In a category C, a morphism f: A ⟶ B will be called λ-presentable if it is λ-presentable as an object of the comma category (A ↓ C). In [H_{1}, 98], we characterized the classes closed under algebraically closed subobjects as the ones which are injective with respect to some class of cones formed by λ-presentable morphisms. As a corollary, we obtained a solution to a problem of L. Fuchs in the context of abelian groups ([Fu, 70]). In this paper, we use this characterization to obtain a different proof of the main result of [RAB, 02], characterizing the λ-injectivity classes as the ones closed under λ-algebraically closed subobjects and λ-reduced products (Theorem 3.5 below).

As for the latter idea, we define a λ-presentable factorization system as a proper factorization system (E,M) in which the morphisms in E between λ-presentable objects are sufficient to determine the morphisms in M. In a locally λ-presentable category C, (Strong Epi, Mono) is λ-presentable, but (Epi, Strong Mono) is not necessarily so. Also, if one sees C as the concrete category of all the models of a λ-limit theory, the embeddings (in the usual classical sense) form the right-hand part of a λ-presentable factorization system. We then propose a definition of existentially closed morphisms which is relative to a given λ-presentable factorization system. Apart from being somewhat simpler then Fakir’s, the definition fits the classical model theoretic one even when the signature contains relation symbols, when one chooses the appropriate factorization system.

Received by the editors 2003-06-28 and, in revised form, 2004-04-16.

Transmitted by Jiri Rosicky. Published on 2004-04-22.

2000 Mathematics Subject Classification: 18A20, 18C35, 03C60, 03C40.

Key words and phrases: pure morphism, algebraically closed, existentially closed.

\begin{matrix}c \\ ◯ \end{matrix} Michel Hébert, 2004. Permission to copy for private use granted.

As an application, Theorem 2.4 below completes the picture regarding the following set of characterizations. [Fa, 75] gives a categorical proof of the well-known characterization of algebraically closed morphisms as those which can be extended to the canonical diagonal morphism into some ultrapower of their domain. The result also holds when “ultrapower” is replaced by “reduced power”, and the proof is given in the infinitary context. Fakir also uses his method to prove that the existentially closed morphisms are those which can be extended by a monomorphism to the canonical diagonal morphism into some ultrapower of their domain. However, in this case ultrapowers cannot be replaced by reduced powers, so that in particular there is no obvious extension available in the infinitary context. Our definition allows considering intermediate types of morphisms (between the algebraically closed and the existentially closed ones), and we identify precisely, in these terms, the morphisms which can be extended by a monomorphism to the canonical diagonal morphism into some reduced power of their domain. We call those weakly λ-existentially closed morphisms, and we also characterize them in terms of preservation of a certain type of formulas.

1. Preliminary concepts

For basic definitions and results in category theory, we refer the reader to [AR, 94] and [Bo, 94]. For convenience, we recall the following.

If λ is a regular infinite cardinal, an object A of a category C is λ-presentable if the hom-functor C(A, −): C ⟶Set preserves λ-directed colimits. Then, C is λ-accessible if it has a (small) set S of λ-presentable objects such that every object is the λ-directed colimit of a diagram with all its vertices in S. Finally, C is locally λ-presentable if it is λ-accessible and cocomplete (or, equivalently, complete). We write finitely presentable for ω-presentable.

Given an object A in a locally λ-presentable (resp. λ-accessible) category C, both the category C^{2} (of all morphisms in C) and the comma category (A ↓ C) (of all morphisms in C with domain A) are also locally λ-presentable (resp. λ-accessible). Our definitions and results will be stated for locally λ-presentable categories, but the interested reader will see easily that all of them apply to λ-accessible categories with pushouts. Throughout the paper we will adopt the following:

1.1. Convention. A morphism f: A ⟶ B in a category C will be called λ-presentable if it is λ-presentable as an object of the comma category (A ↓ C).

Note that being a λ-presentable morphism in our sense is weaker than being a λpresentable object in C^{2} (which turns out to be the same as a morphism between λpresentable objects in C: see [AR, 94], Exercise 2.c). Intuitively (in the context of a variety, say), f: A ⟶ B being λ-presentable means that f provides a way to present B by adding less than λ generators and relations to some presentation of A: see Section 3 for more on that. In categorical terms, this description translates as follows:

1.2. Proposition. Let C be a locally λ-presentable category. Then f: A ⟶ B is λ-presentable if and only if there exists a commutative diagram:

[figure: C; D; A; f; E; r; B; s]

where C and D are λ-presentable, the square is a pushout, and rs = 1_{B}.

Proof. (⇒) Let f: A ⟶ B be λ-presentable. C^{2} being locally λ-presentable, there exists a λ-directed diagram ((a_{ij}, b_{ij}): f_{i} ⟶ f_{j})_{I} of λ-presentable objects of C^{2} with colimit ((a_{i}, b_{i}): f_{i} ⟶ f)_{I}.

Let (f_{i}^{∗}, a_{i}^{∗}) be the pushout of (f_{i}, a_{i}) (i ∈ I), and f_{ij}^{∗} and f′_{i} be the induced morphisms, as in the following diagram:

\xymatrix{ & A_{i} \ar[dddl]_{a_{i}} \ar[d]^{a_{ij}} \ar[rrr]^{f_{i}} &  &  & B_{i} \ar[ddll]_{a_{i}^{∗}} \ar[dddr]^{b_{i}} \ar[d]_{b_{ij}} \\  & A_{j} \ar[ddl]^{a_{j}} \ar[rrr]^{f_{j}} &  &  & B_{j} \ar[dl]_{a_{j}^{∗}} \ar[ddr]_{b_{j}} \\  &  & P_{i} \ar@{-->}[drrr]_{f′_{i}} \ar@{-->}[r]^{f_{ij}^{∗}} & P_{j} \ar@{-->}[drr]^{f′_{j}} \\ A \ar[rrrrr]_{f} \ar[urr]^{f_{i}^{∗}} \ar[urrr]_{f_{j}^{∗}} &  &  &  &  & B}

It is straightforward to verify that in (A ↓ C), one has colim_{I}(f_{ij}^{∗}: f_{i}^{∗} ⟶ f_{j}^{∗})_{I} = (f′_{i}: f_{i}^{∗} ⟶ f). Now, if f is λ-presentable (in (A ↓ C)), then 1_{B}: f ⟶ f must factorize through one of the f′_{i}: f_{i}^{∗} ⟶ f, i.e., there exist i ∈ I and s: f ⟶ f_{i}^{∗} in (A ↓ C) with f′_{i}s = 1_{B}. Taking C = A_{i}, D = B_{i}, P_{i} = E and r = f′_{i}, one gets the required diagram as in the statement of the proposition.

(⇐) The converse implication follows from the following facts, which are either known or easily verified: (i) a morphism between λ-presentable objects in C is a λ-presentable morphism, (ii) the pushout (by any morphism) of a λ-presentable morphism is a λpresentable morphism, and (iii) every retraction in (A ↓ C) of a λ-presentable morphism is a λ-presentable morphism.

Proposition 1.2 says that f: A ⟶ B is λ-presentable if it is, up to a retraction in (A ↓ C), the pushout of a morphism between λ-presentable objects.

1.3. Open Problem. Is every λ-presentable morphism (in a locally λ-presentable category) the pushout of a morphism between λ-presentable objects?

A positive answer to this rather intriguing question would have interesting consequences for what is to come. See the paragraph following Lemma 2.2.

We quickly recall the definition of a factorization system (see [Bo, 94] or [CHK, 85] for more).

Given morphisms p and i, we write p ⊥ i if for every commutative square vp = iu, there exists a unique morphism w making both triangles commute in the following diagram:

[figure: u; p; v; w; i]

If N is a class of morphisms, denote by E(N) and M(N) the classes

E(N) = {p | p ⊥ i for all i ∈ N}
M(N) = {i | p ⊥ i for all p ∈ N}.

Then a factorization system in C is a pair (E,M) of classes of morphisms which are closed under composition and contain all isos, and such that: (1) E(M)= E (or, equivalently, M(E) = M), and (2) every morphism f in C has a factorization f = me with m ∈ M and e ∈ E. The class of all epimorphisms (resp. strong epimorphisms, monomorphisms, etc.) is denoted by Epi (resp. Strong Epi, Mono, etc.). A factorization system (E,M) is proper if E ⊆ Epi and M ⊆ Mono.

In what follows, E_{λ} denotes the class of all morphisms in E with λ-presentable domain and codomain.

1.4. Definition. A proper factorization system (E,M) in a category C is λ-presentable if M(E_{λ}) ⊆ M.

The idea behind 1.4 is that in order to check that a given morphism is in M, it will be sufficient to verify that it has the diagonal property with respect to all morphisms in E between “small” objects only. This will be used crucially in Theorems 2.4 and 2.5 below.

1.5. Examples.

(a) Let Σ be a multisorted λ-ary signature (with or without relation symbols), and C = Mod(Σ) be the category of all Σ-structures and Σ-homomorphisms. Denoting by Emb the class of all (Σ-) embeddings in C (in the usual sense of model theory), and by Sur the class of all surjective homomorphisms, then (Sur, Emb) is a λpresentable factorization system in C. This is readily verified directly, but it also follows from Proposition 1.6 (and its proof) below.

(b) It is well-known that in any locally λ-presentable category, (Strong Epi, Mono) and (Epi, Strong Mono) are (proper) factorization systems. The former is λ-presentable; again, this will follow from Proposition 1.6 (see Example (c) below). However the latter is not, as we will see from the following counterexample.

Let Σ be the signature with a (n+2)-ary relation R_{n} for each positive integer n. Consider the set

T = {ϕ_{n}| n > 0}

where ϕ_{n} is the Σ-sentence

∀x∃^{≤1}(y_{1} … y_{n})∃y_{n+1}(R_{1}(x, y_{1}, y_{2}) ∧ R_{2}(x, y_{1}, y_{2}, y_{3}) ∧ … ∧ R_{n}(x, y_{1}, …, y_{n+1})).

Here, “∃^{≤1}(y_{1} … y_{n})” means “there exists at most one string y_{1} … y_{n}”. The ϕ_{n}’s are equivalent to universal Horn sentences, and consequently the category Mod(T) of all models of T (with all Σ-homomorphisms) is locally finitely presentable (see (c) below). Hence (Epi, Strong Mono) is a factorization system in the category. We show it is not finitely presentable.

Let

A = ⟨{a}; R_{n}^{A} = ∅ for all n⟩

be the Σ-structure with one element a and all relations empty, and

B = ⟨{a, b_{1}, b_{2}, ...}; R_{n}^{B} = {(a, b_{1}, b_{2}, ..., b_{n+1})} for all n⟩.

Both are models of T, and f: A ⟶ B, defined by f(a) = a, is a homomorphism. Then f is easily seen to be an epi in Mod(T), since every homomorphism from B to another T -model is entirely determined by its value on a. Hence it is not a strong mono (otherwise it would be an isomorphism). However it is in M(Epi_{ω}). In order to see this, consider a commutative square

\xymatrix{C \ar[d] \ar[r]^{g} & D \ar[d] \\ A \ar[r]_{f} & B}

with g epi and C and D finitely presentable. First note that the finitely presentable objects of Mod(T) must be finite. Then, the syntactic description of epimorphisms in categories of models due to P.D. Bacsich (see [Ba, 72] or [H_{2}, 98]) says that for every element d in D, there exists a finite conjunction β of atomic formulas such that

T ⊧ ∀x∃^{≤1}y∃z (β(x, y, z ))

and

D ⊧ ∃z (β[g(c), d, z ])

for some string c in C. Let us say that such a d is determined by β. We show that R_{n}^{D} = ∅ for all n.

Suppose that D ⊧ R_{n}[d_{0}, d_{1}, …, d_{n+1}] holds for some n and some d_{i}’s in D. Then all the d_{i}’s must be distinct, since their images in B are distinct. Note also that d_{n+1} cannot be in the image of g. Hence, there must be a greatest integer n for which D ⊧ R_{n}[d_{0}, d_{1}, …, d_{n+1}] holds for some d_{i}’s. But then there is no conjunction of atomic formulas β which can determine d_{n+1}. Hence R_{n}^{D} = ∅. As a consequence, g is surjective, and the required diagonal clearly exists.

(c) Recall that, up to a categorical equivalence, locally λ-presentable categories are the categories of models of λ-limit theories, i.e., set of sentences of the form

∀x(φ(x) ⟶ ∃^{1}y(β(x, y))

where x and y are strings of less than λ variables, “∃^{1}y” means “there exists exactly one string y”, and φ and β are conjunction of less than λ atomic formulas in a multisorted λ-ary signature Σ (see [AR, 94] or [H, 01]).

Let C = Mod(T) be such a (concrete) category of all the models of a λ-limit theory T in some signature Σ. Generalizing Example (a), the following proposition will show that Emb is the right hand part of a λ-presentable factorization system in C. In addition, note that in the categorical equivalence mentioned above, one can always choose Σ to have no relation symbols. Since in this case we have Emb = Mono, the proposition will also imply that (Strong Epi, Mono) is λ-presentable in every locally λ-presentable category, as claimed in the first paragraph of Example (b).

1.6. Proposition. Let C be the category of all models of a λ-limit theory. Then (E(Emb), Emb) is a λ-presentable factorization system in C.

Proof. It follows immediately from [CHK, 85], Lemma 3.1, that (E(Emb), Emb)is a factorization system in C . From [H_{2}, 98], Theorem 20, one deduces that Strong Mono ⊆ Emb, so that E(Strong Mono) = Epi ⊇ E(Emb) (since (Epi, Strong Mono) is a factorization system). Hence (E(Emb), Emb) is proper. We now show that it is λpresentable.

Let Σ be the (λ-ary) signature of the theory. C is a full reflective subcategory of the category Mod(Σ) of all Σ-structures and Σ-homomorphisms. Now, in Mod(Σ), the λ-presentable objects are precisely the structures presentable in the classical sense by less than λ “generators and relations”; this is carefully explained in [AR, 94], 5.5 and 5.28. Let us fix some notations which will be used again later on.

A set of generators and relations can be seen as a set of atomic formulas (identifying the set of generators with a string of variables), so that a typical λ-presentable object of Mod(Σ) can be denoted by

⟨x; φ(x)⟩_{Σ}

for some set φ of less than λ atomic formulas. Note that the length of x must then be < λ. The reflection of ⟨x; φ(x)⟩_{Σ} in C is denoted by ⟨x; φ(x)⟩_{C}, and it is λ-presentable in C. We show that M(E(Emb)_{λ}^{∗}) ⊆ Emb, where E(Emb)_{λ}^{∗} is the class of morphisms in E(Emb) with λ-presentable domain and codomain of the form ⟨x; φ(x)⟩_{C}. This will imply in particular that M(E(Emb)_{λ}) ⊆ Emb, as required.

Given a morphism f: A ⟶ B in M(E(Emb)_{λ}^{∗}), suppose that B ⊧ R[f(a)] for some relation symbol R in Σ∪{=} and some string a in A. We write x_{a} for a string of variables corresponding to a. Then there exists a commutative diagram

\xymatrix{⟨x_{a}; ∅⟩_{Σ} \ar[r]_{i′} \ar[d]^{i} & ⟨x_{a}; R(x_{a})⟩_{Σ} \ar[d]^{j} \\ A \ar[r]_{f} & B}

in Mod(Σ) with i(x_{a} ) = a, i′ is the identity function, and j(x_{a} ) = f(a). In order to show that A ⊧ R[a], we need to show that there exists a diagonal morphism from ⟨x_{a}; R(x_{a})⟩_{Σ} to A making the two triangles commute.

Now, i and j factorize through the reflections:

\xymatrix{⟨x_{a}; ∅⟩_{Σ} \ar[r]_{i′} \ar[d] & ⟨x_{a}; R(x_{a})⟩_{Σ} \ar[d] \\ _{⟨x_{a}; ∅⟩_{C}} \ar[d] \ar[r]_{i′′} & _{⟨x_{a}; R(x_{a})⟩_{C}} \ar[d] \\ A \ar[r]_{f} & B}

It is easily verified that i′ is in E(Emb) (in Mod(Σ)). From the fact that i′′ is the reflection of i′ in C, it follows that it is in E(Emb) (in C). Hence the bottom square of the above diagram has a diagonal morphism with the required property, and the outer square too, as a consequence.

1.7. Convention. From now on, a locally λ-presentable category will always be assumed to come equipped with a locally λ-presentable factorization system (E,M). As is customary, morphisms in E will be denoted by “ [figure: ”] , and morphisms in M by “ [figure]”.

2. Algebraically and existentially closed morphisms

In [Fa, 75], Fakir proposed categorical definitions of algebraically closed and existentially closed morphisms in a locally presentable category C. His definitions generalize the familiar model theoretic concepts (see [CK, 90] or [E, 77]). Algebraically closed morphisms generalize the “pure embeddings” in the theory of abelian groups or, more generally, modules. For that reason they are also called “pure morphisms” (see [AR, 94] and [R, 97]). The following reformulates Fakir’s definitions. An intermediate concept is also introduced; its purpose will appear clearly in Theorem 2.4 below.

2.1. Definitions. Let f: A ⟶ B be a morphism in a locally λ-presentable category C. In (c) below, a λ-presentable cone in E is a cone c = (k_{j}: C [figure: D_{j}] )_{J} where |J| < λ and each k_{j} is a λ-presentable morphism in E.

(a) f is λ-algebraically closed if for every factorization f = hg with g λ-presentable, g has a left inverse.

(b) f is weakly λ-existentially closed if it is λ-algebraically closed and for every factorization f = hg with g λ-presentable, and every λ-presentable morphism k in E from the domain of h, and through which h does not factorize, g has a left inverse which does not factorize through k.

(c) f is λ-existentially closed if for every factorization f = hg with g λ-presentable, and every λ-presentable cone c in E from the domain of h, and through which h does not factorize, g has a left inverse which does not factorize through c.

The three concepts could be defined in a uniform way using (c), by adjusting the size of the cone accordingly: replacing “λ-presentable cone” by “cone c = (k_{j})_{J} of λ-presentable morphisms with |J| < 1, |J| < 2 and |J| < λ” respectively, one gets the definitions (a),

(b) and (c). This will be useful in the proofs below. It also allows the same following diagram to be used to illustrate the three definitions:

[figure: A; D_{j}; } ; ; J; A; D_{j}; } ; ; J; C; (λ); (λ); h; C; A; 1_{A}; (λ); g; f; B; A; 1_{A}; k_{j}]

One reads this as “every diagram of the type on the left can be completed as shown on the right”.

The above definitions are quite different from the formulations in [Fa, 75]. The following lemma shows the connection.

2.2. Lemma. Let f: A ⟶ B be a morphism in a locally λ-presentable category C. The following are equivalent:

(i) f is λ-algebraically closed (resp. weakly λ-existentially closed, λ-existentially closed);

(ii) for every commutative square fl = hg with g λ-presentable (resp., and every cone c = (k_{j}: dom(h) [figure: D_{j}] )_{J} of less than 2, resp. less than λ, λ-presentable morphisms in E, and through which h does not factorize), there exists d: dom(h) ⟶ A such that dg = l (resp., and such that d does not factorize through c).

Proof. Condition (ii) can be represented by the diagram:

[figure: J; J; C; l; (λ); g; D; h; k_{j}; D_{j} } ; ; C; D; d; D_{j} } ; ; A; f; B; A; (λ); (λ)]

(i) ⇒ (ii): Let (g′, l′) be the pushout of (g, l), and then (k′_{j}, l′_{j}) be the pushout of (k_{j}, l′), j ∈ J:

\xymatrix{C \ar[rr]^{g} \ar[dd]_{l} &  & D \ar@{->>}[rr]^{k_{j}} \ar[dl]_{l′} \ar[dd]^{h} &  & D_{j} \ar[dl]^{l′_{j}} \\  & P \ar@{->>}[rr]^{k′_{j}} \ar@{-->}[dr]^{u} &  & P_{j} \\ A \ar[ur]^{g′} \ar[rr]_{f} &  & B}

(u: P ⟶ B is the morphism induced by the commutativity of the square). Then g′ and the k′_{j} are λ-presentable. Furthermore, pushouts (by any morphism) of morphisms in E are in E (see [CHK, 85]). Finally, u does not factorize through the cone c′ = (k′_{j})_{J} (otherwise h would factorize through the cone c = (k_{j})_{J}).

By (i), g′ has a left inverse g′′ which does not factorize through c′. It is easily verified that the diagonal morphism g′′l′ satisfies the conditions required for d in (ii). (ii) ⇒ (i): Take l: C ⟶ A in the square to be 1_{A}.

Let (ii)′ be the statement obtained from 2.2(ii) by replacing all λ-presentable morphisms by morphisms with λ-presentable domain and codomain. Clearly the implication (ii) ⇒ (ii)′ holds. It follows easily from 1.2 above that the reverse implication holds for the λ-algebraically closed case. This shows that our λ-algebraically closed morphisms are just the λ-pure morphisms of [AR, 94] (Note however that pushouts are needed here; the two concepts are different in general accessible categories: see [H_{1}, 98]). Whether the equivalence (ii) ⇔ (ii)′ holds for the other two cases is unknown to us, but one can show that it would be the case if the open problem 1.3 had a positive answer.

Fakir’s definitions ([Fa, 75]) are obtained from 2.2(ii) by replacing (1) the λ-presentable morphism g by a mono from a λ-generated object to a λ-presentable object, (2) the cone c by a set {(m_{j}, n_{j})}_{J} of pairs of morphisms from λ-presentable objects D_{j} to D, and (3) our condition that h (resp. d) does not factorize through c by the condition that hm_{j} ≠ hn_{j} (resp. dm_{j} ≠ dn_{j}). Instead of showing directly that our definitions do generalize the ones of Fakir, we will devote the rest of this section to prove characterization theorems (2.4 and 2.5 below), which will clearly correspond to Fakir’s characterizations of his λ-algebraically closed and λ-existentially closed morphisms (Theorems 5.7 and 5.10 in [Fa, 75]), when we choose the factorization system (Strong Epi, Mono).

At this point, the reader who wishes to see explicitly the link between the categorical definitions and the classical model theoretic formulations, in terms of preservation of formulas of certain types, may have a look at Proposition 3.1 and its proof.

Recall from [CK, 90] that a λ-complete filter on a set I is a non-empty set F of nonempty subsets of I such that all supersets of elements of F are elements of F, and all intersections of less than λ elements of F are elements of F. An ω-complete filter is just called a filter. An ultrafilter on I is a filter F in which every subset of I not in F has its complement in F.

Given a set {A_{i}}_{i∈I} of objects in a locally λ-presentable category C, and a λ-complete filter F on I, the reduced product ∏_{F} A_{i} of {A_{i}}_{i∈I} (with respect to F) is the colimit of the (λ-directed) diagram ( π_{K,J}: ∏_{i∈K} A_{i} ⟶ ∏_{i∈J} A_{i} ; J ⊆ K, J, K ∈ F ) of the canonical

projections between the products.

If C is the category of all models of a λ-limit theory in some signature Σ, as in 1.5 (c) above, one verifies easily that this corresponds to the classical model theoretic concept: ∏_{F} A_{i} is the Σ-structure on the quotient of the product ∏_{i∈I} A_{i}, where for an α-ary relation

symbol R in Σ∪{=}, ∏_{F} A_{i} ⊧ R[(f_{β})_{β<α}] if and only if {i ∈ I | A_{i} ⊧ R[(f_{β}(i))_{β<α}] } ∈ F (see [CK, 90]). ∏_{F} A_{i} is called an ultraproduct if F is an ultrafilter.

In [Fa, 75], Fakir shows that his ω-algebraically closed (resp. ω-existentially closed) embeddings coincide with the classical algebraically closed (resp. existentially closed) embeddings. He then gives a categorical proof of the known facts that a morphism f: A ⟶ B is ω-algebraically closed (resp. ω-existentially closed) if and only if there exists an ultrafilter F and a homomorphism g: B ⟶ ∏_{F} A (resp. a monomorphism) to the ultrapower ∏_{F} A, such that gf is the canonical “diagonal” morphism Δ_{F}: A ⟶ ∏_{F} A, induced by the diagonal morphisms to the powers of A (see [E, 77] for the classical model theoretic proof). He actually proves the first part of this result for λ-algebraically closed morphisms (any λ), replacing “ultrafilters” by “λ-complete filters”. This does not hold for the existential closed morphisms: for example, even the diagonal morphism Δ_{F}: A ⟶ ∏_{F} A into a reduced power is easily seen not to be existentially closed if F is not an ultrafilter. The next theorem fills the gap by showing that the right concept for the extension is the one of weakly λ-existentially closed. In addition, it covers the concrete cases where relation symbols are present and where one uses embeddings instead of monos. First, a few simpler facts.

2.3. Proposition. Let C be a locally λ-presentable category.

(a) Let f: A ⟶ B and g: B ⟶ C be morphisms in C. Then:

(i) if gf is λ-algebraically closed, then so is f;

(ii) if gf is λ-existentially closed (resp. weakly λ-existentially closed), then so is f, whenever g lies in M.

(b) All three concepts in 2.1 are stable under λ-directed colimits.

(c) For any λ-complete filter F, the diagonal morphism Δ_{F}: A ⟶ ∏_{F} A to the reduced power of A is weakly λ-existentially closed.

(d) If λ = ω, then for any ultrafilter U, the diagonal morphism Δ_{U}: A ⟶ ∏_{U} A to the ultrapower of A is ω-existentially closed.

Proof.

(a) (i) is trivial.

(ii) is an immediate consequence of the implication

[figure: A; D_{j}; } ; ; J; A; D_{j}; } ; ; J; C; h; C; A; 1_{A}; f; B; g∈M; D; A; 1_{A}; f; B; g; D; k_{j}∈E]

(i.e., h does not factorize through the cone (k_{j})_{J} implies that gh does not either), which itself follows from the facts that k_{j} ∈E, g ∈M, and (E,M) is a factorization system.

(b) In the diagram

[figure: A; D_{j}; } ; ; J; C; (λ); (λ); h; A; 1_{A}; g; (λ); f; f_{i}; B; A_{i}; f′_{i}; k_{j}]

(f′_{i}: f_{i} ⟶ f) is the colimit of a λ-directed diagram (f_{ij}: f_{i} ⟶ f_{j})_{I} with each f_{i} λ-existentially closed. Then h: g ⟶ f must factorize through some f_{i}, and clearly the induced morphism h_{i}: C ⟶ A_{i} cannot factorize through any of the k_{j}’s. The existence of the required left inverse to g follows since f_{i} is λ-existentially closed.

(c)-(d) We prove (c) and (d) in parallel. Let F be a λ-complete filter on a set I, Δ_{F}: A ⟶ ∏_{F} A the canonical diagonal morphism. Consider a diagram

[figure: D_{j}; } ; ; J; C; (λ); (λ); h; A; g; (λ); Δ_{F}; ∏_{F} A; k_{j}]

with g λ-presentable and |J| < λ, and where h does not factorize through any of the k_{j}’s. ∏_{F} A is the λ-directed colimit of the powers A^{S}, S ∈ F, and from this it follows that Δ_{F} is the colimit in (A ↓ C) of the diagonal morphisms Δ_{S}: A ⟶ A^{S}. g being λ-presentable, h: g ⟶ Δ_{F} must factorize h = c_{S} · h_{S} through one of the colimit components c_{S}: A^{S} ⟶ ∏_{F} A :

\xymatrix{A \ar[d]_{Δ_{S}} \ar[r]^{g} & C \ar[d]^{h} \ar@{-->}[dl]_{h_{S}} \\ A^{S} \ar[r]_{c_{S}} & ∏_{F} A}

Clearly h_{S} does not factorize through any of the k_{j}’s. Let s ∈ S, and π_{s}: A^{S} ⟶ A_{s} = A be the canonical projection. Then π_{s} · h_{S} · g = π_{s} · Δ_{S} = 1_{A}, so that π_{s}h_{S} is a left inverse to g. This takes care of the case J = ∅. If J = {j}, then π_{s}h_{S} cannot factorize through k_{j} for all s, otherwise h_{S} would also factorize through k_{j}. Hence one of the π_{s}h_{S}’s is an adequate left inverse to g. These two cases solve (c).

To complete the proof of (d), we now assume that λ = ω, and that F is an ultrafilter. For each j ∈ J, let

S_{j} : = {s ∈ S | π_{s}h_{S} factorizes through k_{j}} .

Then S_{j} ∉ F, otherwise the colimit component c_{j}: A^{S_{j}} ⟶ ∏_{F} A would provide a factorization of h through k_{j}. Hence its complement S′_{j} in I is in F, and then S′ = S ∩ ( ∩_{j∈J} S′_{j}) is also in F (since |J| < ω). Now S′ = S − ( ∪_{j∈J} S_{j}), so that one can

see that for any s ∈ S′, the morphism π_{s}h_{S}: C → A^{S} → A_{s} = A does not factorize through any of the k_{j}’s.

An example of a diagonal morphisms Δ_{I}: A ⟶ A^{I} which is not λ-existentially closed, is easily constructed.

The following two theorems should be compared to [Fa, 75]’s Theorems 5.7 and 5.10.

2.4. Theorem. Let C be a locally λ-presentable category.

(a) A morphism f: A ⟶ B is λ-algebraically closed iff there exists a λ-complete filter F and a morphism g: B ⟶ ∏_{F} A such that gf = Δ_{F}: A ⟶ ∏_{F} A.

(b) A morphism f: A ⟶ B is weakly λ-existentially closed iff there exists a λ-complete filter F and a morphism g: B [figure]∏_{F} A in M such that gf = Δ_{F}: A ⟶ ∏_{F} A.

Before giving the proof, let us state the following theorem, to emphasize that an analogous characterization for the λ-existentially closed morphisms is known only for λ = ω. The proof of both theorems will follow.

2.5. Theorem. Let C be a locally finitely presentable category.

A morphism f: A ⟶ B is ω-existentially closed iff there exists an ultrafilter U and a morphism g: B [figure]∏_{U} A in M such that gf = Δ_{U}: A ⟶ ∏_{U} A.

Proof.

2.4(a) (⇐) This is immediate from 2.3(a)(i) and (c), since weakly λ-existentially closed morphisms are λ-algebraically closed.

(⇒) The construction will follow a familiar pattern (see the proof of 2.31 in [AR, 94], for instance), but we recall it since it will also be our starting point for the proofs of (b) and of 2.5 below.

Let (f_{ij}: (f_{i}: A ⟶ B_{i}) ⟶ (f_{j}: A ⟶ B_{j}))_{I} be a λ-directed diagram in (A ↓ C) with the f_{i}’s λ-presentable, and with colimit diagram (f′_{i}: (f_{i}: A ⟶ B_{i}) ⟶ (f: A ⟶ B))_{I}.

Let F be the filter defined on (the directed poset) I by

F = {D ⊆ I | D ⊇ i^{∗} for some i ∈ I},

where i^{∗} = {j ∈ I | j ≥ i}. From the fact that I is λ-directed, one sees that F is a λ-complete filter on I. Then the λ-reduced product ∏_{F} B_{n} is the colimit of the diagram (π_{i,j}: ∏_{n∈i^{∗}} B_{n} ⟶ ∏_{n∈j^{∗}} B_{n})_{I} of the canonical projections (i ≤ j) between

products (since ( ∏_{n∈i^{∗}} B_{n} | i ∈ I) is final in ( ∏_{n∈D} B_{n} | D ∈ F)). Now, for each j ∈ I,

there exists a left inverse f̄_{j} to f_{j}, so that we have the following induced diagram

(1) [figure: A; f_{j}; f; f_{i}; B_{i}; f_{ij}; f̄_{i}; _{⟨f_{in}⟩_{n∈i^{∗}}}; B_{j}; f̄_{j}; _{⟨f_{jn}⟩_{n∈j^{∗}}}; …; colim_{i∈I}(B_{i}) = B; f^{∗}; ∏_{n∈i^{∗}} B_{n}; _{∏_{n∈i^{∗}} f̄_{n}}; ∏_{n∈j^{∗}} B_{n}; _{∏_{n∈j^{∗}} f̄_{n}}; …; colim_{i∈I}( ∏_{n∈i^{∗}} B_{n}) = ∏_{F} B_{n}; f̄; A^{i^{∗}}; A^{j^{∗}}; …; colim_{i∈I}(A^{i^{∗}}) = ∏_{F} A]

It is a straightforward exercise to check that f̄f^{∗}f: A → B → ∏_{F} A is the canonical diagonal morphism Δ_{F}.

2.4(b) (⇐) This is immediate from 2.3(a)(ii) and (c).

(⇒) As in the proof of (a), let (f′_{i}: (f_{i} ⟶ f)_{I} = colim_{I}(f_{ij}: f_{i} → f_{j}) in (A ↓ C), with the f_{i}’s λ-presentable. Let H be the set of all λ-presentable morphisms in E

h: B_{i} [figure: C] ,

i ∈ I, through which f′_{i} does not factorize.

First case: H = ∅. Then consider the same construction than in the proof of (a) (⇒), see diagram (1). We show that u = f̄f^{∗} is in M.

(E,M) being λ-presentable, this amounts to show that for all commutative square

(2) \xymatrix{X \ar@{->>}[r]_{v} \ar[d]^{p} & Y \ar[d]^{q} \\ B \ar[r]^{u} & ∏_{F} A}

with v ∈ E_{λ}, there exists a unique diagonal d: Y ⟶ B such that dv = p and ud = q. Since (f′_{i}: B_{i} ⟶ B)_{I} is also the colimit in C of (f_{ij}: B_{i} ⟶ B_{j})_{I}, there exists a factorization p = f′_{i}p_{i} of p for some i ∈ I, p_{i}: X → B_{i}. Let (h, p′_{i}) be the pushout of (v, p_{i})

(3) \xymatrix{ & X \ar@{->>}[rr]^{v} \ar[d]^{p} \ar[ddl]^{p_{i}} & & Y \ar@{-->}[ddl]^{p′_{i}} \\ & B \\ B_{i} \ar[ur]^{f′_{i}} \ar@{->>}[rr]_{h} & & C}

Note that h is a λ-presentable morphism in E, since v is. But H = ∅ means that f′_{i} factorizes through all λ-presentable morphisms in E with domain B_{i}. Hence there exists t: C → B such that th = f′_{i}, and tp′_{i} is easily checked to be the required diagonal.

Second case: H ≠ ∅. The required reduced power will be the product ∏_{h∈H}(∏_{F_{h}} A)

of the reduced powers ∏_{F_{h}} A, h ∈ H, where F_{h} is the filter defined as follows.

Let I_{h} be the set of all pushouts h_{j}: B_{j} [figure: C_{j}] of h by the f_{ij}’s, j ≥ i:

[figure: B_{i}; f_{ij}; h; B_{j}; h_{j}; C; h′_{j}; C_{j}]

Note that

1) I_{h} is a set of λ-presentable morphisms in E,

2) f′_{j}: B_{j} → B does not factorize through h_{j}, and

3) if i ≤ j ≤ k in I, then f_{ij} and h′_{k} induce a morphism h′_{jk} such that the diagram

\xymatrix{B_{i} \ar[r]^{f_{ij}} \ar@{->>}[d]_{h} & B_{j} \ar[r]^{f_{jk}} \ar@{->>}[d]^{h_{j}} & B_{k} \ar@{->>}[d]^{h_{k}} \\ C \ar[r]^{h′_{j}} \ar[rr]_{h′_{k}} & C_{j} \ar[r]^{h′_{jk}} & C_{k}}

commutes.

For j ≥ i, define j^{∗} = {h_{k} | k ≥ j}. We then define F_{h} = {D ⊆ I_{h} | D ⊇ j^{∗} for some j ≥ i}. From the fact that I is λ-directed, one sees that F_{h} is a λ-complete filter on I_{h}.

The λ-reduced product ∏_{F_{h}} B_{n} is the colimit of the diagram (π_{j,k}: ∏_{n∈j^{∗}} B_{n} ⟶

∏_{n∈k^{∗}} B_{n} | i ≤ j ≤ k) of the canonical projections, since ( ∏_{n∈j^{∗}} B_{n} | i ≤ j) is final

in ( ∏_{n∈D} B_{n} | D ∈ F_{h}). Now, for each j ≥ i there exists a left inverse f̄_{h,j} to f_{j}, which

does not factorize through h_{j}

[figure: A; C_{j}; A; C_{j}; B_{j}; f′_{j}; (λ); h_{j}; B_{j}; f̄_{h,j}; h_{j}; A; 1_{A}; f_{j}; (λ); f; B; A; 1_{A}; f_{j}]

For each h: B_{i} = B_{i(h)} [figure]C, we have a diagram similar to (1) in the proof of part (a). Taking the product over all h ∈ H, we get the following diagram

[figure: A; f_{j}; f; f_{i}; B_{i}; f_{ij}; _{⟨f_{in}⟩_{n∈i^{∗}}}; f̄_{h,i}; B_{j}; f̄_{h,j}; _{⟨f_{jn}⟩_{n∈j^{∗}}}; …; colim_{j≥i}(B_{j}) = B; f_{h}^{∗}; _{⟨f_{h}^{∗}⟩_{h∈H}}; ∏_{n∈i^{∗}} B_{n}; _{∏_{n∈i^{∗}} f̄_{h,n}}; ∏_{n∈j^{∗}} B_{n}; _{∏_{n∈j^{∗}} f̄_{h,n}}; …; colim_{j≥i}( ∏_{n∈j^{∗}} B_{n}) = ∏_{F_{h}} B_{n}; f̄_{h}; ∏_{h∈H}(∏_{F_{h}} B_{n}); π_{h}; _{∏_{h∈H} f̄_{h}}; A^{i^{∗}}; g_{ij}; A^{j^{∗}}; …; colim_{j≥i}(A^{j^{∗}}) = ∏_{F_{h}} A; ∏_{h∈H}(∏_{F_{h}} A); π_{h}]

Note that the product ∏_{h∈H}(∏_{F_{h}} A) is actually a reduced power ∏_{F} A for some

λ-complete filter F (see [CK, 90]). The required morphism u: B ⟶ ∏_{F} A is the composition ( ∏_{h∈H} f̄_{h})·⟨f_{h}^{∗}⟩_{h∈H} = ⟨f̄_{h}f_{h}^{∗}⟩_{h∈H}. It is easy to verify that uf: A ⟶ ∏_{F} A

is the canonical diagonal morphism. What remains to be shown is that u is in M.

As in the case H = ∅, we need to consider a commutative square up = qv, with v: X → Y in E_{λ}, and show that there exists a unique diagonal d: Y ⟶ B such that dv = p and ud = q (see diagram (2) above). As before, p factorizes p = f′_{i}p_{i} through some colimit component f′_{i}, and we consider the pushout (h, p′_{i}) of (v, p_{i}) (see diagram (3)). Again we’ll be looking for a factorization of f′_{i} through h.

Suppose there is no such factorization of f′_{i}. Then h ∈ H, by definition of H. Let q_{h} = π_{h}q: Y ⟶ ∏_{F} A = ∏_{h∈H}(∏_{F_{h}} A) ⟶ ∏_{F_{h}} A, where π_{h} is the canonical

projection. Because Y is λ-presentable, there exists j^{∗} ∈ F_{h} such that q_{h} factorizes through the colimit component g_{j}: A^{j^{∗}} ⟶ ∏_{F_{h}} A = colim_{j≥i}(A^{j^{∗}}):

\xymatrix{ & Y \ar[d]^{q} \ar@{-->}[ddl]_{p_{j^{∗}}} \ar[dd]^{q_{h}} \\  & ∏_{F} A \ar[d]^{π_{h}} \\ A^{j^{∗}} \ar[r]_{g_{j}} & ∏_{F_{h}} A}

Put p_{n} = f_{in}p_{i} and p_{n^{∗}} = g_{in}p_{i}. X being λ-presentable, the fact that

g_{j} · ∏_{n∈j^{∗}} f̄_{h,n} · ⟨p_{n}⟩_{n∈j^{∗}} = g_{j} · p_{j^{∗}} · v

implies that there exists k ≥ j such that

\xymatrix{X \ar[d]^{_{⟨p_{n}⟩_{n∈k∗}}} \ar[r]^{v} & Y \ar[d]^{p_{k∗}} \\ ∏_{n∈k^{∗}} B_{n} \ar[r]_{_{∏_{n∈k∗} f̄_{h,n}}} & A^{k^{∗}}}

commutes. Composing with the projection π_{k}: A^{k^{∗}} ⟶ A_{k} = A , we have a commutative diagram

\xymatrix{ & X \ar[r]^{v} \ar[d]^{p_{k}} \ar[dl]_{p_{i}} & Y \ar[d]^{π_{k}p_{k^{∗}}} \ar[ddll]_{p′_{i}} \\ B_{i} \ar@{->>}[d]_{h} \ar[r]^{f_{ik}} & B_{k} \ar[r]_{f̄_{h,k}} & A \\ C}

which induces g: C ⟶ A such that gh = f̄_{h,k}f_{ik} (since (h, p′_{i}) is the pushout of (v, p_{i})). This in turn induces w: C_{k} ⟶ A, which in particular satisfies wh_{k} = f̄_{h,k}:

\xymatrix{B_{i} \ar[r]^{f_{ik}} \ar@{->>}[d]^{h} & B_{k} \ar[r]^{f̄_{h,k}} \ar@{->>}[d]_{h_{k}} & A \\ C \ar[r]_{h′_{k}} \ar[urr]_{g} & C_{k} \ar@{-->}[ur]^{w}}

This contradicts the definition of f̄_{h,k}. Hence there exists t: C ⟶ B such that th = f′_{i}:

\xymatrix{ & X \ar@{->>}[rr]^{v} \ar[d]_{p} \ar[ddl]_{p_{i}} &  & Y \ar@{-->}[dll]_{tp′_{i}} \ar[ddl]^{p′_{i}} \ar[d]^{q} \\  & B \ar[rr]_{u} &  & ∏_{F} A \\ B_{i} \ar[ur]^{f′_{i}} \ar@{->>}[rr]_{h} &  & C \ar@{-->}[ul]^{t}}

We have tp′_{i}v = thp_{i} = f′_{i}p_{i} = p. Since v is epi, we also have utp′_{i} = q, and the unicity of tp′_{i} as a diagonal.

2.5 (⇐) This is immediate from 2.3(a)(ii) and (d).

(⇒) Let f: A → B be existentially closed, and, as in 2.4 (a) and (b), (f′_{i}: (f_{i} ⟶ f))_{I} = colim_{I}(f_{ij}: f_{i} → f_{j}), with the f_{i}’s finitely presentable. Let

R: = {(i, c) | i ∈ I, c ∈ Ĉ_{i}},

where Ĉ_{i} is the set of all finitely presentable cones in E (= finite cones of finitely presentable morphisms in E) with domain B_{i}, and through which f′_{i}: B_{i} → B does not factorize.

We define a preorder on R by

(i, c) ≤ (i′, d)

if and only if i ≤ i′ and for all s: B_{i′} → A, we have:

sf_{ii′} factorizes through c ⟹ s factorizes through d.

Then R is directed: let (i, c) and (i′, d) be in R, with c = (h_{ij} | j ∈ J(c)) and d = (h_{i′j} | j ∈ J(d)). Choose i′′ ≥ i, i′, and let h_{i′′j} be the pushout of h_{ij} by f_{ii′′} (respectively of h_{i′j} by f_{i′i′′}). Then, for e = (h_{i′′j} | j ∈ J(c) ∪ J(d)), we clearly have (i, c), (i′, d) ≤ (i′′, e). Note that e is a finitely presentable cone in E.

For r ∈ R, define r^{∗} = {r′ | r ≤ r′}, and

F = {R′ ⊆ R | R′ ⊇ r^{∗} for some r ∈ R}.

F is a filter on R, which can be extended to an ultrafilter U. For r = (i, c) ≤ (i′, d) = r′ in R, we write f_{rr′}: B_{r} → B_{r′} for the morphism f_{ii′}: B_{i} → B_{i′}; similarly we put f_{r} = f_{i} and f′_{r} = f′_{i}.

Let f̄_{r}: B_{r} → A be a left inverse to f_{r} which does not factorize through c. Then, we get a diagram similar to (1):

[figure: A; f_{s}; f; f_{r}; B_{r}; f_{rs}; f̄_{r}; u_{r}; B_{s}; f̄_{s}; u_{s}; …; colim_{r∈R}(B_{r}) = B; u; A^{r^{∗}}; g_{rs}; A^{s^{∗}}; …; colim_{r∈R}(A^{r^{∗}}) = ∏_{U} A]

where u_{r} = (∏_{t∈r^{∗}} f̄_{t}) · ⟨f_{rt}⟩_{t∈r^{∗}}: B_{r} ⟶ ∏_{t∈r^{∗}} B_{t} ⟶ A^{r^{∗}}. We need to show that uf = Δ_{U}, and u ∈ M. uf = Δ_{U} is straightforward. To show that u ∈ M, we consider, as in 2.4 (b), a commutative square qv = up (see (2) above with F = U), with v ∈ E_{ω}. Then again p factorizes p = f′_{i}p_{i} for some i ∈ I, and we let (h, p′_{i}) be the pushout of (v, p_{i}) (see (3) above). h is a finitely presentable morphism in E, and we now show that f′_{i} factorizes through it. Suppose that there is no such factorization. Then r = (i, {h}) is in R. We rename B_{i} as B_{r}, p_{i} = p_{r} and f′_{i} = f′_{r}:

\xymatrix{ & X \ar@{->>}[r]^{v} \ar[d]^{p} \ar[ddl]_{p_{r}} & Y \ar@{-->}[ddl]^{p′_{r}} \ar[d]^{q} \\  & B \ar[r]^{u} & ∏_{U} A \\ B_{r} \ar[ur]^{f′_{r}} \ar@{->>}[r]_{h} & C}

Y being finitely presentable, there exists s^{∗} ∈ U such that q factorizes q = g_{s}p_{s^{∗}} through the colimit component g_{s}: A^{s^{∗}} ⟶ ∏_{U} A. We can assume s ≥ r, and we have g_{s} · p_{s^{∗}} · v = g_{s} · u_{s} · f_{rs} · p_{r}

\xymatrix{X \ar[d]_{p_{r}} \ar@{->>}[rr]^{v} &  & Y \ar[d]^{p_{s^{∗}}} \\ B_{r} \ar[r]_{f_{rs}} & B_{s} \ar[r]_{u_{s}} & A^{s^{∗}} \ar[r]_{g_{s}} & ∏_{U} A}

X being finitely presentable, there exists t ≥ s and g_{st}: A^{s^{∗}}→ A^{t^{∗}} such that g_{st} · p_{s^{∗}} · v = g_{st} · u_{s} · f_{rs} · p_{r}. Hence the diagram

\xymatrix{X \ar[d]_{p_{r}} \ar@{->>}[rr]^{v} &  & Y \ar[d]^{p_{t^{∗}}} \\ B_{r} \ar[r]_{f_{rt}} & B_{t} \ar[r]_{u_{t}} & A^{t^{∗}}}

commutes. Composing with the projection π_{t}: A^{t^{∗}} → A, we get π_{t}u_{t}f_{rt} = f̄_{t}f_{rt}, and the commutative square π_{t}p_{t^{∗}}v = f̄_{t}f_{rt}p_{r} induces a morphisms g: C → A making

\xymatrix{X \ar[d]_{p_{r}} \ar@{->>}[r]^{v} & Y \ar[d]_{p′_{r}} \ar[ddr]^{π_{t}p_{t^{∗}}} \\ B_{r} \ar@{->>}[r]^{h} \ar[drr]_{f̄_{t}f_{rt}} & C \ar@{-->}[dr]_{g} \\  &  & A}

commute. Now, from the definition of r = (i, {h}) ≤ t = (k, c), the fact that f̄_{t}f_{rt} factorizes through h implies that f̄_{t} factorizes through the cone c = (c_{j}: B_{k} → C_{j} | j ∈ J(c)). But this contradicts the definition of f̄_{t}. Hence there must exist t: C → B such that th = f′_{r}, and one checks as in 2.4 (b) that tp′_{r}: Y → B is the required diagonal in the square qv = up. This completes the proof that u lies in M.

3. Injectivity and syntactic characterizations

In this section, we study the relations between injectivity classes and closure under the three types of subobjects considered so far.

It may be useful first to see explicitly how the categorical Definitions 2.1 specialize in the classical model theoretic context C = Mod(Σ). We already mentioned that if M is the class Emb of all (Σ-) embeddings, then the class E(M), in this category, is the family Sur of all the surjective (Σ-) homomorphisms. Recall also that a basic formula is an atomic formula or the negation of an atomic formula. We have:

3.1. Proposition. Let C = Mod(Σ), Σ a λ-ary signature, with (E,M)=(Sur, Emb). Below, β and δ (resp. γ) are arbitrary sets of less than λ atomic (resp. basic) formulas. Let f: A ⟶ B be a morphism in C. Then

(a) f is λ-algebraically closed iff for every β(x, y) such that

B ⊧ ∃y(∧β[f(a), y])

for some string a in A, we have

A ⊧ ∃y(∧β[a, y]).

(b) f is weakly λ-existentially closed iff for every β(x, y) and δ(x, y) such that

B ⊧ ∃y(∧β[f(a), y] ∧ (¬ ∧ δ[f(a), y ]))

for some string a in A, we have

A ⊧ ∃y(∧β[a, y] ∧ (¬ ∧ δ[a, y ])).

(c) f is λ-existentially closed iff for every γ(x, y) such that

B ⊧ ∃y(∧γ[f(a), y])

for some string a in A, we have

A ⊧ ∃y(∧γ[a, y]).

Proof. For λ = ω, (a) and (c) follow from Theorems 2.4 and 2.5, by the wellknown model theoretic characterizations of the algebraically closed and existentially closed embeddings ([E, 77]). The general statement in (a) and (b) can be derived from 2.4 and 2.5 using the same standard technique than [E, 77]. However, we will sketch a direct proof, in order to make the connection explicit.

We associate to any given family {k_{j}g: A ⟶ C [figure: D_{j}] }_{J} of λ-presentable morphisms, |J| < λ, the following (classical) presentations. First, we present A by its positive diagram

A = ⟨X_{A}; Φ(X_{A})⟩_{Σ} ,

i.e., Φ(X_{A}) := {φ(x_{a}) | A ⊧ φ[a], φ atomic}. Here, as in the proof of 1.6, x_{a} is a string of variables corresponding to a. Capitalized (bold) letters will generally represent strings or sets of unrestricted cardinality; we write φ(x_{a}) to emphasize the fact that only a substring x_{a} of X_{A} of length < λ can actually appear in any given φ (since Σ is λ-ary).

Then, applying the characterization of λ-presentable morphisms (1.2) to g, the fact that λ-presentability in C = Mod(Σ) has the usual meaning (see the proof of Proposition 1.6) allows one to see that C has a presentation of the form

⟨X_{A} ∪ x_{c}; Φ(X_{A}) ∪ β(x_{a}, x_{c})⟩ _{Σ}

where β is some set of less than λ atomic formulas, and c is a string of elements of C of length < λ. In this presentation, g is identified with the function x_{a} ↦ x_{a}; if g(a) = g(a′), the identification x_{a} = x_{a′} (in the presentation of C) follows from the identities in β.

Similarly, for each j there exists a presentation of D_{j} which adds less than λ (variables and) atomic formulas to the ones presenting C. All k_{j}’s are surjective, so that for each generator x_{d} of D_{j}, we have d = k_{j}(c) for some c ∈ C. Because |J| < λ, we can add one x_{c} to the generators of C for each c such that x_{k_{j}(c)} is involved in the presentation of D_{j} (if it is not already there), and the cardinality of the set of all these extra x_{c} will still be < λ. Assuming this has been done in the presentation of C above, we can then present D_{j} by

D_{j} = ⟨X_{A} ∪ x_{c}; Φ(X_{A}) ∪ β(x_{a}, x_{c}) ∪ δ_{j}(x_{a}, x_{c})⟩ _{Σ} ,

for some set δ_{j} of less than λ atomic formulas.

Consider the diagram

[figure: A; D_{j}; } ; ; J; C; h; A; 1_{A}; g; f; B; k_{j}]

where f and h are some Σ-homomorphisms. Then we have

B ⊧ ∧β[f(a), h(c)] ∧ (∧_{j∈J}(¬ ∧ δ_{j}[f(a), h(c)])).

If the diagram can be completed by some appropriate r:

[figure: A; D_{j}; } ; ; J; C; r; A; 1_{A}; g; f; B]

then

A ⊧ ∧β[a, r(c)] ∧ (∧_{j∈J}(¬ ∧ δ_{j}[a, r(c)])).

Using this translation, the details of the proof are now straightforward.

We fix some notations to represent injectivity with respect to specific types of cones and morphisms:

3.2. Definitions. Let C be a locally λ-presentable category.

(a) A λ_{m}-cone is a cone made of λ-presentable morphisms.

(b) If d is a morphism or a cone with domain A, we say that an object C is d-injective,

denoted by

C |⊧ d,

if every morphism f: A ⟶ C factorizes through d.

(c) If D is a class of morphisms or of cones, Mod_{|⊧}(D) will denote the class

{C ∈ C | C |⊧ d for all d ∈ D} .

Then, a class K of objects is a λ_{m}-injectivity class (resp. a λ_{m}-cone-injectivity class) if there exists a class D of λ-presentable morphisms (resp. of λ_{m}-cones) such that K = Mod_{|⊧}(D). If D is a class of morphisms between λ-presentable objects, we say that K is a λ-injectivity class.

Consider again the classical context C = Mod(Σ), with Σ λ-ary, and (E,M) = (Sur, Emb). Proceeding as in the proof of Proposition 3.1, each morphism g_{i} of a λ_{m}-cone d = (g_{i}: A ⟶ B_{i})_{I} can be presented by generators and relations

\xymatrix{⟨X_{A}, Φ(X_{A})⟩_{Σ} \ar[r]^{g_{i}} & ⟨X_{A} ∪ x_{b_{i}}; Φ(X_{A}) ∪ β_{i}(x_{a_{i}}, x_{b_{i}})⟩ _{Σ}}

where g_{i}(x) = x for all x ∈ X_{A}. Then one sees that for C ∈ C,

C |⊧ d

if and only if

C ⊧ ∀X (∧Φ(X ) ⟶ ∨_{i∈I}(∃y_{i}(∧β_{i}(x_{i}, y_{i})))),

where we wrote X for X_{A}, x_{i} for x_{a_{i}} and y_{i} for x_{b_{i}}. Note that the x_{i} ’s are substrings of X , and that for each i ∈ I, the formula ∃y_{i}(∧β_{i}(x_{i}, y_{i} )))) is in L_{λ}(Σ).

The following was proved in [H_{1}, 98].

3.3. Theorem. [H_{1}, 98] Let K be a class of objects in a locally λ-presentable category.

(a) The class of all λ-algebraically closed subobjects of the objects in K is a λ_{m}-cone-injectivity class.

(b) K is a λ_{m}-cone-injectivity class iff it is closed under λ-algebraically closed subobjects.

(c) K is a λ_{m}-injectivity class iff it is closed under λ-algebraically closed subobjects and products.

The proof uses in particular the characterization in Lemma 2.2.

For C = Mod(Σ), Theorem 3.3 says that the classes of Σ-structures closed under λalgebraically closed subobjects are the ones axiomatizable by classes of sentences of the appropriate type, as given after Definition 3.2. Adding closure under products is easily seen to correspond to removing the disjunctions:

3.4. Corollary. A class K of λ-ary Σ-structures is closed in Mod(Σ) under λalgebraically closed substructures (resp. under λ-algebraically closed substructures and products) iff it can be axiomatized by a class of sentences of the form

∀X (∧Φ(X ) ⟶ ∨_{i∈I}(∃y_{i}(∧β_{i}(x_{i}, y_{i}))))

(resp.

∀X (∧Φ(X ) ⟶ ∃y(∧β(x, y))) ),

where Φ is a set (of any cardinality) of atomic formulas, and each β_{i} (resp. β) is a set of less than λ atomic formulas.

The sentences in Corollary 3.4 are in L_{∞}(Σ). What prevent them to be in L_{λ}(Σ), are the sizes of β and of I. Adding closure under λ-directed colimits to closure under products will give the expected preservation theorem in L_{λ}(Σ). In terms of injectivity, this means we will pass from injectivity with respect to λ-presentable morphisms to injectivity with respect to morphisms between λ-presentable objects (i.e., from λ_{m}-injectivity to λinjectivity). This will be shown in the context of locally λ-presentable categories (Theorem 3.5 below), giving us a different proof of the main result of [RAB, 02].

To reduce the class of λ-presentable morphisms in Theorem 3.3 to a set, the additional condition needed of C is precisely to be closed under α-directed colimits for some α. This was shown already in [AR, 93]. We will obtain the refined version of [RAB, 02] by showing that closure under λ-directed colimits allows replacing λ-presentable morphisms by morphisms between λ-presentable objects (of which essentially only a set exists).

It is known from the 60’s that an elementary class K of finitary Σ-structures is closed under directed colimits and (reduced) products if and only if it can be axiomatized by (L_{ω}(Σ)-) sentences of the form

∀x(∧φ(x)) ⟶ ∃y(∧β(x, y))

where φ and β are finite sets of atomic formulas (see [CK, 90]). More recently, Rothmaler ([R, 97]) noticed that the assumption of elementarity of K can be replaced by closure under (ω-) algebraically closed substructures. This amounts to replace closure under elementary substructures by closure under algebraically closed substructures. Krause ([K, 98]) also obtained this characterization, in a context which roughly corresponds to the additive locally finitely presentable categories (note that his locally presented categories are different from the locally presentable categories). Finally, this characterization without the additivity and the finitary assumptions is obtained in [RAB, 02], by very different methods. We reach the same result following a different path altogether, as a consequence of our Theorem 3.3.

3.5. Theorem. [RAB, 02] Let K be a class of objects in a locally λ-presentable category C. Then K is a λ-injectivity class iff it is closed under products, λ-directed colimits and λ-algebraically closed subobjects (or, equivalently, closed under λ-reduced products and λ-algebraically closed subobjects).

Proof. Clearly, closure under λ-directed colimits and products implies closure under λ-reduced products. Hence we need to show that

1) if K is a λ-injectivity class, then it is closed under λ-directed colimits, products and λ-algebraically closed subobjects, and

2) if K is closed under λ-reduced products and λ-algebraically closed subobjects, then it is a λ-injectivity class.

As for 1), λ-injectivity classes are special λ_{m}-injectivity classes, so by 3.3, only closure under λ-directed colimits remains to be shown. This is left as a straightforward exercise. We now prove 2).

Products are special λ-reduced products, so we know from 3.3 that K = Mod_{|⊧}(D) for some class D of λ-presentable morphisms. To be able to replace D by a class (and then, by a set) of morphisms between λ-presentable objects of C, we will use Proposition 1.2.

We first note that for a diagram

[figure: C; m; D; A; m^{∗}; f; E; r; B; s]

satisfying the conditions in the statement of 1.2, we have, for any object X in C,

a) X |⊧ m^{∗} ⇔ X |⊧ f, and

b) X |⊧ m ⇒ X |⊧ m^{∗}.

This follows easily from the facts that the square is a pushout and that rs = 1_{f} in (A ↓ C). Then by a), we can assume that the class D above is made of morphisms of type m^{∗}, i.e., pushouts of morphisms between λ-presentable objects. Using b), we will be done if we prove the following:

3.6. Lemma. Let K be a class of objects in C closed under λ-reduced products. Let K |⊧ f for some f which is the pushout of a morphism between λ-presentable objects. Then f is the pushout of some morphism m_{f} between λ-presentable objects such that K |⊧ m_{f}.

Proof. Let then

\xymatrix{C \ar[d]_{n} \ar[r]^{m} & D \ar[d]^{q} \\ A \ar[r]^{f} & B}

be a pushout diagram, where C and D are λ-presentable and K |⊧ f.

Let (n′_{i}: (n_{i}: C ⟶ A_{i}) ⟶ (n: C ⟶ A))_{I} be the colimit in (C ↓ C) of the λdirected diagram (n_{ij}: n_{i} ⟶ n_{j})_{I}, where the n_{i}’s are λ-presentable morphisms. Taking the pushout (q_{i}, m_{i}) of (n_{i}, m) for each i, we get the commutative diagram

\xymatrix{C \ar[ddd]_{n} \ar[rr]^{m} \ar[dr]^{n_{i}} &  & D \ar[ddd]_{q} \ar[dr]^{q_{i}} \\  & A_{i} \ar[d]^{n_{ij}} \ar[rr]^{m_{i}} &  & P_{i} \ar[d]^{p_{ij}} \\  & A_{j} \ar[dl]^{n′_{j}} \ar[rr]^{m_{j}} &  & P_{j} \ar[dl]^{p_{j}} \\ A \ar[rr]_{f} &  & B}

where the p_{i}’s and the p_{ij}’s are the induced morphisms. We now assume K |⊧ f, and we suppose K |̸⊧ m_{i} for all i ∈ I. We will reach a contradiction.

K |̸⊧ m_{i} means that there exists g_{i}: A_{i} ⟶ K_{i}, K_{i} ∈ K, which does not factorize through m_{i}: A_{i} ⟶ P_{i}. Choose a λ-complete filter F on I which contains all i^{∗} = {j ∈ I | i ≤ j}, i ∈ I. Denote by (c_{i}: ∏_{k∈i^{∗}} A_{k} ⟶ ∏_{F} A_{k})_{I} the colimit diagram (in C) of the

λ-directed diagram of the projections

( π_{ij}: ∏_{k∈i^{∗}} A_{k} ⟶ ∏_{k∈j^{∗}} A_{k} | i ≤ j )_{I}.

Denoting by n_{i^{∗}}: A_{i} ⟶ ∏_{k∈i^{∗}} A_{k} the morphism n_{i^{∗}} = ⟨n_{ij}⟩_{j∈i^{∗}}, the family {n_{i^{∗}}}_{i∈I} induces

a morphism n^{∗}: A = colim_{i∈I}(A_{i}) ⟶ ∏_{F} A_{k} = colim_{i∈I}( ∏_{k∈i^{∗}} A_{k}) such that n^{∗}n′_{i} = c_{i}n_{i^{∗}} for all

i ∈ I. The family {g_{i}: A_{i} ⟶ K_{i}}_{I} induces morphisms ∏_{k∈i^{∗}} g_{k}: ∏_{k∈i^{∗}} A_{k} ⟶ ∏_{k∈i^{∗}} K_{k} and

∏_{F} g_{k}: ∏_{F} A_{k} ⟶ ∏_{F} K_{k}, such that the diagram

\xymatrix{ &  &  & C \ar[dd]^{n} \ar[dr]^{n_{i}} \ar[rr]^{m} &  & D \ar[dd]^{q} \ar[dr]^{q_{i}} \\ K_{i} & ∏_{k∈i^{∗}} K_{k} \ar[l]^{π_{i}} \ar@{-->}[dl]^{c_{i}} & ∏_{k∈i^{∗}} A_{k} \ar[l]^{∏_{k∈i^{∗}} g_{k}} \ar@{-->}[dl]^{c_{i}} &  & A_{i} \ar[llll]_{g_{i}} \ar[ll]^{n_{i^{∗}}} \ar[rr]^{m_{i}} \ar[dl]^{n′_{i}} &  & P_{i} \ar[dl]^{p_{i}} \\ ∏_{F} K_{k} & ∏_{F} A_{k} \ar[l]^{∏_{F} g_{k}} &  & A \ar[ll]^{n^{∗}} \ar[rr]_{f} &  & B}

commutes. Now, ∏_{F} K_{k} ∈ K, so that (∏_{F} g_{k})n^{∗}: A ⟶ ∏_{F} K_{k} must factorize (∏_{F} g_{k})n^{∗} = gf through f, for some g: B ⟶ ∏_{F} K_{k}. We have gq: D ⟶ ∏_{F} K_{k} = colim_{i∈I}( ∏_{k∈i^{∗}} K_{k}) with D λ-presentable, so that there must exist i ∈ I and t_{i}: D ⟶ ∏_{k∈i^{∗}} K_{k} such that

c_{i}t_{i} = gq.

In the diagram

\xymatrix{ &  & C \ar[rr]^{m} \ar[dr]_{n_{i}} \ar[dl]_{t_{i}m} &  & D \\ K_{i} & ∏_{k∈i^{∗}} K_{k} \ar[l]^{π_{i}} \ar[d]^{c_{i}} &  & A_{i} \ar[ll]^{( ∏_{k∈i^{∗}} g_{k})n_{i^{∗}}} \\  & ∏_{F} K_{k}}

we have c_{i} · ( ∏_{k∈i^{∗}} g_{k}) · (n_{i^{∗}}) = c_{i} · (t_{i}m) (by chasing the previous diagram). C being λ-

presentable, we can choose i such that ( ∏_{k∈i^{∗}} g_{k})n_{i^{∗}} = t_{i}m. This equality induces (a unique)

d: P_{i} ⟶ K_{i} such that dm_{i} = g_{i} (and dq_{i} = π_{i}t_{i})

[figure: C; m; n_{i}; D; π_{i}t_{i}; q_{i}; A_{i}; m_{i}; g_{i}; P_{i}; d; K_{i}]

(noting that g_{i} = π_{i} · ( ∏_{k∈i^{∗}} g_{k}) · n_{i^{∗}}). This contradicts the assumption above.

Using 3.6 and the remarks preceding it, it follows that K = Mod_{|⊧}(D′), where D′ = { m_{f} | f ∈ D }. This completes the proof of 3.5.

3.7. Notes.

1) Lemma 3.6 can be seen as the categorical version of an infinitary “compactnesslike” property: following the translation just after Definition 3.2, its meaning in C = Mod(Σ), Σ λ-ary, is that if K is a class of Σ-structures closed under λ-reduced products, then for any given sentence of the form

∀X (∧Φ(X ) ⟶ ∃y(∧β(x, y)))

true in K, where Φ is a set (of any cardinality) of atomic formulas, x ⊆ X , and β is a set of less than λ atomic formulas, there exists a subset φ of Φ of cardinality less than λ, and x′ ⊇ x such that K satisfies

∀x′(∧φ(x′) ⟶ ∃y(∧β(x, y))).

Clearly this and Corollary 3.4 imply that the classes of Σ-structures closed under products, λ-directed colimits and λ-algebraically closed substructures are the ones axiomatizable by sets of L_{λ}(Σ)-sentences as above (called λ-regular sentences). In [RAB, 02], a counterexample shows that the analogous reduction is not possible if the class is not closed under products.

2) What was done in Theorem 3.3 and Corollary 3.4 for algebraically closed embeddings can be done for existentially closed and weakly existentially closed embeddings.

First, instead of λ_{m}-cones, one considers diagrams of the form ( \xymatrix{A \ar[r]^{(λ)}_{g_{i}} & B_{i} \ar@{->>}[r]_{h_{ij}}^{(λ)} & C_{ij}} |

i ∈ I, j ∈ J_{i} ), with each |J_{i}| < λ (resp., |J_{i}| < 2), and then defines injectivity with respect to these diagrams in an appropriate way. Then, using a very similar technique than the proof in [H_{1}, 98], and using the relevant parts of Lemma 2.2, one shows that the classes of objects closed under λ-existentially closed (resp., weakly λ-existentially closed) subobjects are precisely the injectivity classes with respect to the above special types of diagrams. A syntactic characterization (in the context of C = Mod(Σ)) follows easily. However, serious difficulties arise when one adds closure under λ-directed colimits, in trying to obtain the analogous of the characterization 3.5. This is not surprising, considering the difficulty of the proofs of the classical preservation theorem for reduced products, even for ∀∃-theories (see [CK, 90], Section 3.5). The infinitary case also seems to raise additional problems, see [HS, 81]. Actually, the very fact that the known proof of the classical preservation theorem for reduced products is so intricate may be seen as an incentive to look for a different (more categorical?) line of attack.

References

[AR, 93] J. Adámek, J. Rosický, On injectivity in locally presentable categories, Trans. Amer. Soc. 336 (1993), 785-804.

[AR, 94] J. Adámek, J. Rosický, Locally Presentable and Accessible Categories, Cambridge Univ. Press, Cambridge 1994.

[AN, 79] H. Andréka, I. Németi, Injectivity in categories to represent all first order formulas, Demonstratio Math. 12 (1979), 717-732.

[Ba, 72] P.D. Bacsich, Injectivity in model theory, Coll. Math. 25 (1972), 165-176.

[Bo, 94] F. Borceux, Handbook of Categorical Algebra, Cambridge Univ. Press, Cambridge 1994.

[CHK, 85] C. Cassidy, M. Hébert, G.M. Kelly, Reflective subcategories, localizations and factorization systems, J. Austr. Math. Soc. 38 (Series A) (1985), 287-329.

[CK, 90] C.C. Chang, H.J. Keisler, Model Theory, North-Holland, Amsterdam 1990.

[E, 77] P.C. Eklof, Ultraproducts for Algebraists, in Handbook of Mathematical Logic, ed. by J. Barwise, Studies in Logic 90, North-Holland, Amsterdam 1977.

[Fa, 75] S. Fakir, Objets algébriquement clos et injectifs dans les catégories localement présentables, Bull. Soc. Math. France, Mem. 42, 1975.

[Fu, 70] L. Fuchs, Infinite Abelian Groups I, Academic Press, New York 1970.

[H_{1}, 98] M. Hébert, Purity and injectivity in accessible categories, JPAA 129 (1998), 143-147.

[H_{2}, 98] M. Hébert, On generation and implicit partial operations in locally presentable categories, Appl. Cat. Struct. 6 (1998), 473-488.

[H, 01] M. Hébert, Concretely locally presentable and locally generated categories, Appl. Cat. Struct. 9 (2001), 595-624.

[HS, 81] W. Hodges, S. Shelah, Infinite games and reduced products, Ann. Math. Logic 20 (1981), 77-108.

[K, 98] H. Krause, Exact definable categories, J. Algebra 201 (1998), 456-492.

[RAB, 02] J. Rosický, J. Adámek, F. Borceux, More on injectivity in locally presentable categories, Theory and Applications of Categories 10 (2002), 148-161.

[R, 97] P. Rothmaler, Purity in model theory, Advances in algebra and model theory (Essen, 1994; Dresden, 1995), 445–469, Algebra Logic Appl. 9, Gordon and Breach, Amsterdam, 1997.

Department of Mathematics, The American University in Cairo, Cairo 11511, EGYPT

Email: mhebert@aucegypt.edu

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