Theory and Applications of Categories, Vol. 12, No. 12, 2004, pp. 355–371.

K-PURITY AND ORTHOGONALITY

MICHEL HÉBERT

Abstract. Adámek and Sousa recently solved the problem of characterizing the subcategories K of a locally λ-presentable category C which are λ-orthogonal in C, using their concept of K_{λ}-pure morphism. We strenghten the latter definition, in order to obtain a characterization of the classes defined by orthogonality with respect to λ-presentable morphisms (where f: A [figure] B is called λ-presentable if it is a λ-presentable object of the comma category (A ↓ C)). Those classes are natural examples of reflective subcategories defined by proper classes of morphisms. Adámek and Sousa’s result follows from ours. We also prove that λ-presentable morphisms are precisely the pushouts of morphisms between λ-presentable objects of C.

1. Introduction

We say that a morphism f: A [figure] B in a category C is λ-presentable if it is λ-presentable as an object of the comma category (A ↓ C). In [H_{1}, 98] and [H, 04], we demonstrated the usefulness of this concept to study purity. We start our paper by characterizing these morphisms as precisely the pushouts of the morphisms between λ-presentable objects (Theorem 2.3), a problem left open in [H, 04]. Then we compare subcategories defined by

(a) orthogonality with respect to morphisms between λ-presentable objects (called λ-orthogonality classes), to those defined by

(b) orthogonality with respect to λ-presentable morphisms (called λ_{m}-orthogonality classes).

In [AS, 04], Adámek and Sousa introduce the concept of K_{λ}-pure morphism, and use it to characterize the λ-orthogonality classes as the ones closed under products, λdirected colimits and K_{λ}-pure subobjects. This completed the solution to a problem which arose from the works of Coste ([C, 79]) and Volger ([V, 79]). Using λ-presentable morphisms, we define strongly K_{λ}-pure morphisms, and we obtain a characterization of the λ_{m}-orthogonality classes as those which are closed under products and strongly K_{λ}-pure subobjects (Theorem 4.1). We derive the Adámek-Sousa’s result from our characterization, by proving a lemma giving closure conditions on K under which all K_{λ}-pure morphisms are strong.

This parallels the results and methods of [H_{1}, 98] and [H, 04]. In the former paper, we obtained a characterization of λ_{m}-injectivity classes as those closed under products and (strongly) λ-pure subobjects, and in the latter we showed how to derive from this the characterization of λ-injectivity classes of [RAB, 02].

Received by the editors 2004-09-17 and, in revised form, 2004-10-04.

Transmitted by Walter Tholen. Published on 2004-10-21.

2000 Mathematics Subject Classification: 18A20, 18C35, 03C60, 18G05.

Key words and phrases: pure morphism, othogonality, injectivity, locally presentable categories,

accessible categories.

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

On our way, we observe that all λ_{m}-orthogonality classes are reflective in C. Note that, in contrast with the morphisms between λ-presentable objects, there is a proper class of λ-presentable morphisms. Similarly, λ_{m}-cone-orthogonality classes are multireflective, but despite partial results in this direction, we could not determine whether the “expected” characterization for them holds, namely as the classes closed under pullbacks and strongly K_{λ}-pure subobjects. However, an example will show that closedness under pullbacks, K_{λ}- pure subobjects and λ-directed colimits is not sufficient for being a λ-cone-orthogonality class.

Much of the usefulness of the λ-presentable morphisms rests on the fact that if C is a λ-accessible category with pushouts, then all the associated comma categories (A ↓ C) are also λ-accessible (Proposition 2.6). Whether this holds without the assumption that pushouts exist is left as an open problem.

Finally, we also consider general orthogonality classes in locally presentable categories. We find the orthogonality hull of any subcategory K to be the one obtained by adding to K all the objects which, for every α ≥ λ, is a strongly K_{α}-pure subobject of some product of objects in K (Corollary 4.5).

2. K_{λ}-purity and strong K_{λ}-purity

For basic definitions and results on locally presentable and accessible categories, we refer the reader to [AR, 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 [figure] Set preserves λ-directed colimits. Then, C is λ-accessible if it has all λ-directed colimits, as well as a (small) set S of λ-presentable objects such that every object in C 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.

Throughout the paper we will need to distinguish, for a given category C, between the λ-presentable objects of the comma categories (A ↓ C) (A ∈ C), and those of C^{2} (= the category of morphisms in C). In order to do this, we denote by C_{λ} the full subcategory of C on the λ-presentable objects of C, and we recall, from [H_{1}, 98], the following definition.

2.1. Definition. A morphism f: A [figure] B in a category C will be called λ-presentable if it is a λ-presentable object of the comma category (A ↓ C).

2.2. Remarks and Examples.

K-PURITY AND ORTHOGONALITY  357

(a) In a λ-accessible category, being a λ-presentable morphism is easily seen to be strictly weaker than being λ-presentable as an object of C^{2}: this is clear from the fact that the latter is the same than being a morphism in C_{λ} (see [AR, 94], Exercise 2.c), and from Theorem 2.3 below. For example, a function f: X [figure] Y is finitely presentable in Set^{2} just when X and Y are finite, but it is a finitely presentable morphism in Set if and only if the sets Y − X and {(x, x′) | x ≠ x′, f(x) = f(x′)} are finite.

(b) Let C be the category of all right R-modules, for some ring R. One can verify directly (or using Theorem 2.3 below) that the embedding A ↪ B of a submodule is a finitely presentable morphism in C if and only if the quotient B/A is a finitely presentable module.

(c) The intuition might suggest that a morphism f: A [figure] B is λ-presentable when f provides a way to “present B” in the usual sense of classical algebra, by adding less than λ generators and relations to some presentation of A. This is indeed correct in the category of all structures of some λ-ary signature (see [H, 04]). In categorical terms, there is a translation of this idea in terms of pushouts that seems reasonable, and that the following theorem shows to be valid in a wider context.

2.3. Theorem.  Let C be a locally λ-presentable category. Then f: A  [figure]B is

λ-presentable if and only if there exists a pushout diagram

[figure: A; B; f; C; A; C; D; D; B]

where C and D are λ-presentable

Proof. Proposition 1.2 of [H, 04] already shows that this is true up to a retraction in (A ↓ C):

2.4. Lemma.  ([H, 04]) In a λ-accessible category with pushouts, f: A  [figure]B is

λ-presentable if and only if there exists a commutative diagram

[figure: A; B′; f′; C; A; m; C; D; h; D; B′; n; B′; B; r; B; B′; s; A; B; f]

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

We need another, technical lemma, which will be used again later:

2.5. Lemma. Any pushout square

[figure: A; B′; f′; C; A; m; C; D; h; D; B′; n]

with C and D λ-presentable can be completed as follows:

[figure: A_{i}; B_{i}; f_{i}; C; A_{i}; m_{o,i}; C; D; h; D; B_{i}; n_{o,i}; A_{j}; B_{j}; f_{j}; A_{i}; A_{j}; m_{i,j}; A_{i}; B_{i}; B_{i}; B_{j}; n_{i,j}; C; A; m; D; B′; n; A; B′; f′; A_{j}; A; m_{j}; A_{j}; B_{j}; B_{j}; B′; n_{j}]

where all A_{i}’s and B_{i}’s are λ-presentable, all squares are pushouts, and (m_{i}: A_{i} [figure] A)_{I} and (n_{i}: B_{i} [figure] B′)_{I} are λ-directed colimit diagrams.

Proof. Let (m_{i,j}: A_{i} [figure]A_{j})_{I} be a λ-directed diagram with the A_{i}’s λ-presentable, and with colimit cocone (m_{i}: A_{i} [figure] A)_{I}. m must factorize m = m_{i}m_{o,i} for some i ∈ I, and then A is the colimit of (m_{j,k}: A_{j} [figure]A_{k})_{k≥j≥i}. For each j ≥ i, form the pushout (f_{j}, n_{o,j}) of (h, m_{o,j}), and let the n_{j}, (j ≥ i) and the n_{j,k} (j ≤ k) be the induced morphisms, as in the diagram above. Then all the B_{i}’s are λ-presentable, and the squares in the diagram are pushouts. The fact that (n_{j}: B_{j} [figure] B′)_{j≥i} is the colimit diagram of (n_{j,k}: B_{j}

[figure] B_{k})_{k≥j≥i} is a straighforward computation.

End of the proof of Theorem 2.3:

Given f: A [figure]B λ-presentable, there exist diagrams as in Lemmas 2.4 and 2.5. If srn = n (in the diagram of Lemma 2.4), it follows easily that r is an isomorphism, and we are done. Otherwise, srn must factorize through some n_{j}, srn = n_{j}n′. Let g: B_{j} [figure: E] be the coequalizer of n′ and n_{o,j}, and (g′: B′ [figure]F, n′_{j}: E [figure]F) be the pushout of (g, n_{j}), as in the diagram below. Then E is λ-presentable, and (g′f′, n′_{j}) is the pushout of (gf_{j}, m_{j}).

We have rn_{j}n_{o,j} = rn = rsrn = rn_{j}n′, so that there exists a unique q: E [figure] B such that qg = rn_{j}. r and q then induce a unique t: F [figure] B such that tn′_{j} = q and tg′ = r:

[figure: A_{j}; B_{j}; f_{j}; C; A_{j}; m_{o,j}; C; D; h; D; B_{j}; n_{o,j}; A; B′; f′; A_{j}; A; m_{j}; A_{j}; B_{j}; B_{j}; B′; n_{j}; D; B_{j}; n′; B′; F; g′; B_{j}; B′; B_{j}; E; g; E; F; n′_{j}; B′; B; s; F; B; t; B; B′; r; A; B; f; E; B; q]

To complete the proof, we need only to show that t is an isomorphism: the required pushout will then be (f)(m_{j}) = (q)(gf_{j}).

We have t(g′s) = rs = 1_{B}, and we need to show that (g′s)t = 1_{F}. The lower right hand square being a pushout, this amounts to show that

(1) g′stg′ = g′, and

(2) g′stn′_{j} = n′_{j}.

As for (1), we have g′stg′ = g′sr, and showing that g′sr = g′ amounts to show that (a) g′srf′ = g′f′ and (b) g′srn = g′n, because the (vertical) left rectangle nh = f′m is a pushout. Now, (a) is clear since srf′ = sf = f′; for (b), we have g′srn = g′n_{j}n′ = n′_{j}gn′ = n′_{j}gn_{o,j} = g′n_{j}n_{o,j} = g′n.

For the second equality (2), the top left square being a pushout and g being epi, it will be sufficient to show that (i) g′stn′_{j}gn_{o,j} = n′_{j}gn_{o,j} and (ii) g′stn′_{j}gf_{j} = n′_{j}gf_{j}. For (i), we have g′stn′_{j}gn_{o,j} = g′stg′n_{j}n_{o,j} = g′n_{j}n_{o,j} = n′_{j}gn_{o,j} (using (1)). Finally, (ii) follows from g′stn′_{j}gf_{j} = g′stg′f′m_{j} = g′f′m_{j} = n′_{j}gf_{j} (using (1) again).

Theorem 2.3 solves a problem left open in [H, 04].

Recall that, since finitely accessible categories have filtered colimits, finitely accessible categories with finite colimits are locally finitely presentable. In the following, we will often work with the slightly weaker assumption of the existence of pushouts. The next result gives the main reason:

2.6. Proposition. Let A be any object of a λ-accessible category C with pushouts. Then (A ↓ C) is λ-accessible.

Proof. From Lemma 2.4, one sees easily that there is essentially only a set of λpresentable objects in (A ↓ C). To obtain f: A [figure]B as the colimit of a λ-directed diagram of λ-presentable morphisms, one proceeds as in the proof of 2.4 (see [H, 04]): C^{2} being locally λ-presentable, consider any λ-directed diagram ((a_{ij}, b_{ij}): f_{i} [figure]f_{j})_{I} of λ-presentable objects of C^{2} with colimit ((a_{i}, b_{i}): f_{i} [figure]f)_{I}; let then (f_{i}^{∗}, a_{i}^{∗}) be the

pushout (in C) of (f_{i}, a_{i}) (i ∈ I), and f_{ij}^{∗}: f_{i}^{∗}  [figure]f_{j}^{∗} and f′_{i}: f_{i}^{∗}  [figure]f be the induced
morphisms. One can verify that colim_{I}(f_{ij}^{∗}: f_{i}^{∗}  [figure] f_{j}^{∗})_{I} = (f′_{i}: f_{i}^{∗}  [figure] f) in (A ↓ C).

2.7. Remarks.

(a) One can compare Proposition 2.6 to Corollary 2.44 (and its proof) in [AR, 94], which shows that if C is λ-accessible and A is α-presentable, then (A ↓ C) is β-accessible for any β sharply greater than λ and α.

(b) As opposed to what we erroneously claimed in [H_{1}, 98], we do not know if Proposition 2.6 holds in every λ-accessible category. As a consequence, all categories in [H_{1}, 98] should be assumed to have pushouts.

We now consider some generalizations of the concept of purity.

2.8. Definition. Let C be a λ-accessible category C with pushouts, K a full subcategory of C. Let f: A [figure] B be a morphism in C.

(a) f is a K-epi if for all pairs B [figure: g; h] K, K ∈ K and gf = hf imply g = h.

(b) ([AS, 04]) f is K_{λ}-pure if for every commutative square

[figure: A; B; f; C; A; m; C; D; h; D; B; n]

where h is a K-epi in C_{λ}, there exists d: D [figure] A such that dh = m.

(c) f is strongly K_{λ}-pure if for every factorization A [figure: h]D [figure: n]B of f, h is a split mono whenever it is a λ-presentable K-epi.

2.9. Remarks.

(a) If one replaces, in 2.8(b), the condition “h is a K-epi in C_{λ}” by “h is a λ-presentable K-epi”, the definition obtained is equivalent to the one in (c). This is completely analogous to the equivalence of strong monos and extremal monos in categories with pushouts (take K = C).

(b) For K empty, every morphism in C is K-epi, and the (strongly) K_{λ}-pure morphisms are just the (strongly) λ-pure morphisms of ([H_{1}, 98]) [AR, 94]. Recall from [H_{1}, 98] (but see Remark 2.7(b) above) that in the context of our definition, f: A [figure] B is strongly λ-pure if and only if it is the λ-directed colimit in (A ↓ C) of split monos. As a consequence, they are just the same than the λ-pure morphisms. For the context in which K-purity and strong K-purity are equivalent, see Proposition 2.11 below.

(c) In general λ-accessible categories, strong purity is not equivalent to purity (as defined in [AR, 94]): [AHT, 96] provides an example of an ω-pure morphism in a finitely accessible category, which is not a strong mono. From the description of the category, the morphism is easily seen to be finitely presentable. Hence it is not strongly ω-pure, otherwise it would be a split mono.

2.10. Examples.

(a) In the category of abelian groups, where λ-purity was originally defined, L. Fuchs ([Fu, 70]) calls a subgroup G of A λ-pure if G is a direct summand of any group B such that G ≤ B ≤ A and |B/G| < λ. Note first that the condition |B/G| < λ can be replaced by B/G being λ-generated: this is clear for λ > ω, and follows from standard results in abelian group theory for λ = ω. Also known are the facts that an abelian group G is a direct summand of its extension B if and only if the embedding is a split mono, and that λ-generated abelian groups are necessarily λ-presentable. Using 2.2(b), we see that Fuchs’ definition corresponds to our formulation of strong purity (i.e., 2.8(c) with K empty).

(b) Let C be the category StrΣ of all Σ-structures and Σ-homomorphisms, where Σ is a (multi-sorted) λ-ary signature. It is well-known that the λ-pure morphisms in C are the homomorphisms reflecting the λ-ary positive primitive (or pp-) formulas, i.e., the existentially quantified conjunctions of less than λ atomic formulas. One can compare this with the embeddings (reflecting the atomic formulas), and the λelementary embeddings (reflecting all the λ-ary formulas). More generally, using the usual syntactic translation of presentability in this context (see [AR, 94] or [H, 04]), as well as the syntactic characterization of epimorphisms in locally λ-presentable categories (see [H_{2}, 98]), one can prove that for a reflective subcategory K of C closed under λ-directed colimits, a morphism f: A [figure]B is K_{λ}-pure (or, equivalently, strongly K_{λ}-pure: see the next proposition) iff for every set ϕ of less than λ atomic formulas in Σ such that K ⊧ ∀x∃^{≤1}y∃z (∧ϕ(x, y, z )), and for every string a in A, the condition

B ⊧ ∃yz (∧ϕ[f(a), y, z ]) ⟹ A ⊧ ∃yz (∧ϕ[a, y, z ])

is satisfied. If K is empty, the condition K ⊧ ∀x∃^{≤1}y∃z (∧ϕ(x, y, z )) is void, and we obtain the above characterization of λ-purity. This actually applies if C is any variety of Σ-structures. For example, if C is the category of abelian groups and K is the subcategory of all groups with trivial 2torsion, then a subgroup A of B is (strongly) K_{ω}-pure iff every element of A which has a 2^{n}-th root in B for some positive integer n, must have one in A itself. This follows from the fact that K ⊧ ∀x∃^{≤1}y(2^{n} · y = x) for all n.

In 2.10(b), we stated in particular that for sufficiently nice subcategories K of varieties, the K_{λ}-pure morphisms are always strongly so. The following result is more precise:

2.11. Proposition. Let K be a full subcategory of a locally λ-presentable category closed under products and λ-directed colimits. Then the K_{λ}-pure morphisms and the strongly K_{λ}-pure morphisms coincide.

Proof. That a strongly K_{λ}-pure morphism is K_{λ}-pure is true for every subcategory K of any λ-accessible category with pushouts. This follows easily from the fact that the pushout of a λ-presentable K-epi is also a λ-presentable K-epi, and because morphisms in C_{λ} are λ-presentable.

To prove the reverse implication, we first show that under the conditions of the proposition, any λ-presentable K-epi is the pushout of a K-epi in C_{λ}.

Let f: A [figure] B be a λ-presentable K-epi. By Theorem 2.3, there exists a diagram as in the statement of Lemma 2.5, with f′ and B′ replaced by f and B. We show that one of the f_{i}’s must be a K-epi.

Suppose not. Then for each i ∈ I, there exist distinct morphisms u_{i}, v_{i}: B_{i} [figure]K_{i} with K_{i} ∈ K, such that u_{i}f_{i} = v_{i}f_{i}. We will use the categorical version of the standard model-theoretic construction of reduced product (with respect to a λ-complete filter: see [CK, 90] or [H, 04]). For each i ∈ I, let i^{∗} = {j ∈ I | j ≥ i}. Consider the set

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

The diagram (π_{i,j}: ∏_{k∈i^{∗}} K_{k} [figure] ∏_{k∈j^{∗}} K_{k})_{I} of the canonical projections (i ≤ j) between

the products is λ-directed, and we will denote by (π_{i}: ∏_{k∈i^{∗}} K_{k} [figure]∏_{F} K_{k})_{I} its colimit

diagram (∏_{F} K_{k} is just the reduced product with respect to the λ-complete filter F). For i ∈ I, the set of all u_{k}n_{i,k}: B_{i} [figure]B_{k} [figure]K_{k}, k ≥ i, induces a morphism u′_{i}: B_{i}

[figure]∏_{k∈i^{∗}} K_{k}, and similarly for v′_{i}, defined from the v_{k}n_{i,k}’s. Let t_{i}: A_{i} [figure] ∏_{k∈i^{∗}} K_{k} be

the morphism u′_{i}f_{i} = v′_{i}f_{i}. The t_{i}’s, u′_{i}’s and v′_{i}’s respectively induce morphisms t, u′, v′ between the colimits, as in the diagram:

[figure: A_{k}; B_{k}; f_{k}; A_{i}; A_{k}; m_{i,k}; A_{i}; B_{i}; f_{i}; B_{i}; B_{k}; n_{i,k}; A_{k}; A; m_{k}; B_{k}; B; n_{k}; A; B; f; B_{i}; K_{i}; u_{i}; B_{i}; K_{i}; v_{i}; B_{k}; K_{k}; u_{k}; B_{k}; K_{k}; v_{k}; A_{i}; ∏_{k∈i^{∗}} K_{k}; t_{i}; A_{i}; B_{i}; f_{i}; B_{i}; ∏_{k∈i^{∗}} K_{k}; B_{i}; ∏_{k∈i^{∗}} K_{k}; u′_{i}; B_{i}; ∏_{k∈i^{∗}} K_{k}; v′_{i}; A_{i}; A; m_{i}; B_{i}; B; n_{i}; ∏_{k∈i^{∗}} K_{k}; ∏_{F} K_{k}; π_{i}; B; ∏_{F} K_{k}; u′; B; ∏_{F} K_{k}; v′; A; B; f; ∏_{F} K_{k}; A; t; ∏_{F} K_{k}; B]

Clearly u′_{i} ≠ v′_{i}, since u_{i} ≠ v_{i}. But we actually have more: u′_{k}n_{i,k} ≠ v′_{k}n_{i,k} for all k ≥ i, because of the top left pushout square. Now u′f = t = v′f implies u′ = v′, since ∏_{F} K_{k} ∈ K and f is K-epi. Hence π_{i}u′_{i} = π_{i}v′_{i}. But B_{i} is λ-presentable, and ∏_{F} K_{k} = colim_{j≥i}( ∏_{k∈j^{∗}} K_{k}), so there must exist j ≥ i such that π_{i,j}u′_{i} = π_{i,j}v′_{i}. This means

u′_{j}n_{i,j} = v′_{j}n_{i,j}, a contradiction.

The proposition now follows easily, using again the fact that λ-presentable K-epis are preserved by pushouts.

The details of the proof show that the closure conditions in the statement of the proposition could actually be relaxed to the closure under λ-complete reduced products.

2.12. Facts. For convenience, we collect a few facts about purity, some of them already mentioned. References are given for most. The others are obvious or left as straightforward exercises. Let C be a λ-accessible category with pushouts, K any full subcategory of C. Then

(1) the following implications hold:

λ-pure ⇒ strongly K_{λ}-pure ⇒ K_{λ}-pure;

(2) if α ≥ λ, then (any sort of) α-purity implies (the same sort of) λ-purity;

(3) all three types of purity are closed under composition ([AR, 94] and [H, 04]);

(4) for any α, the (multiple) equalizer of any set of morphisms with codomain in K is strongly K_{α}-pure;

(5) λ-pure morphisms are regular monos ([AHT, 96]);

(6) the λ-pure morphisms with domain A are precisely the λ-directed colimits in (A ↓ C) of split monos ([H_{1}, 98]);

(7) λ-pure morphisms are stable under pushouts ([AHT, 96]);

(8) the class of all K_{λ}-pure morphisms in C is closed under λ-directed colimits in C^{2} (adapt the proof of [AR, 94], 2.30);

(9) for each object A in C, the class of all strongly K_{λ}-pure morphisms from A in C is closed under λ-directed colimits in (A ↓ C) (adapt the proof of [H, 04], 2.3(b)).

(10) the right-hand implication in (1) is an equivalence if C has and K is closed under products and λ-directed colimits (Proposition 2.11 above);

(11) all three types of purity are closed under products in C^{2} (use pushouts);

(12) if C has finite copowers, K_{λ}-pure morphisms are monos for every K ([AS, 04]).

Accordingly, in the right context, we will speak of (K_{λ}-) pure subobjects. Note finally that λ-pure subobjects are sometimes called λ-algebraically closed subobjects (for example in [Fa, 75] and [H, 04]).

3. Injectivity, orthogonality, reflectivity: earlier results

In this section, we briefly survey a body of known facts, in order to provide the context for our main results (in Section 4).

Recall that an object K in C is injective with respect to (respectively orthogonal to) a morphism f: A [figure] B if

C(f, K): C(B, K) [figure] C(A, K)

is surjective (respectively bijective). We write A ⊥ f to mean that A is orthogonal to f. For a class M of morphisms in C, Inj(M) (resp. M^{⊥}) denotes the class of all the objects which are injective with respect to (resp. orthogonal to) all the morphisms in M. A class of objects of the form Inj(M) (resp. M^{⊥}) is called an injectivity class (resp. an orthogonality class). Some of the following might be less familiar:

3.1. Definition. A class K of objects in C is a λ_{m}-injectivity class (resp. a λ-injectivity class) if K = Inj(M) for some class M of λ-presentable morphisms (resp. of morphisms in C_{λ}). λ_{m}-orthogonality classes and λ-orthogonality classes are defined similarly, replacing Inj(M) by M^{⊥}.

3.2. Related Characterizations. Our main Theorem (4.1) will complete the following array of known results: Let K be a class of objects in a locally λ-presentable category C. Then,

(1) ([RAB, 02]) K is a λ-injectivity class in C if and only if it is closed under (i) products, (ii) λ-pure subobjects, and (iii) λ-directed colimits.

(2) ([H_{1}, 98]) K is a λ_{m}-injectivity class in C if and only if it is closed under (i) products, and (ii) λ-pure subobjects.

(3) ([AS, 04]) K is a λ-orthogonality class in C if and only if it is closed under (i) products, (ii) K_{λ}-pure subobjects, and (iii) λ-directed colimits.

Note that in statement (3), “products” can be replaced by “limits”, since by 2.12 (4), closure under K_{λ}-pure subobjects and products implies closure under all limits.

3.3. Remarks.

(a) Lemma 3.6 in [H, 04] showed how a class M of λ_{m}-morphisms such that K = Inj(M), can be replaced by a class (and hence, a set) of morphisms in C_{λ} when K is closed under λ-directed colimits. This lemma was interpreted as a compactness-type result, and gave a different proof of the Rosický-Adámek-Borceux characterization (3.2(1)) above.

(b) One may feel that closure under K_{λ}-pure subobjects is not a very pleasant property, as the definition of K-purity depends on K globally. But this unpleasant feature might be unavoidable, at least in the finitary case, if one recalls that there is no possible uniform syntactic characterization of subcategories of varieties of finitary structures closed under limits and directed colimits. The latter means the following: given a finitary signature Σ, H. Volger showed in [V, 79] that every class of Σ-structures, closed in StrΣ under limits and directed colimits, is finitarily axiomatizable, but that no set Δ of (first-order, finitary) sentences exists such that a class of structures is closed under these operators if and only if it can be axiomatized by a subset of Δ. [V, 79] does provide a syntactic characterization for these classes, but its formulation involves, as expected, a “global” property of the theory.

(c) Note however that there is a simple uniform syntactic characterization of orthogonality classes in StrΣ. Those are the ones definable by classes of (L_{∞}(Σ)-) sentences of the form

∀x(∧ϕ(x) [figure] ∃!y(∧ψ(x, y))),

where ϕ and ψ are sets (of any cardinality) of atomic formulas. These subcategories correspond to the prevarieties of [AS, 04], except that here we do not allow Σ to be a proper class (in order for the ambient category to be locally presentable). If Σ is λ-ary, the λ_{m}-orthogonality classes and the λ-orthogonality classes are those among them for which the cardinality of ψ, respectively of ϕ and ψ, are smaller than λ. Only in the latter case the classes of sentences are actually sets (of L_{λ}(Σ)-sentences). More details can be found [AR, 94], [AS, 04] and [H, 04].

3.4. Orthogonality vs Reflectivity.

(a) One sees easily that for any subcategory of any category, we have

reflective ⇒ orthogonal ⇒ limit-closed.

Orthogonality classes have generated a fair amount of literature in the last 30 years or so, in particular concerning the conditions under which the arrows above can be reversed. In the context of locally presentable categories, note first that under the Weak Vopenka’s Principle (a large-cardinal axiom), limit-closed subcategories are automatically α-orthogonal for some α (and hence reflective). In particular, both arrows above are equivalences under this assumption. Note also that this Principle is actually equivalent to the implication on the right being reversible (see [AR, 94]).

(b) The so-called “Orthogonal Subcategory Problem” is about the left arrow in (a): characterize the classes M of morphisms such that (the corresponding full subcategory) M^{⊥} is reflective. In the seminal paper [FK, 72] of Freyd and Kelly, one of the main theorems implies that, in a locally presentable category C, if M is the union of a (small) set and of a class of epis from some proper factorization system in C, then M^{⊥} is reflective. In particular, λ-orthogonality classes are reflective.

(c) The characterizations in 3.2(3) and Theorem 4.1 are about the right arrow in (a): what closure conditions should be added to reverse this implication? First note that adding closure under α-directed colimits for some α implies reflectivity (see [AR, 94]), and hence orthogonality. However, from the result of Volger already mentioned in 3.3(b), there exist reflective subcategories of locally finitely presentable categories which are closed under directed colimits, and not ω-orthogonal (but note that the proof of the main result in [HR, 01] shows that they must be ω_{1}-orthogonal). The main result in [HR, 01] states that no such counterexample exists for higher cardinalities. The remaining problem of characterizing the ω-orthogonality classes was finally solved in [AS, 04] (see 3.2(3)). We can summarize all this as follows (for a subcategory K of a locally λ-presentable category):

ω-orthogonal ⇔ [limits, K_{ω}-pure subobjects, directed colimits]-closed

max(λ, ω_{1})-orthogonal ⇔ [limits, λ-directed colimits]-closed

Note that closure under limits and directed colimits alone does not imply closure under K_{ω}-pure subobjects, since it does not imply ω-orthogonality, a crucial fact proved by H. Volger from the same example in [V, 79] (p. 37) used to prove the result mentioned in 3.3(b). However, we did not find an example showing that closure under limits and K_{ω}-pure subobjects does not imply closure under directed colimits. (Closure under limits and strongly K_{ω}-pure subobjects does not imply closure under directed colimits: see 4.3(c) below).

4. Main results

4.1. Theorem. Let K be a class of objects in a locally λ-presentable category C. Then K is a λ_{m}-orthogonality class in C if and only if it is closed under

(i) products, and (ii) strongly K_{λ}-pure subobjects.

Proof. That the closure conditions are necessary is a straightforward exercise. To show that they are sufficient, we follow the same path than the one in [AS, 04] for the λ-orthogonality case. The proof will flow in a more natural way here, but we must show first that the conditions still imply that K is reflective. This is a consequence of the following lemma, which we will use again later:

4.2. Lemma. Let C be a λ-accessible category with pushouts, and K be a subcategory of C closed under α-pure subobjects, for some α ≥ λ. Then the inclusion functor of K in C satisfies the Solution Set Condition.

Proof. Let A ∈ C. Using Theorem 2.33 of [AR, 94], one can find γ large enough so that A is γ-presentable, C is γ-accessible, K is closed under γ-pure subobjects, and

any morphism f: A  [figure] B factorizes through a γ-pure morphism f′: A′  [figure] B with A′

γ-presentable (see [AR, 94]). Then the set

S_{A} = {f: A [figure] K | K ∈ K, K is γ-presentable in C}

is easily seen to be a solution set for A.

End of the proof of Theorem 4.1: From 2.12 (4), K is closed under limits, and hence is reflective in C, by the lemma.

Let M be the class of all λ-presentable K-epis f such that K ⊥ f for all K in K. We prove that for A ∈ M^{⊥}, the reflection η_{A}: A [figure] rA in K is strongly K_{λ}-pure. This will show that M^{⊥} ⊆ K, and hence that M^{⊥} = K.

Consider then any factorization A [figure: h] D [figure: n] rA of η_{A}, with h a λ-presentable K-epi. Then h ∈ M. Indeed, any g: A [figure]K ∈ K extends (uniquely) to some g′: rA [figure]K such that g′η_{A} = g, and g′n is then unique such that (g′n)h = g, since h is a K-epi. But then 1_{A} factorizes through h (since A ∈ M), i.e., h is a split mono.

4.3. Remarks.

(a) The Adámek-Sousa characterization of λ-orthogonality classes (3.2(3) above) can be obtained from Theorem 4.1. The same Lemma 3.6 of [H, 04] used in the injectivity case (see 3.3(a)) actually holds if all the morphisms in its statement are replaced by K-epis, with essentially the same proof. As a consequence, the replacement can be done in the same way if K = M^{⊥} (instead of Inj(M)). The proof of the lemma is rather technical, so this certainly does not provide a simpler proof of 3.2(3). However, it may help organizing a body of related results, and clarifying the role of each closure operator.

(b) Theorem 4.1 shows in particular that M^{⊥} is reflective for every class M of λpresentable morphisms. It would be interesting to see if the Freyd-Kelly theorem refered to in 3.4(b), or the related results in W. Tholen’s [T, 83], could be generalized as to include this case. A related problem is the following: among the factorization systems (E,M) determining the λ_{m}-orthogonality class M^{⊥} = K as a reflective subcategory, is there one for which E and M can be constructed in a natural way from the λ-presentable K-epis and the strongly K_{λ}-pure morphisms respectively?

(c) 3.3(c) provides an easy way to construct examples. In the category of groups, let K be the class of all groups in which any element which has a p-root for all odd primes p must have a unique square root. K is definable by the sentence

∀xx_{2}x_{3}...( ∧_{i>1} x = x_{i}^{p_{i}} [figure] ∃!y(x = y^{2})),

where p_{i} is the i-th prime number. By 3.3(c), K is an ω_{m}- (and ω_{1}-) orthogonality class, hence closed under strongly K_{ω}-pure subobjects and under K_{ω_{1}}-pure subobjects. However it is not closed under directed colimits (and hence not ω-orthogonal): the group of rational numbers with odd denominators in reduced form is not in K, but it is the direct limit of its subgroups A_{i} = {a/p_{2}^{k_{2}}p_{3}^{k_{3}}...p_{i}^{k_{i}} | a, k_{j} ∈ ℤ}, i ≥ 2, which are all in K. Hoping to exhibit a λ_{m}-orthogonality class which is not a small-orthogonality class (i.e., not α-orthogonal for any α), one could try writing down a proper class of sentences as in 3.3(c). However, by 3.4(a), one could not prove, without negating the Vopenka’s Principle, that this class is not equivalent to a set of such sentences.

If K and N are subcategories of C, Π(N) and K_{λ}^{s}(N) will denote respectively the class of all products and the class of all strongly K_{λ}-pure subobjects of the members of N.

4.4. Theorem. Let K be a subcategory of a locally λ-presentable category C. Then K_{λ}^{s}(Π(K)) is the λ_{m}-orthogonality hull of K.

Proof. Let N = K_{λ}^{s}(Π(K)). Using Theorem 4.1, we need to prove that:

(a) Π(N) ⊆ N,

(b) N_{λ}^{s}(N)) ⊆ N, and

(c) N ⊆ S for every λ_{m}-orthogonality class S containing K.

(a) This follows from the fact that the product of a set of strongly K_{λ}-pure morphisms is also strongly K_{λ}-pure (Facts 2.12(11)).

(b) We first show that a morphism is K-epi if and only if it is N -epi. That N -epis are K-epis is trivial from K ⊆ N. Let g: A [figure]D be a K-epi, and u, v: D [figure]N ∈ N be morphisms such that ug = vg. Then there exists a strongly K_{λ}-pure morphism h: N

[figure]∏_{i∈I} K_{i} for some K_{i}’s in K. If π_{i}: ∏_{i∈I} K_{i} [figure]K_{i} is the canonical projection, then

π_{i}hug = π_{i}hvg implies π_{i}hu = π_{i}hv for each i ∈ I (since g is a K-epi), which implies hu = hv, and hence u = v since h is mono, by 2.12 (12). Now, the fact that K-epi = N -epi implies that strongly K_{λ}-pure = strongly N_{λ}-pure. Let then A be in N_{λ}^{s}(N). There exist B ∈ N and a diagram

[figure: A; B; f; B; ∏_{i∈I} K_{i}; g]

with f strongly N_{λ}-pure, g strongly K_{λ}-pure, and K_{i} ∈ K for each i. Hence both morphisms are strongly K_{λ}-pure, so that by the Fact 2.12 (3), gf is also strongly K_{λ}-pure. We conclude that A ∈ K_{λ}^{s}(Π(K)) = N.

(c) If S is an λ_{m}-orthogonality class containing K, then it contains Π(K), and hence S_{λ}^{s}(Π(K)), by Theorem 4.1. But strongly K_{λ}-pure morphisms are obviously strongly S_{λ}- pure, and hence S contains K_{λ}^{s}(Π(K)) = N

4.5. Corollary. Let K be a subcategory of a locally λ-presentable category C. Then

(1) K is an orthogonality class if and only if (i) it is closed under products, and (ii) ∩_{α≥λ} K_{α}^{s}(K) = K.

(2) ∩_{α≥λ} K_{α}^{s}(Π(K)) is the orthogonality hull of K.

Proof. (1)(⇒) Let K = M^{⊥}. Orthogonality classes are closed under all limits, as mentioned above, and ∩_{α≥λ}K_{α}^{s}(K) ⊇ K is clear. We need to show that ∩_{α≥λ}K_{α}^{s}(K) ⊆ K.

Note that the conclusions of the Theorems 4.4 and 4.1 clearly hold for any α ≥ λ. Let A ∈ ∩_{α≥λ}K_{α}^{s}(K). Now, any given m: C [figure] D in M is α-presentable for α large enough. We show that A ⊥ m. There exists a strongly K_{α}-pure morphism f: A [figure] K for some K ∈ K, so that for any g: C [figure] A, fg factorizes through m, by a unique h: D [figure] K. We form the pushout (g′, m′) of g and m, and let n be the induced morphism:

[figure: A; P; m′; C; A; g; C; D; m; D; P; g′; A; K; f; P; K; n; D; K; h]

m′ being α-presentable, it has a left inverse r, and rg′m = rm′g = g. We only need to show that rg′ is unique to factorize g through m. We use the fact that f is mono (see 2.12 (12)): if hm = (rg′)m for some h: D [figure]A, then fhm = f(rg′)m, which implies fh = f(rg′) because m is K-epi. Hence h = rg′.

(⇐) K being closed under products, we have K = ∩_{α≥λ}K_{α}^{s}(K) = ∩_{α≥λ}K_{α}^{s}(Π(K)), i.e., by Theorem 4.4, an intersection of orthogonality classes, and hence an orthogonality class itself.

(2) This is proved from (1) in the same way that Theorem 4.4 is proved from Theorem 4.1

4.6. Remark. One can compare Corollary 4.5 to the characterization given in Exercise 2.o of [AR, 94]. Correctly restated (personal communication from the authors), it says that the orthogonality hull of a subcategory K of a locally presentable category C is ⋂_{α∈Ord} K(α), where K(α) is the smallest subcategory of C containing K and closed under limits and α-directed colimits in C. This is proved using mainly the facts that each K(α) is a small-orthogonality class, and that every orthogonality class is the intersection of small-orthogonality classes.

For the rest of the paper, we briefly consider cone-orthogonality in locally λ-presentable categories. An object K in C is injective with respect to the cone c = (c_{i}: C [figure]D_{i})_{I} if every g: C [figure]K factorizes through c (i.e., through at least one of the c_{i}’s). K is orthogonal to c if every g: C [figure] K factorizes through exactly one c_{i}, and furthermore this factorization is unique. A λ_{m}-cone (λ-cone) is a cone made of λ-presentable morphisms (resp. of morphisms in C_{λ}). If M is a class of (λ_{m}-, λ-) cones, Inj(M) and M^{⊥} will be called (λ_{m}-, λ-) cone-injectivity and cone-orthogonality classes respectively.

Example 3.1 of [RAB, 02] shows that classes closed under λ-pure subobjects and λdirected colimits are not necessarily λ-cone-injectivity classes. In contrast, Theorem 4 of [H_{1}, 98] proves that λ_{m}-cone-injectivity classes are precisely the ones closed under λ-pure subobjects (the characterization is actually proved in every λ-accessible category with pushouts).

Using the same technique than in [H_{1}, 98], one can show that for every classes K and N of objects in C, K_{λ}^{s}(N) = Inj(M) for a class M of cones made of λ-presentable K-epis. (One can use this to give another proof of Theorem 4.1, without using the fact that K is reflective.) As a consequence, a class K closed under strongly K_{λ}-pure subobjects comes close to be a λ_{m}-cone-orthogonality class: what is missing is that a given morphism may factorize through several branches of the same cone of M. The expected cure would be to add closure under pullbacks. Note that Lemma 4.2 implies that such a K, closed under strongly K_{λ}-pure subobjects and pullbacks, will be multireflective in C. However, we were not able to deduce (nor to disprove) that this was enough to insure that K is a λ_{m}-cone-orthogonality class.

As for λ-cone-orthogonality, a natural conjecture from the characterization of λorthogonality in [AS, 04] (3.2(3) above), would be that they are the classes closed under (i) pullbacks, (ii) K_{λ}-pure subobjects, and (iii) λ-directed colimits. However this is not true, the same Example 3.1 of [RAB, 02] being actually a counterexample. First, the category B described there is easily found to be closed (in the category StrΣ of all Σstructures) under pullbacks. Also, every morphism in StrΣ is a B-epi, for the simple reason that there are no two distinct morphisms from any given Σ-structure to any object in B. Hence purity and B-purity are the same. Finally, the fact that B is not a ω-cone-injectivity class (proved there) implies that it is not a ω-cone-orthogonality class either (see [AR, 94], 4.23).

Hence the following three problems remain:

4.7. Open Problems. In locally λ-presentable categories, characterize, by closure properties:

(1) the λ_{m}-cone-orthogonality classes,

(2) the λ-cone-orthogonality classes, and

(3) the cone-orthogonality classes.

References

[AHT, 96] J. Adámek, H. Hu, W. Tholen, On pure morphisms in accessible categories, J. Pure Appl. Algebra 107 (1996), 1-8.

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

[AS, 04] J. Adámek, L. Sousa, On reflective subcategories of varieties, J. Algebra 276 (2004), 685-705.

[C, 79] M. Coste, Localisation, spectra and sheaf representation, Appl. of Sheaves, Lect. Notes in Math. 753, Springer-Verlag, Berlin 212-238.

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

[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, Vol. I, Academic Press, New York 1970.

[FK, 72] P.J. Freyd, G.M. Kelly, Categories of continuous functors I, J. Pure Appl. Algebra 2 (1972), 169-192.

[H, 04] M. Hébert, Algebraically closed and existentially closed substructures in categorical context, Theory Appl. Categ. 12 (2004), 270-298.

[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.

[HR, 01] M. Hébert, J. Rosický, Uncountable orthogonality is a closure property, Bull. London Math. Soc. 33 (2001), 685-688.

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

[T, 83] W. Tholen, Factorizations, localizations, and the Orthogonal Subcategory Problem, Math. Nachr. (1983), 63-85.

[V, 79] H. Volger, Preservation theorems for limits of structures and global sections of sheaves, Math. Zeitschr. 166 (1979), 27-53.

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

Email: mhebert@aucegypt.edu

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