Categorical deﬁnitions and properties via generators

. In the present work, we show how the study of categorical con-structions does not have to be done with all the objects of the category, but we can restrict ourselves to work with families of generators. Thus, universal properties can be characterized through iterated families of generators, which leads us in particular to an alternative version of the adjoint functor theorem. Similarly, the properties of relations or subobjects algebra can be investigated by this method. We end with a result that relates various forms of compactness through representable functors of generators.


Introduction
Category theory studies objects externally, through the relationships they establish with their environment. That is why most of the definitions and categorical theorems are reduced to proving that, given a fixed object, for all the other objects of the category that satisfy certain hypotheses there is a morphism between them with certain properties. However, in this article, we show that this verification can be reduced to generator families. Let's review some background of this idea.
Generators were introduced in [11] to simplify the work with the injective objects in the context of the abelian categories. We recall that an object I is injective if for each monomorphism s : V → U and each morphism g : V → I, there exists a morphism h : U → I such that hs = g. In [11, p. 136] Grothendieck proves that an object I in an abelian category AB5 is injective if it satisfies the previous definition only when U is a generator. This theorem allowed him to prove his famous theorem about the existence of enough injectives in abelian categories. These results were generalized in [2] to more general categories, among which are the Grothendieck topos.
The same restriction idea can be applied to study the semantics associated with a topos E. In 1972 Mitchell discovered that each topos has an internal language, which is a generalization of the first-order classical language, where the variables have types associated with the objects of the topos. A formula ϕ with free variables x 1 , . . . , x n of respective types A 1 , . . . , A n can be interpreted in E either by using the subobjects lattice structure (or the presence of a classifier Ω) as a subobject |[ϕ]| → A 1 ×. . .×A n (or as a morphism ϕ : A 1 ×. . .×A n → Ω). There are two ways to define the validity of the formula ϕ. In the first place, we can say that ϕ is valid, written |= ϕ, when |[ϕ]| = A 1 × . . . × A n (in other words, when ϕ factors through : 1 → Ω). Secondly, given X ∈ ob(E) and a family of morphisms a i : X → A i (i = 1, . . . , n), we say that (a 1 , . . . , a n ) satisfies or forces ϕ (denoted X ϕ(a 1 , . . . , a n )), if im(a 1 , . . . , a n ) ≤ |[ϕ]|.
The relation between both semantics is known since the origins of categorical logic [17]: |= ϕ iff for every X ∈ ob(E) and every a i : X → A i (i = 1, . . . , n), X ϕ(a 1 , . . . , a n ). However, it is proved in [4, Volume 3, section 6.6] that the previous result continues to hold if we consider only the case in which X belongs to a family of generators G.
Finally, generators have been used to characterize the properties of the lattice of subobjects. If C is a category and A is a certain subclass of Heyting algebras, we say that C is A-Heyting if for every object B in C, Sub(B) ∈ A.
It was proved in [5,Lemma 6.3] that a Grothendieck topos E is bi-Heyting if and only if for every G in a family of generators G, Sub(G) is bi-Heyting. In section 4 we generalize this result.
Despite this background, this method has not been exploited to its full potential. For example, the important case of universal properties has not been investigated. In this paper, we would like to offer some guidelines in this regard.

Generators
Over the years, different variants of the original generator concept have been presented. In the following definition, we mention the most used ones.
Definition 2.1. Let G be a family of objects in a category C. We say that G is a family of (1) generators if for each pair of different morphisms a, b : A ⇒ B in C there is a G ∈ G and a morphism t : G → A such that at = bt [4]; (2) extremal generators if 1. holds and, furthermore, for all proper subobject s : U → A there is a G ∈ G and a morphism t : G → A which don't factor through s [7]; (3) strong generators if 1. holds and, furthermore, for any morphism f : A → B and any subobject s : U → B, if for all t : G → A, with G ∈ G, f t factors through s then f factors through s [19]; (4) iterated generators (colimit-generator in [18]) if every object A in C is the colimit of (t i : N i → A, δ ij : N i → N j ), where each N i is, in turn, the colimit of a family (n l i : The following proposition, known in the literature, justifies the nomenclature used in 2.1.
Proposition 2.2. Let C be a category with coproducts and G a family of objects of C. For every object A ∈ C, we define (1) G is a family of generators iff each γ A is an epimorphism.
(2) G is a family of extremal generators iff each γ A is an extremal epimorphism.
(3) G is a family of strong generators iff each γ A is a strong epimorphism.

Proposition 2.3. [22]
Every family of iterated generators is strong; every family of strong generators is extremal.
None of the converse implications holds in general (the singleton is an extremal generator in the category of topological spaces that is not strong [4,Example 4.5.17.f]; the generator given in [6, 4.3] is strong but not is iterated). However, Proposition 2.4. [22] Assume that C has coproducts and every epimorphism is regular. Then every family of generators is iterated.
Example 2.5. (1) In every category C, ob(C) is a family of iterated generators in C. Therefore, the categorical definitions given in terms of "all objects" of C constitute a particular case of those given in terms of "generator families." (2) In the category of abelian groups, Z is an iterated generator. In general, in every category of modules on a ring A, A is an iterated generator. This is a consequence of the fact that every module supports a free presentation.
(3) In the category of functors, the representable functors constitute a family of iterated generators [4, Volume 1, Teorema 2.15.6]. That a Grothendieck topos has a family of (iterated) generators is one of the conditions of Giraud's theorem.
(4) In the category Sh(L) of sheaves on a locale L, the classifier Ω is a cogenerator [3]; by 2.4 this cogenerator is iterated.
(5) The singleton is an iterated generator in the category of compact Hausdorff spaces (proposition 2.4).

Universal properties
Universal properties are at the heart of category theory since many of the most important categorical constructs are defined in terms of them. The following is its usual definition.
In short, the universal properties are of the form ". . . for every pair (A, c) . . . there is a unique morphism . . . ". Therefore, there are basically two ways to soften this notion: we can remove the requirement of uniqueness in the definition 3.1, obtaining the so-called weak universal properties [10]; or we can restrict to work with only a certain type of couples (A, c). This second option will be the one we will take in this paper, where we will consider the case in which A belongs to a family of generators.
by the generator definition and the uniqueness of the G-universal.
To see the existence of such d, we consider a family (a i : N i → A, δ ij : N i → N j ) whose colimit is A, and families (n r i : ; applying twice the notion of colimit and that F preserves epimorphic families we can build d.
In the rest of this section, we will analyze the most important examples of G-universal constructions.

Adjoint functors
A functor F : A → B has a right adjoint if for each B ∈ B, there is a universal morphism from F to B [13, p. 81]. Therefore, the above results apply directly in this case.
Proposition 3.5. Consider a category A with a family of iterated generators G and a functor F : A → B. The following conditions are equivalent (1) F has a right adjoint functor.
(2) The following conditions hold: (a) Each object of B admits a G-coreflection along F ; Proof. If F has a right adjoint functor, then it satisfies the condition (a) by definition; that it must also satisfy the condition (b) is known in the literature [14,Lemma 2,p. 395]. For the other implication, it is enough to apply the proposition 3.3.
It may be interesting to compare the previous result with the dual of the adjoint functor's theorem. Both start from weakening the concept "all object of A has a coreflection" (the latter weakening the concept of coreflection, the former the class of objects that have coreflection), to later restore it by conjugating some kind of glue (cocompleteness, iterated generators) and of continuity of F (preservation of colimits, of epimorphic families). It is also worth noting that the proposition 3.5 is not a consequence of this theorem, as the following examples show.
Example 3.6. (1) Let k be a finite field, F inSet the category of finite sets and F inV ect k the category of finite-dimensional vector spaces over k.
The free functor F : F inSet → F inV ect k is defined on objects by taking F (X) to be a vector space with basis X. Since F inSet is not cocomplete, we can not use the adjoint functor theorem for proving that F has a right adjoint. However, the proposition 3.5 can be used for this purpose (the singleton is an iterated generator in F inSet and F preserves epimorphic families).
). The problem is that when f is arbitrary, the previous definition is not always a sheaf. However, this can be guaranteed by requiring, for example, that the morphism f * preserves arbitrary joins. If this is the case, the rest of the demonstration is relatively simple. Since that for each u : We then have that the following diagram commutes where c is the only natural transformation given by Yoneda lemma on It remains to prove that D f preserves epimorphic families. Let {t i : H i → A} i∈I be an epimorphic family in Sh(L), C a sheaf in Sh(M), x, y : in the category of sets. Thus, x u = y u for each u, because {t i } is epimorphic (note that this argument is simpler than to prove that D f preserves colimits and to apply the dual of the adjoint functor theorem).

Limits
Let C, D be two categories. If D is small, we denote the category of functors from D to C by C D . We define the diagonal functor : C → C D by: • for every c ∈ ob(C), c is the constant functor (the functor which sends each i ∈ ob(D) to c and each f ∈ mor(D) to the identity id c : c → c).
• for every f : c → c , f is the natural transformation which has the value f at each i ∈ ob(J ).
Given a functor F : D → C, the limit of F is the universal morphism from to F [13, Section III.4]. Thus, in this case, we can express the G-universality in the following way. Proof. Suppose that {e k : u k → b} k∈K is an epimorphic family in C and that s, t : b → H are natural transformations such that s (e k ) = t (e k ) for all k ∈ K. Given d ∈ D, we have for all k ∈ K and hence s d = t d . Proof. The definition of limit implies the existence of a unique morphism l : L G → L such that q D l = p D for each object D of D. Let a, b : C → L G be two different morphisms; then there exists G ∈ G and a morphism m : G → C such that am = bm. Since L G is a G-limit, there exists D such that p D am = p D bm and so q D lam = q D lbm. In particular, la = lb, which proves that l is a monomorphism.
Now, if L G is a proper subobject of L, since G is a family of extremal generators, there exists a G in G and a morphism t : G → L such that t does not factor through l. Given the family of morphisms {q D t : G → F (D)}, the notion of G-limit implies the existence of e : G → L G such that p D e = q D t for all D.
Since L is a limit, le = t, which contradicts that t does not factor through l.
Unlike 3.10, in the previous proposition, we need to assume the existence of the limit of F . Let's see now how by demanding a little more of the family G, each G-equalizer is an equalizer.  For the case of the product, it seems that it is necessary to assume that the family G is iterated. We present direct proof of this fact.
Proposition 3.13. Let C be a category with a family of iterated generators G.
Proof. Analoguous to that of 3.3.
Note that the propositions 3.12, 3.13 provide other demonstration of 3.10, because every limit can be obtained from products and equalizers.
Another way to express the previous results is through the good behavior of representable functors originating from G with respect to limits. The following definition is well-known in category theory.
Definition 3.14. [4, Volume 1, Definition 2.9.6] A family of functors (G k : C → D) k∈K collectively reflects limits when, given a F : I → C and a cone (S → F i) i∈ob(I) on F if for each k, G k (S) → G k (F i) is the limit of G k F , then (S → F i) i∈ob(I) is the limit of F . When the family is reduced to a single functor G : C → D we say that G reflects limits. (2) If G is a family of iterated generators then the functors C(G, ) : C → Set, with G ∈ G, collectively reflect limits.
Proof. If (S → F i) i∈ob(I) is a cone in C then for each G ∈ G we have in the category of sets a diagram of the form where L C(G, )•F is the limit of C(G, ) • F , p i are the usual projections in sets and (s i ) i∈obI is the induced morphism, which takes the form (si ) i∈obI The hypothesis, that (s i ) i∈obI is an isomorphism for each G ∈ G, means that (S → F i) i∈ob(I) is a G-limit of F . Applying 3.11 and 3.10, we get the result.
Example 3.16. Let {F k } k∈K be a family of sheaves in Sh(L). We define for each u ∈ L shown from the sheaf definition that C ∈ ob(Sh(L)). We prove only that C is the coproduct of {F k } k∈K . Given i ∈ K, u ∈ L, we define the i-th canonical To prove that the previous diagram is a coproduct, by 2.5 (4) and 3.10, it suffices to consider families of natural transformations of the form {s k : F k → Ω}. We which proves that o is a natural transformation. By definition, for all k ∈ K, oI k = s k . To prove the uniqueness of o, consider another transformation o : C → Ω such that o I k = s k for every k ∈ K. By the naturality of o , we have

Relations
There are several ways to define relations in a category C, equivalent to each other provided that C has good exactness properties (see [4, Volume 2, section 2.5] for a detailed discussion). Here, we will use the most general of them all.
Definition 3.17. [4, Volume 2, pp. 101, 102] A relation (R, r 1 , r 2 ) on an object X is an object R with a monomorphic pair r 1 , r 2 : R ⇒ X (i.e. for any a, b : A ⇒ R, a = b iff r 1 a = r 1 b and r 2 a = r 2 b).
For every A ∈ Ob(C), we define a relation (in the usual sense) generated by (R, r 1 , r 2 ) on C(A, X) as: Since that the R A are usual relations, we can say that (R, r 1 , r 2 ) has a certain property if all R A have it. So, we say that (R, r 1 , r 2 ) is reflexive (respectively, symmetric, antisymmetric, transitive, . . . ) if for every object A in C, the relation R A is reflexive (respectively, symmetric, antisymmetric, transitive, . . . ).
We will show that we can restrict the verification of the properties of a relation of all the objects in the category to work with only families of generators. The idea is that these can be seen as universal properties.
Proposition 3.18. Let C be a category with a family of iterated generators G and (R, r 1 , r 2 ) a relation on X. If for all G ∈ G, R G is reflexive (respectively, symmetric, transitive) then (R, r 1 , r 2 ) is reflexive (respectively, symmetric, transitive).

Proof. (1) Reflexivity.
We must prove that for each A ∈ ob(C) and each f : A → X there exists b : A → R such that r 1 b = f, r 2 b = f . Note that this b is unique because r 1 , r 2 is monomorphic. As this is valid by hypothesis when A ∈ G, the proposition 3.3 implies that (R, r 1 , r 2 ) is reflexive.
(2) Symmetry. This can be established as the following universal property: for every A ∈ ob(C) and every pair of morphisms f 1 , f 2 : A ⇒ X such that (f 1 , f 2 ) ∈ R A , there exists a unique b : A → R such that r 1 b = f 2 , r 2 b = f 1 .
(3) Transitivity. The universal property, in this case, is: for every A ∈ ob(C) and every pair of morphisms f 1 , f 2 : A ⇒ X such that there exists f 3 : The case of antisymmetry is much simpler since the Proposition 3.3 does not need to be applied nor the hypothesis that G is iterated. Proposition 3.19. Let C be a category with a family of generators G and (R, r 1 , r 2 ) a relation on X. If for each G ∈ G, R G is antisymmetric then (R, r 1 , r 2 ) is antisymmetric.
Given an arbitrary generator G and a arbitrary morphism g : G → A, the equations r 1 ag = f 1 g, r 2 ag = f 2 g, r 1 bg = f 2 g, r 2 bg = f 1 g imply that (f 1 g, f 2 g), (f 2 g, f 1 g) ∈ R G and thus f 1 g = f 2 g. Since G, g are arbitrary, the generator definition implies that f 1 = f 2 .
For an application of this method to a concrete example see [1, Proposition 2.3.5].

Subobjects lattice
We will show that to prove that C is A-Heyting, it is enough to see that for all G in a family of generators G, Sub(G) ∈ A. This procedure is a generalization of [5,Section 6].
Each morphism f : Y → X in C induces the inverse image function f * : Sub(X) → Sub(Y ). The least that we demand to the operations in Sub( ) is that they respect this function.
Definition 4.1. [16] Let T X : I Sub(X) → Sub(X) be a family of I-ary operations on Sub(X), for each X in a category C. We say that {T X } X∈C is natural if for every f : Y → X and every {S i } I ⊆ Sub(X), the function Proposition 4.2. Let C be a regular category with disjoint coproducts; A be a subclass of Heyting algebras closed under isomorphisms, subalgebras and arbitrary products; and G be a family of generators in C. Suppose that all language operations of A are natural in C. If Sub(G) ∈ A for each G ∈ G then Sub(X) ∈ A for each X ∈ C.
f Dom(f ) X γ and, by the uniqueness of the epi-mono factorization, we conclude that A = B.
On the other hand, we have Sub( f Dom(f )) = f Sub(Dom(f )). Indeed, if we denote with i f : Dom(f ) → f Dom(f ) the injections of the coproduct, we can define Let's see that α and β are inverse functions. First, given any g ∈ G∈G C(G, X) we have Dom(g) ∩ , because the coproducts are disjoint. So, the diagram Let X be an object of a category C such that Sub(X) is a Heyting algebra and T X : I Sub(X) → Sub(X) is a natural operation along monomorphism. Then T X is a compatible operation on Sub(X).

Compactness
We say that a family of generators G P is projective if each G ∈ G P is projective, i.e. when for any morphism f : G → B and any epimorphism q : A B, f factors through q by some morphism G → A. The representable functors indexed by families of projective generators allow us to capture forms of compactness in some categories.
Proposition 5.1. Let C be a category with a family of projective generators G P and F : I → C a functor with limit {l i : L → F i} i∈I . Given a cone {s i : S → F i} i∈I , we consider the following statements (1) the canonical morphism s : S → L is epimorphic; (2) for any G ∈ G P and for any choice of Then 1. always implies 2., and 2. implies 1. in the case where I is cofiltered and the s i epimorphic.
Note that in statement 1. objects live in the category C while in 2. they all live in the category of sets. In effect, we have that in the diagram Proof. 1.⇒ 2. Suppose that every finite intersection of is not empty. Given any F g : F j → F k, since (s j ) −1 (x j ) ∩ (s k ) −1 (x k ) = ∅ then there exists y : G → S such that s j y = x j and s k y = x k , and thus F gx j = F gs j y = s k y = x k . Since representable functors preserve limits and by the form that limits have in the category of sets, we conclude that (x i ) i∈ob(I) ∈ C(G, L). By hypothesis s : S → L is epimorphic and G is projective so that s : C(G, S) → C(G, L) is also an epimorphism. Therefore, there exists x : G → S such that sx = (x i ) i∈I and by construction x ∈ I (s i ) −1 (x i ).
2.⇒ 1. Suppose by contradiction that s : S → L is not epimorphic. Then there is some g : G → L that can not be factored through s. We consider l i g ∈ C(G, F i). Since G is projective and the s i are epimorphic, we have that the s i are also epimorphic and thus (s i ) −1 (p i g) = ∅. Let J ⊂ I be any finite subset. By cofiltered hypothesis, there exists F k with g kj : k → j for all j ∈ J. If y ∈ (s k ) −1 (l k g) then s j y = F g kj s k y = F g kj l k g = l j g and thus y ∈ J (s j ) −1 (l j g). By 2. there exists x ∈ I (s i ) −1 (x i ). Then sx = (s )x = (l i g) i∈ob(I) = g, seen as elements of C(G, L), which contradicts that g is not factored through s. This proposition is known in the literature [21, p. 243] and the modules that satisfy these conditions are called linearly compact. They coincide with those that satisfy the AB5 * axiom of [11].
(3) Since in the category of sets the functor Set({ * }, ) does not generally satisfy the finite intersection property, the proposition 5.1 also helps to understand why there exists functors F : I → Set, with • I cofiltered, • F (i) = ∅, for all i ∈ ob(I), • F (g) surjective, for all g ∈ mor(I), such that lim F = ∅ (for some examples of this type of functors, see [12], [20]).