Abstract
In the present work, we show how the study of categorical constructions 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.
Highlights
Category theory studies objects externally, through the relationships they establish with their environment
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
(3) G is a family of strong generators iff each γA is a strong epimorphism
Summary
Category theory studies objects externally, through the relationships they establish with their environment. 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. 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. We would like to offer some guidelines in this regard
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