Prekernels of topologically axiomatized subcategories of concrete categories

Abstract

In the present work we introduce the notion of proper Moore-concrete subcategory of a given concrete category C over a base category X and show that it agrees with that of reflective modification of C. This allows us to associate with any object C of C an object in B that we call the B-closure of C, and with any morphism in C a unique morphism between the corresponding B-closures that we call the B-closure extension.Next, we provide some general properties of proper Moore-concrete subcategories which will be useful to deeply investigate prekernels, precokernels and pretorsion theories in proper Moore-concrete subcategories with respect to some given class of trivial objects. More in detail, we analyze some different conditions to be put on short sequences of morphisms or on short pre-exact sequences in C in order to get informations on the corresponding B-closure extensions and on the B-closure of the various terms of the associated short sequences of B-closure extensions in B. Furthermore, we induce pretorsion theories on proper Moore-concrete subcategories starting from pretorsion theories on the ambient category C and, more in general, we study the interrelations between pretorsion theories on C and proper Moore-concrete subcategories.Finally, we provide a characterization of trivial morphisms and of prekernels of a functor-structured category when we choose Z to be the class of all projective objects. Next, restricting our attention to set-functor structured categories, we obtain a stable pointed category B/R (and a corresponding subcategory (B/R)⁎ as a quotient of B with respect to a suitable congruence relation R on the hom-sets and use the associated quotient functor to induce a correspondence between prekernels in the subcategory B⁎ without the objects with empty ground set and kernels in (B/R)⁎, and between precokernels in B⁎ and weak cokernels in (B/R)⁎.