In the present work we introduce the notion of {\em proper Moore-concrete subcategory} of a given concrete category $\mfC$ over a base category $\mfX$ and show that it agrees with that of reflective modification of $\mfC$. This allows us to associate with any object $C$ of $\mfC$ an object in $\mfB$ that we call the $\mfB$-{\em closure} of $C$, and with any morphism in $\mfC$ a unique morphism between the corresponding $\mfB$-closures that we call the $\mfB$-{\em closure extension}. Next, we provide some general properties of proper Moore-concrete subcategories which will be useful to deeply investigate {\em prekernels}, {\em precokernels} and {\em pretorsion theories} in proper Moore-concrete subcategories with respect to some given class of {\em 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 $\mfC$ in order to get informations on the corresponding $\mfB$-closure extensions and on the $\mfB$-closure of the various terms of the associated short sequences of $\mfB$-closure extensions in $\mfB$. Furthermore, we induce pretorsion theories on proper Moore-concrete subcategories starting from pretorsion theories on the ambient category $\mfC$ and, more in general, we study the interrelations between pretorsion theories on $\mfC$ and proper Moore-concrete subcategories. Finally, we provide a characterization of trivial morphisms and of prekernels of a functor-structured category when we choose $\mfZ$ to be the class of all projective objects. Next, restricting our attention to set-functor structured categories, we obtain a stable pointed category $\mfB/\mfR$ (and a corresponding subcategory $(\mfB/\mfR)^*$ as a quotient of $\mfB$ with respect to a suitable congruence relation $\mfR$ on the hom-sets and use the associated quotient functor to induce a correspondence between prekernels in the subcategory $\mfB^*$ without the objects with empty ground set and kernels in $(\mfB/\mfR)^*$, and between precokernels in $\mfB^*$ and weak cokernels in $(\mfB/\mfR)^*$.
Prekernels of topologically axiomatized subcategories of concrete categories
F. Infusino
2023-01-01
Abstract
In the present work we introduce the notion of {\em proper Moore-concrete subcategory} of a given concrete category $\mfC$ over a base category $\mfX$ and show that it agrees with that of reflective modification of $\mfC$. This allows us to associate with any object $C$ of $\mfC$ an object in $\mfB$ that we call the $\mfB$-{\em closure} of $C$, and with any morphism in $\mfC$ a unique morphism between the corresponding $\mfB$-closures that we call the $\mfB$-{\em closure extension}. Next, we provide some general properties of proper Moore-concrete subcategories which will be useful to deeply investigate {\em prekernels}, {\em precokernels} and {\em pretorsion theories} in proper Moore-concrete subcategories with respect to some given class of {\em 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 $\mfC$ in order to get informations on the corresponding $\mfB$-closure extensions and on the $\mfB$-closure of the various terms of the associated short sequences of $\mfB$-closure extensions in $\mfB$. Furthermore, we induce pretorsion theories on proper Moore-concrete subcategories starting from pretorsion theories on the ambient category $\mfC$ and, more in general, we study the interrelations between pretorsion theories on $\mfC$ and proper Moore-concrete subcategories. Finally, we provide a characterization of trivial morphisms and of prekernels of a functor-structured category when we choose $\mfZ$ to be the class of all projective objects. Next, restricting our attention to set-functor structured categories, we obtain a stable pointed category $\mfB/\mfR$ (and a corresponding subcategory $(\mfB/\mfR)^*$ as a quotient of $\mfB$ with respect to a suitable congruence relation $\mfR$ on the hom-sets and use the associated quotient functor to induce a correspondence between prekernels in the subcategory $\mfB^*$ without the objects with empty ground set and kernels in $(\mfB/\mfR)^*$, and between precokernels in $\mfB^*$ and weak cokernels in $(\mfB/\mfR)^*$.I documenti in IRIS sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione.
