Hom$_*$ commuting with filtered products

TL;DR

This paper investigates conditions under which the Hom functor commutes with filtered products in categories like R-modules, focusing on classes of objects characterized by this commuting property relative to a cardinal .

Contribution

It characterizes classes of objects in rich categories where Hom functors commute with -filtered products, extending understanding of such interactions in module categories.

Findings

Identifies conditions for Hom to commute with -filtered products.

Provides a characterization of classes _* of objects with this property.

Extends known results to broader classes of objects in module categories.

Abstract

In a sufficiently rich category, such as a category of R-modules, and a given infinite cardinal $\kappa$, we examine classes $\Cal H^\kappa_*$ of objects M, such that the following natural monomorphism is an isomorphism: $$\prod_{i\in I}}^{\kappa}\hbox{\rm Hom}\,(M, A_i)\cong \hbox{\rm Hom}\,(M, \prod_{i\in I}}^{\kappa} A_i),$$ for every family of objects $\{A_i, i\in I\}$ ($\prod^\kappa$ denotes the subproduct of all the vectors with support $<\kappa$).