Hom$_*$ commuting with filtered products

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 $<\kappa$).