Volume 47_1_(2026)1_28

R-Mod-enriched categories are left R-module objects of Cat(Ab) and Cat(Ab)-enriched functors   

Matteo Doni

Abstract. We establish the feasibility of investigating the theory of R-Mod-enriched categories for any unitary ring R, through the framework of Ab-enriched category theory. We prove that the category of R-Mod-enriched categories, denoted by Cat(R-Mod), and the category of R-modules inside the category of Ab-enriched categories, Cat(Ab), denoted by LModR(Cat(Ab)), are equivalent. Here, R denoted the Ab-enriched category with only one object {*}, whose only non-trivial hom-Ab-object is Hom(*, *) = R, and whose composition is given by the multiplication in R. Moreover, we prove that the latter category LModR(Cat(Ab)) is equivalent to the category of Cat(Ab)-enriched functors with target Cat(Ab) and source the Cat(Ab)-enriched category R with only one object {*} and whose only non-trivial hom-Cat(Ab)-object is Hom(*, *) = R, denoted by FunCat(Ab)(R,Cat(Ab)). 

Rend. Mat. Appl. (7) 47 (2026) 1-28;  pdf