$R\text{-}\mathrm{Mod}$-enriched categories are left $\underline{R}$-module objects of $Cat(\mathbb{A}\mathrm{b})$ and $Cat(\mathbb{A}\mathrm{b})$-enriched functors

TL;DR

This paper demonstrates that $R$-module enriched categories can be studied within abelian-enriched category theory, establishing equivalences among various categories of enriched categories and functors for any commutative ring $R$.

Contribution

It proves the equivalence between $R$-module enriched categories, modules inside abelian-enriched categories, and enriched functor categories, extending the framework for studying $R$-enriched categories.

Findings

Establishes the equivalence of categories of $R$-enriched categories and modules.

Shows the category of $R$-enriched categories is equivalent to modules inside $Cat(Ab)$.

Demonstrates the category of enriched functors is equivalent to $R$-enriched categories.

Abstract

We establish the feasibility of investigating the theory of $R\text{-}\mathrm{Mod}$-enriched categories, for any commutative and unitary ring $R$, through the framework of $\mathbb{A}\mathrm{b}$-enriched category theory. In particular, we prove that the category of $R$-$\mathrm{Mod}$-enriched categories, $Cat(R$-$\mathrm{Mod})$, the category of $\underline{R}$-modules inside $Cat(\mathbb{A}\mathrm{b})$, $\mathrm{LMod}_{\underline{R}}(Cat(\mathbb{A}\mathrm{b}))$, and the category of $Cat(\mathbb{A}\mathrm{b})$-enriched functors, $Fun^{Cat(\mathbb{A}\mathrm{b})}(\underline{\underline{R}},Cat(\mathbb{A}\mathrm{b}))$ are equivalent.