Comodules and contramodules over coalgebras associated with locally finite categories

TL;DR

This paper explores how to associate coalgebras to small locally finite categories and studies the relationships between comodules, contramodules, and module categories, providing explicit descriptions and functor properties.

Contribution

It introduces a method to attach coalgebras to locally finite categories and characterizes comodules and contramodules as subcategories of module categories, with a focus on functor full-and-faithfulness.

Findings

Explicit description of comodules and contramodules as subcategories of module categories

Conditions for full-and-faithfulness of the contramodule forgetful functor

Construction of coalgebras from locally finite categories

Abstract

We explain how to attach a coalgebra $\mathcal C$ over a field $k$ to a small $k$-linear category $\mathsf E$ satisfying suitable finiteness conditions. In this context, we study full-and-faithfulness of the contramodule forgetful functor, and describe explicitly the categories of locally finite left $\mathcal C$-comodules and left $\mathcal C$-contramodules as certain full subcategories of the category of left $\mathsf E$-modules.