Categories of modules, comodules and contramodules over representations

TL;DR

This paper develops a categorical framework relating modules, comodules, and contramodules over representations of small categories, unifying classical properties and exploring their structures akin to sheaves over schemes.

Contribution

It introduces a unified categorical approach to modules, comodules, and contramodules over algebraic representations, incorporating adjoint functors and classical properties within a Grothendieck category context.

Findings

Categorical framework unifies modules, comodules, and contramodules.

Classical properties of coalgebras naturally emerge in the framework.

Analysis of generators and Grothendieck categories as noncommutative space analogs.

Abstract

We study and relate categories of modules, comodules and contramodules over a representation of a small category taking values in (co)algebras, in a manner similar to modules over a ringed space. As a result, we obtain a categorical framework which incorporates all the adjoint functors between these categories in a natural manner. Various classical properties of coalgebras and their morphisms arise naturally within this theory. We also consider cartesian objects in each of these categories, which may be viewed as counterparts of quasi-coherent sheaves over a scheme. We study their categorical properties using cardinality arguments. Our focus is on generators for these categories and on Grothendieck categories, because the latter may be treated as replacements for noncommutative spaces.