General Comodule-Contramodule Correspondence

TL;DR

This paper explores the deep algebraic and homotopical relationships between comodules and contramodules over a comonoid within symmetric closed monoidal categories, establishing enriched structures and adjunctions.

Contribution

It introduces a comprehensive framework connecting comodules and contramodules through enriched categories and adjunctions, advancing understanding in both algebraic and homotopical contexts.

Findings

Enriched categories of comodules and contramodules are constructed and related.

Enriched functors and adjunctions between these categories are established.

Homotopical analysis of comodules and contramodules over simplicial sets and spaces is performed.

Abstract

This paper is a fundamental study of comodules and contramodules over a comonoid in a symmetric closed monoidal category. We study both algebraic and homotopical aspects of them. Algebraically, we enrich the comodule and contramodule categories over the original category, construct enriched functors between them and enriched adjunctions between the functors. Homotopically, for simplicial sets and topological spaces, we investigate the categories of comodules and contramodules and the relations between them.