Smooth and rough modules over self-induced algebras

TL;DR

This paper introduces smoothening and roughening functors for self-induced algebras in monoidal categories, generalizing previous work on group representations and exploring their adjoint relationships.

Contribution

It defines new functors for modules over self-induced algebras and studies their adjoint pairs, extending existing theories to a broader categorical context.

Findings

Defined smoothening and roughening functors for self-induced algebras

Established adjoint relationships between categories of smooth and rough modules

Generalized previous constructions from group representation theory

Abstract

A non-unital algebra in a closed monoidal category is called self-induced if the multiplication induces an isomorphism between A\otimes_A A and A. For such an algebra, we define smoothening and roughening functors that retract the category of modules onto two equivalent subcategories of smooth and rough modules, respectively. These functors generalise previous constructions for group representations on bornological vector spaces. We also study the pairs of adjoint functors between categories of smooth and rough modules that are induced by bimodules and Morita equivalence.