On the tensor product of modules over skew monoidal actegories

TL;DR

This paper investigates the structure of modules over skew monoidal categories, establishing conditions for monoidality and introducing the concept of self-cocomplete subcategories to understand module categories.

Contribution

It provides a detailed analysis of skew monoidal module categories, constructs a skew monoidal forgetful functor, and introduces self-cocomplete subcategories for understanding monoidality.

Findings

Conditions for the forgetful functor to be strong monoidal

Criteria for the module category to be monoidal

Introduction of self-cocomplete subcategories

Abstract

This paper is about skew monoidal tensored V-categories (= skew monoidal hommed V-actegories) and their categories of modules. A module over <M,*,R> is an algebra for the monad T = R * _ on M. We study in detail the skew monoidal structure of M^T and construct a skew monoidal forgetful functor from M^T to the category of E-objects in M where E=M(R,R) is the endomorphism monoid of the unit object R. Then we give conditions for the forgetful functor to be strong monoidal and for the category M^T of modules to be monoidal. In formulating these conditions a notion of `self-cocomplete' subcategories of presheaves appears to be useful which provides also some insight into the problem of monoidality of the skew monoidal structures found by Altenkirch, Chapman and Uustalu on functor categories [C,M].