# On the tensor product of modules over skew monoidal actegories

Source: [On the tensor product of modules over skew monoidal actegories](<https://www.sota2.com/research/paper/68fa531b1e209131ce98073a>)

Collection index: [llms.txt](<https://www.sota2.com/research/llms.txt>)

Data updated: 2025-10-25T12:16:06.364Z

Authors: K. Szlachanyi

Published: 2015-12-04T09:46:18.000Z

## 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 \[C,M\].

## Recorded benchmark results

A research result applies to its stated dataset, split, metric and evaluation configuration. Compare like-for-like settings, preserve the metric's unit and direction, and distinguish paper-reported results from independent reproduction. The best result recorded here is not a claim of exhaustive, current global coverage.

## Original sources and implementation

[Original paper](<https://arxiv.org/abs/1512.01353>)

[PDF](<https://arxiv.org/pdf/1512.01353.pdf>)