Skew monoidal structures on actegories

TL;DR

This paper introduces a method to construct skew monoidal structures from strong actions on actegories, unifying various examples and exploring conditions for braidings and closedness.

Contribution

It provides a new construction framework for skew monoidal structures from strong actions and analyzes how braidings and closedness are transferred.

Findings

Construction of skew monoidal structures from strong actions

Conditions under which braidings induce braidings on actegories

Criteria for closedness of the resulting skew monoidal structures

Abstract

We present a construction of skew monoidal structures from strong actions. We prove that the existence of a certain adjoint allows one to equip the actegory with a skew monoidal structure and that this adjunction becomes monoidal. This construction provides a unifying framework for the description of several examples of skew monoidal categories. We also demonstrate how braidings on the original monoidal category of a given action induce braidings on the resulting skew monoidal structure on the actegory and describe sufficient conditions for closedness of the resulting skew monoidal structure.