Actegories for the Working Amthematician

TL;DR

This paper provides a comprehensive overview of actegories, including definitions, structures, and classification results, aiming to fill a gap in the literature and support applications in applied category theory.

Contribution

It offers new definitions for monoidal, braided, and symmetric monoidal actegories and presents Cayley-like classification theorems for these structures.

Findings

Explicit definitions of tensor product and hom-tensor adjunction for actegories

Introduction of new definitions for monoidal and braided actegories

Three classification results for actegories and biactegories

Abstract

Actions of monoidal categories on categories, also known as actegories, have been familiar to category theorists for a long time, and yet a comprehensive overview of this topic seems to be missing from the literature. Recently, actegories have been increasingly employed in applied category theory, thereby encouraging an effort to fill this gap according to the new needs of these applications. This work started as an investigation of the notion of monoidal actegory, a compatible pair of monoidal and actegorical structures, and ended up including a sizable reference on the elementary theory of actegories. We cover basic definitions and results on actegories and biactegories, spelling out explicitly many folkloric definitions, including their tensor product and their hom-tensor adjunction. We give new definitions of actegories with monoidal, braided monoidal and symmetric monoidal structure. In the last section, we provide three Cayley-like classification results for these structures.