Twisted actions of categorical groups

TL;DR

This paper develops a theory of twisted actions of categorical groups using semidirect products, introduces representations respecting vector spaces, and establishes an analog of Schur's lemma, supported by numerous examples.

Contribution

It introduces a novel framework for twisted actions of categorical groups and extends representation theory with a Schur's lemma analog.

Findings

Development of a semidirect product of categories

Introduction of representations respecting vector space structures

Establishment of an analog of Schur's lemma in this context

Abstract

We develop a theory of twisted actions of categorical groups using a notion of semidirect product of categories. We work through numerous examples to demonstrate the power of these notions. Turning to representations, which are actions that respect vector space structures, we establish an analog of Schur's lemma in this context. Keeping new terminology to a minumum, we concentrate on examples exploring the essential new notions introduced.