Partial Actions on Categories

TL;DR

This paper extends the concept of partial actions from algebras to small categories, introduces partial skew categories, and explores conditions for globalizing such actions, broadening the theoretical framework in category theory.

Contribution

It generalizes partial actions to small categories and defines partial skew categories, providing foundational results and conditions for globalization.

Findings

Defined partial actions on small categories

Introduced partial skew categories

Established conditions for globalization

Abstract

In this paper we introduce the definition of partial action on small $k$-categories generalizing the similar well known notion of partial actions on algebras. The point of view of partial action which we use in this paper is the one which was introduced by Exel in his work on $C^*$-algebras, see \cite{E}. Various generalizations were done afterward, see \cite{CJ, DEP, DE, DFP}. Also we define the notion partial skew category. We prove similar results to the ones in \cite{CM}. Finally we show a result given conditions for a partial action to have a globalization.