Partial Actions on Categories

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.