$C^{\ast }-$crossed product of groupoid actions on categories

Abstract

Suppose that $G$ is a groupoid acting on a small category $H$ in the sense of \cite[Definition 4]{NOT} and $H{\times }_{\hat{I}}\pm G$ is the resulting semi-direct product category (as in \cite[Proposition 8]{NOT}). We show that there exists a subcategory $H_r\subseteq H$ satisfying some nice property called ``regularity'' such that $H_r{\times }_{\hat{I}}\pm G=H{\times }_{\hat{I}}\pm G$. Moreover, we show that there exists a so-called ``quasi action'' (see Definition \ref{quasi}) ${\hat{I}}^2$ of $G$ on $C^{\ast }(H_r)$ (where $C^{\ast }(H_r)$ is the semigroupoid $C^{\ast }$-algebra as defined in \cite{EXE}) such that $C^{\ast }(H_r{\times }_{\hat{I}}\pm G)=C^{\ast }(H_r){\times }_{\hat{I}}^{2}G$ (where the crossed product for ${\hat{I}}^2$ is as defined in Definition \ref{cross}).

14 pages