$C^*-$crossed product of groupoid actions on categories

Suppose that $G$ is a groupoid acting on a small category $H$ in the sense of \cite[Definition 4]{NOT} and $H\times_\alpha G$ 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 $H_r \times_\alpha G = H\times_\alpha G$. Moreover, we show that there exists a so-called quasi action’’ (see Definition \ref{quasi}) $\beta$ of $G$ on $C^(H_r)$ (where $C^(H_r)$ is the semigroupoid $C^$-algebra as defined in \cite{EXE}) such that $C^(H_r\times_\alpha G) = C^*(H_r)\times_\beta G$ (where the crossed product for $\beta$ is as defined in Definition \ref{cross}).

💡 Research Summary

The paper investigates the interplay between a groupoid (G) acting on a small category (H) and the associated (C^{*})-algebraic constructions. Starting from the notion of a groupoid action introduced in NOT (Definition 4), the authors recall that such an action assigns to each arrow (g\in G) a functor (\alpha_{g}) on (H) satisfying the usual compatibility conditions with composition and identities. Using this data they form the semi‑direct product category (H\times_{\alpha}G) (Proposition 8 in NOT). Objects are pairs ((x,g)) with (x\in H^{0}) and arrows are pairs ((h,g)) where (h) is an arrow of (H) and the source and range are twisted by the action of (g). The composition law \