C-crossed product of groupoid actions on categories

Suppose that G is a groupoid acting on a small category H in the sense of [9, Definition 4] and H ×α G is the resulting semi-direct product category (as in [9, Proposition 8]). We show that there exists a subcategory Hr ⊆ H satisfying some nice property called “regularity” such that Hr ×α G = H ×α G. Moreover, we show that there exists a so-called “quasi action” (see Definition 3.1) β of G on C(Hr) (where C(Hr) is the semigroupoid C-algebra as defined in [3]) such that C(Hr ×α G) = C(Hr) ×β G (where the crossed product for β is as defined in Definition 3.6). 2000 Mathematics Subject Classification: Primary 46L05; Secondary 18B40.