C*-crossed product of groupoid actions on categories

الأصل · EN

Suppose that G is a groupoid acting on a small category H in the sense of [Definition 4]NOT and H×αG is the resulting semi-direct product category (as in [Proposition 8]NOT). We show that there exists a subcategory Hᵣ H satisfying some nice property called ``regularity'' such that Hᵣ ×αG = H×αG. Moreover, we show that there exists a so-called ``quasi action'' (see Definition quasi) β of G on C*(Hᵣ) (where C*(Hᵣ) is the semigroupoid C*-algebra as defined in EXE) such that C*(Hᵣ×αG) = C*(Hᵣ)×βG (where the crossed product for β is as defined in Definition cross).