Given a left-cancellative small category C (in the sense of Spielberg), a discrete group G, and a functor from C into G, we construct a skew product category C x G. This represents a ``coaction one can see'' in the sense that the Cuntz-Krieger algebra O(C x G) of the skew product is isomorphic to the crossed product of O(C) by a coaction of G. Moreover, the skew product carries a natural free action of G that corresponds to the dual action on the crossed product under this isomorphism, and that allows us to recover C as the quotient category. As a sort of converse, we also have a “Gross-Tucker”-type theorem that says that any LCSC that carries a free action of G can be realized as a skew product category. This theory fits into a broader context that goes back over 25 years to ideas of Kumjian, Pask, and Raeburn for graph C∗-algebras.

In this talk, I’ll present these results in as elementary a fashion as possible, and say what I can about their proofs. Time permitting, I’ll also try to explain how we can use our results to study Cuntz-Krieger algebras arising from free actions of groups on finitely-aligned LCSCs, and to construct coactions of groups on Exel-Pardo algebras. The talk is based on joint work with Erik Bedos and John Quigg.