Given a group cocycle on a finitely aligned left cancellative small category (LCSC), we investigate the associated skew product category and its Cuntz–Krieger algebra, which we describe as the crossed product of the Cuntz–Krieger algebra of the original category by an induced coaction of the group. We 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. Finally, we discuss the universal group of a small category and connectedness of skew product categories
