We consider Toeplitz and Cuntz-Krieger $C^*$-algebras associated with fin\-itely aligned left cancellative small categories. We pay special attention to the case where such a category arises as the Zappa-Szep product of a category and a group linked by a one-cocycle. As our main application, we obtain a new approach to Exel-Pardo algebras in the case of row-finite graphs. We also present some other ways of constructing $C^*$-algebras from left cancellative small categories and discuss their relationship.
Keywords: Groups, graphs, self-similarity, category of paths, left cancellative small categories, Zappa-Szep products, Toeplitz algebras, Cuntz-Krieger algebras
2010 MSC: 46L05, 46L55
Theory and Applications of Categories, Vol. 33, 2018, No. 42, pp 1346-1406.