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We consider Toeplitz and Cuntz-Krieger $C^*$-algebras associated withfinitely aligned left cancellative small categories. We pay special attentionto the case where such a category arises as the Zappa-Sz\'ep product of acategory and a group linked by a one-cocycle. As our main application, weobtain 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 leftcancellative small categories and discuss their relationship.

On finitely aligned left cancellative small categories, Zappa-Sz\'ep products and Exel-Pardo algebras