Homology and Twisted \(C^*\)-Algebras for Self-similar Actions and Zappa–Szép Products

We study the categorical homology of Zappa–Szép products of small categories, which include all self-similar actions. We prove that the categorical homology coincides with the homology of a double complex, and so can be computed via a spectral sequence involving homology groups of the constituent categories. We give explicit formulae for the isomorphisms involved, and compute the homology of a class of examples that generalise odometers. We define the \(C^*\) $C^{\ast }$ -algebras of self-similar groupoid actions on k-graphs twisted by 2-cocycles arising from this homology theory, and prove some fundamental results about their structure.