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

Abstract 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 $$C^*$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mi>C</mml:mi> <mml:mo>∗</mml:mo> </mml:msup> </mml:math> -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.