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arXiv · 2311.09600

Homology and twisted C*-algebras for self-similar actions and Zappa-Sz\'ep products

Alexander Mundey·Aidan Sims
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Abstract

We study the categorical homology of Zappa-Sz\'ep 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*-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.

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Alexander Mundey, Aidan Sims. 2023-11-16. Homology and twisted C*-algebras for self-similar actions and Zappa-Sz\'ep products. https://arxiv.org/abs/2311.09600

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