Self-similar groupoid actions on k-graphs, and invariance of K-theory for cocycle homotopies

Abstract We establish conditions under which an inclusion of finitely aligned left-cancellative small categories induces inclusions of twisted $$C^*$$ <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. We also present an example of an inclusion of finitely aligned left-cancellative monoids that does not induce a homomorphism even between (untwisted) Toeplitz algebras. We prove that the twisted $$C^*$$ <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 a jointly faithful self-similar action of a countable discrete amenable groupoid on a row-finite k -graph with no sources, with respect to homotopic cocycles, have isomorphic K -theory.