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Meyer and Nest showed that the Baum--Connes map is equivalent to a map on$K$-theory of two different crossed products. This approach is stronglycategorial in method since its bases is to regard Kasparov's theory $KK^G$ as atriangulated category. We have tried to translate this approach to the realm ofinverse semigroup equivariant $C^*$-algebras but can prove the existence of aBaum--Connes map only under some unverified additional assumptions which wehowever strongly motivate. Some of our results may be of independent interest,for example Bott periodicity, the definition of induction functors, thedefinition of a completely novel compatible $L^2(G)$-space, a Cuntz picture of$KK^G$, and the verification that $KK^G$ is a triangulated category.

Attempts to define a Baum--Connes map via localization of categories for inverse semigroups