Aspects of equivariant $KK$-theory in its generators and relations picture

Abstract

We give a new proof of the universal property of $K{K}^G$-theory with respect to stability, homotopy invariance and split-exactness for $G$ a locally compact group, or a locally compact (not necessarily Hausdorff) groupoid, or a countable inverse semigroup which is relatively short and conceptual. Morphisms in the generators and relations picture of $K{K}^G$-theory are brought to a particular simple form.