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

TL;DR

This paper provides a new, simplified proof of the universal property of $KK^G$-theory, emphasizing its generators and relations, applicable to various algebraic structures like groups, groupoids, and inverse semigroups.

Contribution

It introduces a concise, conceptual proof of the universal property of $KK^G$-theory and simplifies the form of morphisms in its generators and relations framework.

Findings

New proof of the universal property of $KK^G$-theory

Simplified form of morphisms in generators and relations picture

Applicable to groups, groupoids, and inverse semigroups

Abstract

We give a new proof of the universal property of $KK^G$-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 $KK^G$-theory are brought to a particular simple form.