An Elliott intertwining approach to classifying actions of ${\mathrm{C}}^{\ast }$-tensor categories

Abstract

We introduce a categorical approach to classifying actions of ${\mathrm{C}}^{\ast }$-tensor categories on ${\mathrm{C}}^{\ast }$-algebras up to cocycle conjugacy. We show that, in this category, inductive limits exist, and there is a natural notion of approximate unitary equivalence. Then, we generalise classical Elliott intertwining results to the tensor category-equivariant case, in the same fashion as done by Szabó for the group equivariant case.