Some remarks on representations up to homotopy
The notion of representation up to homotopy of a topological or Lie groupoid was introduced and studied by \textit{C. A. Abad} and \textit{M. Crainic} [J. Reine Angew. Math. 663, 91--126 (2012; Zbl 1238.58010)]. In the paper under review, this concept is explored for its links with classical representation theory. It is shown that such `homotopy representations' separate points on any locally trivial bundle of compact abelian groups (torus bundle) and that any locally trivial bundle of compact groups admits enough homotopy representations. In the second part of the paper, it is shown that, for any compact group, \(G\), the category of finite dimensional (ordinary) representations of \(G\) can be recovered from the dg-category of homotopy representations of \(G\) using simple standard methods.
scientific article; zbMATH DE number 6573992