Some remarks on representations up to homotopy

TL;DR

This paper explores the properties of representations up to homotopy on bundles of compact groups and establishes their sufficiency for point separation and their categorical equivalence to ordinary representations.

Contribution

It demonstrates that representations up to homotopy can distinguish points on bundles of compact groups and shows the derived representation category is equivalent to classical representations.

Findings

Representations up to homotopy separate points on locally trivial bundles of compact groups.

Derived representation category of any compact group is equivalent to its finite-dimensional representation category.

Provides insights into the algebraic structure of quantum field theories.

Abstract

Motivated by the study of the interrelation between functorial and algebraic quantum field theory, we point out that on any locally trivial bundle of compact groups, representations up to homotopy are enough to separate points by means of the associated representations in cohomol- ogy. Furthermore, we observe that the derived representation category of any compact group is equivalent to the category of ordinary (finite- dimensional) representations of the group.