Homotopy of Planar Lie Group Equivariant Presheaves

TL;DR

This paper develops a new framework for studying equivariant presheaves with planar Lie group actions using model category theory, enabling a better understanding of equivariant cohomology theories as derived mapping spaces.

Contribution

It introduces local Quillen model structures on simplicial presheaves with planar Lie group actions, connecting equivariant cohomology to derived mapping spaces.

Findings

Established model structures for equivariant presheaves

Characterized equivariant cohomology theories as derived mapping spaces

Applied crossed simplicial groups to equivariant homotopy theory

Abstract

We utilise the theory of crossed simplicial groups to introduce a collection of local Quillen model structures on the category of simplicial presheaves with a compact planar Lie group action on a small Grothendieck site. As an application, we give a characterisation of equivariant cohomology theories on a site as derived mapping spaces in these model categories.