# An isovariant Elmendorf's theorem

## TL;DR

This paper establishes a model category structure for isovariant maps between G-spaces and proves an Elmendorf's theorem variant, linking it to diagram categories through Quillen equivalence.

## Contribution

It introduces a Quillen model structure for the category of G-spaces with isovariant maps and proves an isovariant Elmendorf's theorem via model-theoretic methods.

## Key findings

- Established a Quillen model structure for isovariant G-spaces
- Proved a Quillen equivalence to a diagram category
- Extended Elmendorf's theorem to the isovariant setting

## Abstract

An isovariant map between spaces with a group action is an equivariant map which preserves isotropy groups. In this paper, we show that for a finite group $G$, the category of $G$-spaces with isovariant maps has a Quillen model structure. We prove a Piacenza-style model theoretic proof of an isovariant Elmendorf's theorem, showing that this model structure is Quillen equivalent to a model category of diagrams.

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Source: https://tomesphere.com/paper/1907.12135