On the homotopy theory of $\mathbf{G}$ - spaces

TL;DR

This paper demonstrates an equivalence between the elementary homotopy theory of G-spaces and a homotopy theory of simplicial sets over BG, using relative categories without assuming model structures.

Contribution

It establishes a strict homotopy equivalence between the homotopy theory of G-spaces and simplicial sets over BG, without relying on model category assumptions.

Findings

Homotopy theories of G-spaces and simplicial sets over BG are equivalent.

Constructs a strict homotopy equivalence between the two relative categories.

No model category structure is needed for the equivalence.

Abstract

The aim of this paper is to show that the most elementary homotopy theory of $\mathbf{G}$-spaces is equivalent to a homotopy theory of simplicial sets over $\mathbf{BG}$, where $\mathbf{G}$ is a fixed group. Both homotopy theories are presented as Relative categories. We establish the equivalence by constructing a strict homotopy equivalence between the two relative categories. No Model category structure is assumed on either Relative Category.