Model category of diffeological spaces

TL;DR

This paper establishes a model structure on the category of diffeological spaces, enabling the development of smooth homotopy theory by defining weak equivalences via smooth homotopy groups.

Contribution

It constructs a compactly generated model structure on diffeological spaces with weak equivalences based on smooth homotopy groups, introducing specific diffeologies on simplices.

Findings

Model structure on diffeological spaces is constructed.

Weak equivalences are characterized by smooth homotopy groups.

Diffeologies on simplices are designed to include horns as deformation retracts.

Abstract

The existence of a model structure on the category $\mathcal{D}$ of diffeological spaces is crucial to developing smooth homotopy theory. We construct a compactly generated model structure on the category $\mathcal{D}$ whose weak equivalences are just smooth maps inducing isomorphisms on smooth homotopy groups. The essential part of our construction of the model structure on $\mathcal{D}$ is to introduce diffeologies on the sets $\Delta^{p}$ $(p \geq 0)$ such that $\Delta^{p}$ contains the $k^{th}$ horn $\Lambda^{p}_{k}$ as a smooth deformation retract.