Model category of diffeological spaces

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 \(\varDelta ^{p}\)\((p \ge 0)\) such that \(\varDelta ^{p}\) contains the \(k\mathrm{th}\) horn \(\varLambda ^{p}_{k}\) as a smooth deformation retract.