The homotopy theory of diffeological spaces

Diffeological spaces are generalizations of smooth manifolds. In this paper, we study the homotopy theory of diffeological spaces. We begin by proving basic properties of the smooth homotopy groups that we will need later. Then we introduce the smooth singular simplicial set S(X) associated to a diffeological space X, and show that when S(X) is fibrant, it captures smooth homotopical properties of X. Motivated by this, we define X to be fibrant when S(X) is, and more generally define cofibrations, fibrations and weak equivalences in the category of diffeological spaces using the smooth singular functor. We conjecture that these form a model structure, but in this paper we assume little prior knowledge of model categories, and instead focus on concrete questions about smooth manifolds and diffeological spaces. We prove that our setup generalizes the naive smooth homotopy theory of smooth manifolds by showing that a smooth manifold without boundary is fibrant and that for fibrant diffeological spaces, the weak equivalences can be detected using ordinary smooth homotopy groups. We also show that our definition of fibrations generalizes Iglesias-Zemmour’s theory of diffeological bundles. We prove enough of the model category axioms to show that every diffeological space has a functorial cofibrant replacement. We give many explicit examples of objects that are cofibrant, not cofibrant, fibrant and not fibrant, as well as many other examples showing the richness of the theory. For example, we show that both the free and based loop spaces of a smooth manifold are fibrant. One of the implicit points of this paper is that the language of model categories is an effective way to organize homotopical thinking, even when it is not known that all of the model category axioms are satisfied.