  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 $S^D(X)$ associated to a
diffeological space $X$, and show that when $S^D(X)$ is fibrant, it captures
smooth homotopical properties of $X$. Motivated by this, we define $X$ to be
fibrant when $S^D(X)$ is, and more generally define cofibrations, fibrations
and weak equivalences in the category of diffeological spaces using the smooth
singular simplicial set functor. We conjecture that these form a model category
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 the free loop space of a smooth manifold is
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.
