Smooth singular complexes and diffeological principal bundles

In previous papers, we used the standard simplices ${\mathrm{\Delta }}^p$ $(p\ge 0)$ endowed with diffeologies having several good properties to introduce the singular complex $S^{\text{dcal}}(X)$ of a diffeological space $X$. On the other hand, Hector and Christensen-Wu used the standard simplices ${\mathrm{\Delta }}_{\mathrm{s}\mathrm{u}\mathrm{b}}^p$ $(p\ge 0)$ endowed with the sub-diffeology of ${\text{rbb}}^{p+1}$ and the standard affine $p$-spaces ${\text{abb}}^p$ $(p\ge 0)$ to introduce the singular complexes $S_{\mathrm{s}\mathrm{u}\mathrm{b}}^{\text{dcal}}(X)$ and $S_{\mathrm{a}\mathrm{f}\mathrm{f}}^{\text{dcal}}(X)$, respectively, of a diffeological space $X$. In this paper, we prove that $S^{\text{dcal}}(X)$ is a fibrant approximation both of $S_{\mathrm{s}\mathrm{u}\mathrm{b}}^{\text{dcal}}(X)$ and $S_{\mathrm{a}\mathrm{f}\mathrm{f}}^{\text{dcal}}(X)$. This result easily implies that the homotopy groups of $S_{\mathrm{s}\mathrm{u}\mathrm{b}}^{\text{dcal}}(X)$ and $S_{\mathrm{a}\mathrm{f}\mathrm{f}}^{\text{dcal}}(X)$ are isomorphic to the smooth homotopy groups of $X$, proving a conjecture of Christensen and Wu. Further, we characterize diffeological principal bundles (i.e., principal bundles in the sense of Iglesias-Zemmour) using the singular functor $S_{\mathrm{a}\mathrm{f}\mathrm{f}}^{\text{dcal}}$. By using these results, we extend characteristic classes for $\text{dcal}$-numerable principal bundles to characteristic classes for diffeological principal bundles.