Quillen equivalences between the model categories of smooth spaces, simplicial sets, and arc-gengerated spaces

In the preceding paper, we have constructed a compactly generated model structure on the category $\text{dcal}$ of diffeological spaces together with the adjoint pairs $\mathrm{\mid }{\mathrm{\mid }}_{\text{dcal}}:\text{scal}\rightleftarrows \text{dcal}:S^{\text{dcal}}$ and $\tilde{\cdot }:\text{dcal}\rightleftarrows {\text{ccal}}^0:R$, where $\text{scal}$ and ${\text{ccal}}^0$ denote the category of simplicial sets and that of arc-generated spaces, respectively. In this paper, we show that $(\mathrm{\mid }{\mathrm{\mid }}_{\text{dcal}},S^{\text{dcal}})$ and $(\tilde{\cdot },R)$ are pairs of Quillen equivalences. Since our approach developed in the preceding paper applies to the category $\text{ccal}h$ of Chen spaces as well, $\text{ccal}h$ is also a compactly generated model category. We also show that the adjoint pair $\mathfrak{S}_\mathfrak{o} : \ccal h \rightleftarrows \dcal : \mathfrak{Ch}^\sharp$ introduced by Stacey is a pair of Quillen equivalences.