whose value $\mathbf{\Delta}[n]$ at $n \in \mathbb{N}$ is a simplicial set that models the $n$-simplex but is much bigger than the standard $n$-simplex $\Delta[n] = Hom_{\Delta}(-,[n])$. This is such that $\mathbf{\Delta}[-]$ is a cofibrant replacement of $*$ and of $\Delta[-] = Hom_\Delta(-,-)$ in the projective model structure on functors$\Delta \to sSet_{Quillen}$.

The fat simplex can be used to express the homotopy colimit over simplicialdiagrams in terms of coends of the form $\int^{[n] \in \Delta} \mathbf{\Delta}[n] \cdot F_n$. This construction is originally due to Bousfield and Kan.

Definition

Write $\Delta$ for the simplex category. For $[n] \in \Delta$ write $\Delta/[n]$ for the corresponding overcategory. Finally write

This exhibits $\mathbf{\Delta}$ as a cofibrant resolution of $\Delta$ and of $*$ in the projective model structure on functors on $[\Delta, sSet_{Quillen}]$.