nLab fat simplex

Idea

The fat simplex functor is a cosimplicial simplicial set

$$\mathbf{\Delta} : \Delta \to sSet$$

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 simplicial diagrams 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

$$\mathbf{\Delta}[n] := N(\Delta/[n])$$

(in sSet) for the nerve of this overcategory.

This construction is functorial in $[n]$:

$$\mathbf{\Delta}(-) = N(\Delta/(-)) : \Delta \to sSet \,.$$

Examples

• The fat 0-simplex is $\mathbf{\Delta}[0] = N(\Delta)$, the nerve of the simplex category (because $[0] \in \Delta$ is the terminal object).

Properties

There is a canonical morphism

$$\mathbf{\Delta} \to \Delta$$

of cosimplicial simplicial set, called the Bousfield-Kan map.

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

See the discussion at Reedy model structure and at Bousfield-Kan map for details.