Contents

Idea

The fat simplex functor is a cosimplicial simplicial set

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

The fat simplex can be used to express the homotopy colimit over simplicial diagrams in terms of coends of the form ${\int }^{[n]\in \Delta }\Delta [n]\cdot F_n$. 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 $

(in sSet) for the nerve of this overcategory.

This construction is functorial in $[n]$:

Examples

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

Properties

There is a canonical morphism

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

This exhibits $\Delta$ as a cofibrant replacement of $\Delta$ and of $*$ in the projective model structure on functors on $[\Delta ,{\mathrm{sSet}}_{\mathrm{Quillen}}]$.