nLab
fat simplex

Idea

The fat simplex functor is a cosimplicial simplicial set

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

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

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

(in sSet) for the nerve of this overcategory.

This construction is functorial in $[n]$:

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

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

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

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

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

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