model category

for ∞-groupoids

# Contents

## Idea

The fat simplex functor is a cosimplicial simplicial set

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

whose value $\Delta \left[n\right]$ at $n\in ℕ$ is a simplicial set that models the $n$-simplex but is much bigger than the standard $n$-simplex $\Delta \left[n\right]={\mathrm{Hom}}_{\Delta }\left(-,\left[n\right]\right)$. This is such that $\Delta \left[-\right]$ is a cofibrant replacement of $*$ and of $\Delta \left[-\right]={\mathrm{Hom}}_{\Delta }\left(-,-\right)$ 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 ${\int }^{\left[n\right]\in \Delta }\Delta \left[n\right]\cdot {F}_{n}$. This construction is originally due to Bousfield and Kan.

## Definition

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

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

(in sSet) for the nerve of this overcategory.

This construction is functorial in $\left[n\right]$:

$\Delta \left(-\right)=N\left(\Delta /\left(-\right)\right):\Delta \to \mathrm{sSet}\phantom{\rule{thinmathspace}{0ex}}.$\mathbf{\Delta}(-) = N(\Delta/(-)) : \Delta \to sSet \,.

## Examples

• The fat 0-simplex is $\Delta \left[0\right]=N\left(\Delta \right)$, the nerve of the simplex category (because $\left[0\right]\in \Delta$ 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 $\left[\Delta ,{\mathrm{sSet}}_{\mathrm{Quillen}}\right]$.

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

Revised on February 13, 2011 00:16:33 by Urs Schreiber (62.28.152.34)