model category

for ∞-groupoids

# Contents

## Idea

The model structure on reduced simplicial sets is a presentation of the full sub-(∞,1)-category

∞Grpd${}_{\ge 1}^{*/}↪$ ∞Grpd${}^{*/}$ $\simeq$ Top${}^{*/}$

of pointed ∞-groupoids on those that are connected.

By the looping and delooping-equivalence, this is equivalent to the (∞,1)-category of ∞-groups and this equivalence is presented by a Quillen equivalence to the model structure on simplicial groups.

## Definition

###### Definition

A reduced simplicial set is a simplicial set $S$ with a single vertex:

${S}_{0}=*\phantom{\rule{thinmathspace}{0ex}}.$S_0 = * \,.

Write ${\mathrm{sSet}}_{0}\subset$ sSet for the full subcategory of the category of simplicial sets on those that are reduced.

###### Proposition

There is a model category structure on ${\mathrm{sSet}}_{0}$ whose

• weak equivalences

• and cofibrations

are those in the standard model structure on simplicial sets.

This appears as (GoerssJardine, ch V, prop. 6.2).

## Properties

###### Proposition

The simplicial loop space functor $G$ and the delooping functor $\overline{W}\left(-\right)$ (discussed at simplicial group) constitute a Quillen equivalence

$\left(G⊣\overline{W}\right):\mathrm{sGr}\stackrel{\stackrel{G}{←}}{\underset{\overline{W}}{\to }}{\mathrm{sSet}}_{0}$(G \dashv \bar W) : sGr \stackrel{\overset{G}{\leftarrow}}{\underset{\bar W}{\to}} sSet_0

with the model structure on simplicial groups.

This appears as (GoerssJardine, ch. V prop. 6.3).

###### Proposition

Under the forgetful functor $U:{\mathrm{sSet}}_{0}↪\mathrm{sSet}$

• a fibration $f:X\to Y$ maps to a fibration precisely if it has the right lifting property against $*\to {S}^{1}:=\Delta \left[1\right]/\partial \Delta \left[1\right]$;

In particular

• every fibrant object maps to a fibrant object.

The first statment appears as (GoerssJardine, ch. V, lemma 6.6.). The second (an immediate consequence) appears as (GoerssJardine, ch. V, corollary 6.8).

## References

A standard textbook reference is chapter V of

Revised on April 14, 2012 10:23:31 by Urs Schreiber (82.113.106.163)