model category

for ∞-groupoids

# Contents

## Idea

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

∞Grpd${}^{*/}_{\geq 1} \hookrightarrow$ ∞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:

$S_0 = * \,.$

Write $sSet_0 \subset$ sSet for the full subcategory of the category of simplicial sets on those that are reduced.

###### Proposition

There is a model category structure on $sSet_0$ 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 $\bar W(-)$ (discussed at simplicial group) constitute a Quillen equivalence

$(G \dashv \bar W) : sGr \stackrel{\overset{G}{\leftarrow}}{\underset{\bar W}{\to}} sSet_0$

with the model structure on simplicial groups.

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

###### Proposition

Under the forgetful functor $U : sSet_0 \hookrightarrow sSet$

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

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)