model category

for ∞-groupoids

group theory

# Contents

## Idea

The model structure on simplicial groups is a presentation of the ∞-groups in ∞Grpd $\simeq$ Top. See group object in an (∞,1)-category.

## Definition

There is a model category structure on the category $sGrp$ of simplicial groups where a morphism is

## Properties

Forming loop space objects and classifying spaces provides a Quillen equivalence

$(\Omega \dashv \bar W) : sGrp \stackrel{\overset{}{\leftarrow}}{\to} sSet_0$

## References

The general theory is in chapter V of

• Paul Goerss and J. F. Jardine, 1999, Simplicial Homotopy Theory, number 174 in Progress in Mathematics, Birkhauser. (ps)

The Quillen equivalence is in proposition 6.3.

Revised on April 15, 2014 03:35:32 by Tim Porter (2.26.24.125)