#
nLab

model structure on simplicial groups

### Context

#### Model category theory

**model category**

## Definitions

## Morphisms

## Universal constructions

## Refinements

## Producing new model structures

## Presentation of $(\infty,1)$-categories

## Model structures

### for $\infty$-groupoids

for ∞-groupoids

### for $n$-groupoids

### for $\infty$-groups

### for $\infty$-algebras

#### general

#### specific

### for stable/spectrum objects

### for $(\infty,1)$-categories

### for stable $(\infty,1)$-categories

### for $(\infty,1)$-operads

### for $(n,r)$-categories

### for $(\infty,1)$-sheaves / $\infty$-stacks

#### Group Theory

**group theory**

### Classical groups

### Finite groups

### Group schemes

### Topological groups

### Lie groups

### Super-Lie groups

### Higher groups

### Cohomology and Extensions

# 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

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

with the model structure on reduced simplicial sets.

## 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)