Contents

model category

for ∞-groupoids

group theory

# Contents

## Idea

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

## Definition

###### Proposition

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

## Properties

###### Proposition

Every object in the projective model structure (Prop. ) is fibrant.

###### Proof

This statement amounts to saying that the underlying simplicial set of any simplicial group is a Kan complex. That this is the case is Moore’s theorem (here).

###### Proposition

(Quillen equivalence between simplicial groups and reduced simplicial sets)

Forming simplicial loop space objects and simplicial classifying spaces gives a Quillen equivalence

$\big( \Omega \dashv \overline{W} \big) \;\colon\; sGrp \stackrel{\overset{}{\longleftarrow}}{\longrightarrow} sSet_0$

## References

The model structure on simplicial groups is due to

Further discussion is in

Last revised on July 4, 2021 at 08:19:46. See the history of this page for a list of all contributions to it.