nLab
model structure on simplicial groups

Contents

Context

Model category theory

model category

Definitions

Morphisms

Universal constructions

Refinements

Producing new model structures

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

Model structures

for \infty-groupoids

for ∞-groupoids

for rational \infty-groupoids

for nn-groupoids

for \infty-groups

for \infty-algebras

general

specific

for stable/spectrum objects

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

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

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

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

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

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 model category structure on the category sGrpsGrp of simplicial groups where a morphism is

(Quillen 67, II 3.7, see also Goerss-Jardine 99, V)

Properties

Proposition

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

(ΩW¯):sGrpsSet 0 \big( \Omega \dashv \overline{W} \big) \;\colon\; sGrp \stackrel{\overset{}{\longleftarrow}}{\longrightarrow} sSet_0

with the model structure on reduced simplicial sets.

(Goerss-Jardine 99, Chapter V, Prop. 6.3)

References

The model structure on simplicial groups is due to

Further discussion is in

Last revised on October 26, 2020 at 05:29:35. See the history of this page for a list of all contributions to it.