nLab
model structure on simplicial groups

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 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 ∞-groups in ∞Grpd \simeq Top. See group object in an (∞,1)-category.

Definition

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

Properties

Forming loop space objects and classifying spaces provides a Quillen equivalence

(ΩW¯):sGrpsSet 0 (\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)