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 equivariant \infty-groupoids

for rational \infty-groupoids

for rational equivariant \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 projective 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

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

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