nLab model structure on simplicial groups

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

• is a weak equivalence if the underlying morphism is a weak equivalence in the standard model structure on simplicial sets;

• is a fibration if the underlying morphism is a Kan fibration of simplicial sets;

• is a cofibration if it has the left lifting property with respect to all trivial fibrations.

(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

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

with the model structure on reduced simplicial sets.

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

Related concepts

• Borel model structure (presenting infinity-actions of simplicial groups)

References

The model structure on simplicial groups is due to

• Daniel Quillen, p. II 3.7 of: Axiomatic homotopy theory in: Homotopical Algebra, Lecture Notes in Mathematics 43, Springer 1967(doi:10.1007/BFb0097438)

Further discussion is in

• Paul Goerss, J. F. Jardine, chapter V of: Simplicial homotopy theory, Progress in Mathematics, Birkhäuser (1999) Modern Birkhäuser Classics (2009) (doi:10.1007/978-3-0346-0189-4, webpage)