nLab Quillen equivalence between simplicial groups and reduced simplicial sets

Statement

Write

• $Groups(sSet)$ for the category of simplicial groups

• $Groups(sSet)_{proj}$ for the model structure on simplicial groups whose weak equivalences and fibrations are those of the underlying morphisms in the classical model structure on simplicial sets (hence the underlying simplicial weak equivalences and Kan fibrations, respectively);

• $sSet_{\geq 1}$ for the category of reduced simplicial sets;

• $(sSet_{\geq 1})$ for the model structure on reduced simplicial sets whose weak equivalences and cofibrations are those of the underlying morphisms in the classical model structure on simplicial sets.

Proposition

(Quillen equivalence between simplicial groups and reduced simplicial sets)

The

• simplicial classifying space-construction $\overline{W}(-)$ is the right adjoint

• simplicial loop space-construction $\Omega(-)$ is the left adjoint

in a Quillen equivalence between the projective model structure on simplicial groups and the injective model structure on reduced simplicial sets.

$$Groups(sSet)_{proj} \underoverset {\underset{\overline{W}}{\longrightarrow}} {\overset{ \Omega }{\longleftarrow}} {\;\;\;\;\; \simeq_{\mathrlap{Qu}} \;\;\;\;\;} (sSet_{\geq 0})_{inj} \,.$$

(e.g. Goerss & Jardine 09, V Prop. 6.3)

Related concepts

• looping and delooping

• May recognition theorem

References

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