Contents

# Contents

## Idea

In simplicial homotopy theory there is a standard construction (Kan 58), traditionally denoted $G$, for the simplicial analog of the homotopy type of the loop space $\Omega X$ of a connected topological space, now incarnated as a simplicial group: It is the left adjoint in an adjunction

$(G \dashv \overline{W}) \;\colon\; sSet_* \underoverset {\underset{\overline{W}}{\longleftarrow}} {\overset{G}{\longrightarrow}} {\bot} sGrp$

between pointed simplicial sets and simplicial groups, whose right adjoint is the simplicial classifying space construction.

This adjunction is a Quillen adjunction for the Kan–Quillen model structure on pointed simplicial sets and its transferred model structure on simplicial groups.

When restricted to reduced simplicial sets, this Quillen adjunction becomes a Quillen equivalence.

The generalization of this construction to non-reduced simplicial sets and simplicial groupoids is the Dwyer-Kan loop groupoid functor.

## Properties

### Quillen equivalence between simplicial groups and reduced simplicial sets

###### Proposition

(Quillen equivalence between simplicial groups and reduced simplicial sets)
The simplicial classifying space-construction $\overline{W}(-)$ (Def. ) is the right adjoint in a Quillen equivalence between the projective model structure on simplicial groups and the injective model structure on reduced simplicial sets.

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

The left adjoint $\Omega$ is the simplicial loop space-construction.

## References

For a more complete list of references, see the article simplicial delooping functor.

More details and relation to décalage:

Last revised on May 21, 2022 at 02:25:12. See the history of this page for a list of all contributions to it.