nLab
simplicial loop space

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.

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

Related concepts

• simplicial delooping functor

References

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

• Daniel Kan, On homotopy theory and c.s.s. groups, Ann. of Math. 68 (1958), 38-53 (jstor:1970042) (Sections 10–11.)

• Peter May, chapter VI of Simplicial objects in algebraic topology, Van Nostrand, Princeton (1967) (ISBN:9780226511818, djvu)

• 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)

More details and relation to décalage:

• Danny Stevenson, Décalage and Kan’s simplicial loop group functor, Theory and Applications of Categories, Vol. 26, 2012, No. 28, pp 768-787 (arXiv:1112.0474, tac:26-28)