nLab simplicial loop space

Idea

In simplicial homotopy theory there is a standard construction [Kan (1958a), Kan (1958b)] – 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 (more on which at model structure on simplicial groups).

When restricted to reduced simplicial sets, this Quillen adjunction becomes a Quillen equivalence which exhibits the homotopy theory of simplicial groups as equivalent to the classical homotopy theory of pointed connected homotopy type homotopy type (cf. at looping and delooping).

The generalization of the Kan loop group construction to non-reduced simplicial sets – and then taking values in 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 $\overline{W}(-)$ 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

• Dwyer-Kan loop groupoid

References

The original articles:

• Daniel M. Kan, Section 9 of: A combinatorial definition of homotopy groups, Annals of Mathematics 67 2 (1958) 282–312 [doi:10.2307/1970006]

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

Early review:

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

A modern monograph account:

• Paul Goerss, J. F. Jardine, Section V.5 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]