nLab simplicial loop space



Homotopy theory

homotopy theory, (∞,1)-category theory, homotopy type theory

flavors: stable, equivariant, rational, p-adic, proper, geometric, cohesive, directed

models: topological, simplicial, localic, …

see also algebraic topology



Paths and cylinders

Homotopy groups

Basic facts




In simplicial homotopy theory there is a standard construction [Kan (1958a), Kan (1958b)] – traditionally denoted “GG” – for the simplicial analog of the homotopy type of the loop space ΩX\Omega X of a connected topological space, now incarnated as a simplicial group: It is the left adjoint in an adjunction

(GW¯):sSet *W¯GsGrp (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.


Quillen equivalence between simplicial groups and reduced simplicial sets


(Quillen equivalence between simplicial groups and reduced simplicial sets)
The simplicial classifying space-construction W¯()\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 QuW¯Ω(sSet 0) inj. Groups(sSet)_{proj} \underoverset {\underset{\overline{W}}{\longrightarrow}} {\overset{ \Omega }{\longleftarrow}} {\;\;\;\;\; \simeq_{\mathrlap{Qu}} \;\;\;\;\;} (sSet_{\geq 0})_{inj} \,.

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

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


The original articles:

Early review:

A modern monograph account:

More details and relation to décalage:

Last revised on May 28, 2023 at 20:07:07. See the history of this page for a list of all contributions to it.