nLab
simplicial loop space

Contents

Context

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

Introductions

Definitions

Paths and cylinders

Homotopy groups

Basic facts

Theorems

Contents

Idea

In simplicial homotopy theory there is a standard construction (Kan 58), 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.

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 W¯()\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 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)

References

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

More details and relation to décalage:

Last revised on July 4, 2021 at 07:38:55. See the history of this page for a list of all contributions to it.