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.

References

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

More details and relation to décalage:

Last revised on June 5, 2021 at 19:28:09. See the history of this page for a list of all contributions to it.