nLab
simplicial group action

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

Representation theory

Contents

Definition

Given a simplicial group 𝒢Groups(sSet)\mathcal{G} \in Groups(sSet), hence a group object internal to SimplicialSets, a group action of 𝒢\mathcal{G} on a simplicial set XX is a 𝒢\mathcal{G}-action object internal to SimplicialSets, hence a morphism of simplicial sets of the form

𝒢×XρX \mathcal{G} \times X \xrightarrow{\;\; \rho \;\;} X

satisfying the action property.

The category of simplicial 𝒢\mathcal{G}-action may be understood as the sSet-enriched functor category from the one-object sSet-category B𝒢\mathbf{B}\mathcal{G} to sSet:

(1)𝒢Actions(sSet)sSetCat(B𝒢,,sSet) \mathcal{G}Actions(sSet) \;\; \simeq \;\; sSetCat \big( \mathbf{B}\mathcal{G}, \,, sSet \big)

Properties

Model category structure

Proposition

(model structure on simplicial group actions)
There is a model category-structure on simplicial group actions (1) whose weak equivalences and fibrations are those in the underlying classical model structure on simplicial sets, hence are the simplicial weak equivalences and Kan fibrations of the underlying simplicial sets.

(DDK 80, Prop. 2.2. (ii), \; Guillou, Prop. 5.3, \; Goerss & Jardine 09, V Lem. 2.4).

References

Created on July 4, 2021 at 14:18:44. See the history of this page for a list of all contributions to it.