nLab simplicial group action

Contents

Context

Representation theory

representation theory

geometric representation theory

Contents

Definition

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

$\mathcal{G} \times X \xrightarrow{\;\; \rho \;\;} X$

satisfying the action property.

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

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

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.