nLab simplicial group action

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).

Related concepts

• Borel model structure

• G-set, topological G-space

• ∞-action

References

• Emmanuel Dror, William Dwyer, Daniel Kan, Equivariant maps which are self homotopy equivalences, Proc. Amer. Math. Soc. 80 (1980), no. 4, 670–672 (jstor:2043448)

• Paul Goerss, J. F. Jardine, Section V.2 of: Simplicial homotopy theory, Progress in Mathematics, Birkhäuser (1999) Modern Birkhäuser Classics (2009) (doi:10.1007/978-3-0346-0189-4, webpage)

• Bert Guillou, A short note on models for equivariant homotopy theory, 2006 (pdf, pdf)