nLab
infinity-permutation representation

Idea

An ∞-representation on objects of ∞Grpd.

The analog on (∞,1)-category theory of a permutation representation.

Properties

Tannaka duality

See at Tannaka duality the section For ∞-permutation representations.

Grothendieck construction

By the (∞,1)-Grothendieck construction $\mathrm{\infty }$-permutation representations

$$BG\to \mathrm{\infty }\mathrm{Grpd}$$\mathbf{B}G \to \infty \mathrm{Grpd}

correspond to ∞-functors

$$V//G\to BG$$V//G \to \mathbf{B}G

sitting in fiber sequences

$$V\to V//G\to BG,$$V \to V//G \to \mathbf{B}G \,,

where $V//G$ is the corresponding action ∞-groupoid?.