nLab
invariant

Idea

For $A$ a monoid equipped with an action on an object $V$, an invariant of the action is an element of $V$ which is taken by the action to itself.

Definitions

For $\mathrm{\infty }$-group actions

For $H$ an (∞,1)-topos, $G\in \mathrm{Grp}(H)$ an ∞-group and

$$*:BG⊢:V(*):\mathrm{Type}$$* : \mathbf{B} G \vdash : V(*) : Type

an ∞-action of $G$ on $V\in H$, the type of invariants is the absolute dependent product

$$⊢\prod _{*:BG}V(*):\mathrm{Type}.$$\vdash \prod_{* : \mathbf{B}G} V(*) : Type \,.

The connected components of this is equivalently the group cohomology of $G$ with coefficients in the infinity-module $V$.

Related concepts

• stabilizer subgroup

• invariant theory, invariant polynomial

• gauge invariance