# nLab invariant

### Context

#### Algebra

higher algebra

universal algebra

# Contents

## 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 $\infty$-group actions

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

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

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

$⊢\prod _{*:BG}V\left(*\right):\mathrm{Type}\phantom{\rule{thinmathspace}{0ex}}.$\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 $V$.

Revised on December 15, 2012 21:40:52 by Urs Schreiber (71.195.81.250)