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

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

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

$\vdash \prod_{* : \mathbf{B}G} V(*) : Type \,.$

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

homotopy type theoryrepresentation theory
pointed connected context $\mathbf{B}G$∞-group $G$
dependent type∞-action/∞-representation
dependent sum along $\mathbf{B}G \to \ast$coinvariants/homotopy quotient
context extension along $\mathbf{B}G \to \ast$trivial representation
dependent product along $\mathbf{B}G \to \ast$homotopy invariants/∞-group cohomology
dependent sum along $\mathbf{B}G \to \mathbf{B}H$induced representation
context extension along $\mathbf{B}G \to \mathbf{B}H$
dependent product along $\mathbf{B}G \to \mathbf{B}H$coinduced representation

Revised on November 17, 2013 00:17:00 by Urs Schreiber (82.113.98.128)