nLab
invariant

Contents

Idea

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

Definitions

For \infty-group actions

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

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

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

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

The connected components of this is equivalently the group cohomology of GG with coefficients in the infinity-module VV.

representation theory and equivariant cohomology in terms of (∞,1)-topos theory/homotopy type theory:

homotopy type theoryrepresentation theory
pointed connected context BG\mathbf{B}G∞-group GG
dependent type∞-action/∞-representation
dependent sum along BG*\mathbf{B}G \to \astcoinvariants/homotopy quotient
context extension along BG*\mathbf{B}G \to \asttrivial representation
dependent product along BG*\mathbf{B}G \to \asthomotopy invariants/∞-group cohomology
dependent product of internal hom along BG*\mathbf{B}G \to \astequivariant cohomology
dependent sum along BGBH\mathbf{B}G \to \mathbf{B}Hinduced representation
context extension along BGBH\mathbf{B}G \to \mathbf{B}H
dependent product along BGBH\mathbf{B}G \to \mathbf{B}Hcoinduced representation
spectrum object in context BG\mathbf{B}Gspectrum with G-action (naive G-spectrum)

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