nLab
coinvariant

Contents

Idea

A coinvariant is dual to an invariant.

Definitions

Of group representations

Let GG be a discrete group, kk a commutative unital ring, k[G]k[G] the group ring of GG and VV a left k[G]k[G]-module, hence a linear GG-representation over kk.

Then there is well-defined quotient kk-module V G=V/vgvV_G = V/\langle{v - g v}\rangle called the module of GG-coinvariants. Here vgv\langle{v - g v}\rangle denotes the smallest kk-submodule of VV containing all expressions of the form vgvv - g v where gGg\in G and vVv\in V.

Notice that here vv and gvg v are two points on same orbit of GG and so the coinvariants are essentially the orbits of GG in VV

Of coalgebras

Let CC be a kk-coalgebra, χ\chi a group-like element, that is, an element such that Δ C(χ)=χχ\Delta_C(\chi)=\chi\otimes\chi, and ρ:VVC\rho:V\to V\otimes C a right CC-coaction. Any element vVv\in V such that ρ(v)=vχ\rho(v)=v\otimes\chi is called a (ρ,χ)(\rho,\chi)-coinvariant element in the CC-comodule (V,ρ)(V,\rho). Suppose HH is a bialgebra, AA an algebra and ρ:AAH\rho:A\to A\otimes H a coaction making AA into a right HH-comodule algebra. The unit element 1 H1_H is a group-like element, and we call (ρ,1)(\rho,1)-coinvariants simply ρ\rho-coinvariants. The subset of ρ\rho-coinvariants in AA is a subalgebra, called the subalgebra of coinvariants.

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 December 21, 2013 00:47:53 by Toby Bartels (75.88.42.104)