nLab coinvariant

Idea

The notion of coinvariant is dual to that of invariant.

Definitions

Of group representations

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

Then there is well-defined quotient $k$-module $V_G = V/\langle{v - g v}\rangle$ called the module of $G$-coinvariants. Here $\langle{v - g v}\rangle$ denotes the smallest $k$-submodule of $V$ containing all expressions of the form $v - g v$ where $g\in G$ and $v\in V$.

Notice that here $v$ and $g v$ are two points on same orbit of $G$ and so the coinvariants are essentially the orbits of $G$ in $V$.

Of coalgebras

Let $C$ be a $k$-coalgebra, $\chi$ a group-like element, that is, an element such that $\Delta_C(\chi)=\chi\otimes\chi$, and $\rho:V\to V\otimes C$ a right $C$-coaction. Any element $v\in V$ such that $\rho(v)=v\otimes\chi$ is called a $(\rho,\chi)$-coinvariant element in the $C$-comodule $(V,\rho)$. Suppose $H$ is a bialgebra, $A$ an algebra and $\rho:A\to A\otimes H$ a coaction making $A$ into a right $H$-comodule algebra. The unit element $1_H$ is a group-like element, and we call $(\rho,1)$-coinvariants simply $\rho$-coinvariants. The subset of $\rho$-coinvariants in $A$ is a subalgebra, called the subalgebra of coinvariants.

Related concepts

• invariant

• norm map

• homotopy coinvariant functor?

representation theory and equivariant cohomology in terms of (∞,1)-topos theory/homotopy type theory (FSS 12 I, exmp. 4.4):

homotopy type theory	representation theory
pointed connected context $\mathbf{B}G$	∞-group $G$
dependent type on $\mathbf{B}G$	$G$-∞-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 product of internal hom along $\mathbf{B}G \to \ast$	equivariant cohomology
dependent sum along $\mathbf{B}G \to \mathbf{B}H$	induced representation
context extension along $\mathbf{B}G \to \mathbf{B}H$	restricted representation
dependent product along $\mathbf{B}G \to \mathbf{B}H$	coinduced representation
spectrum object in context $\mathbf{B}G$	spectrum with G-action (naive G-spectrum)