coinvariant

A *coinvariant* is dual to an invariant.

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 $V$

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**.

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

homotopy type theory | representation 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 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 $\mathbf{B}G \to \mathbf{B}H$ | |

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) |

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