symmetric monoidal (∞,1)-category of spectra
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.
For $\mathbf{H}$ an (∞,1)-topos, $G \in Grp(\mathbf{H})$ an ∞-group and
an ∞-action of $G$ on $V \in \mathbf{H}$, the type of invariants is the absolute dependent product
The connected components of this is equivalently the group cohomology of $G$ with coefficients in the infinity-module $V$.
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) |