homotopy theory, (∞,1)-category theory, homotopy type theory
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The refinement of the notion of fixed point/invariant to homotopy theory. Also homotopy invariant
See at
Generally, for $G$ an ∞-group and $X$ equipped with an ∞-action, then the homotopy fixed points is the base change along/dependent product over the delooping $\mathbf{B}G$:
By the general discussion at dependent product this is given by forming sections, which by the discussion at ∞-action here means forming sections of the $X$-associated ∞-bundle to the universal principal ∞-bundle
Such sections
are equivalently homotopy-equivariant function out of the point equipped with the trivial $G$-action:
For geometrically discrete ∞-group ∞-actions, hence for $\mathbf{H} =$ ∞Grpd, it is the Borel model structure which presents the homotopy theory. By the discussion there, the derived hom space is computed as the hom-space out of plain actions of simplicial groups out of the total space of the simplicial universal principal bundle $\mathbf{E}G = W G$. Therefore in this case one finds
This is the form in which homotopy fixed points are often defined in traditional literature (going back to Thomason 83 and as in the historical discussion of the Sullivan conjecture).
In the context of complex oriented cohomology theory
… (Hill-Hopkins-Ravenel, theorem 8.4)…
categorical fixed point spectrum?
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) |
The original article:
Further discussion, for groupoids and stacks:
In the context of complex oriented cohomology theory:
Last revised on December 15, 2021 at 17:45:52. See the history of this page for a list of all contributions to it.