nLab homotopy fixed point



Homotopy theory

homotopy theory, (∞,1)-category theory, homotopy type theory

flavors: stable, equivariant, rational, p-adic, proper, geometric, cohesive, directed

models: topological, simplicial, localic, …

see also algebraic topology



Paths and cylinders

Homotopy groups

Basic facts


Representation theory



The refinement of the notion of fixed point/invariant to homotopy theory. Also homotopy invariant

See at



Generally, for GG an ∞-group and XX equipped with an ∞-action, then the homotopy fixed points is the base change along/dependent product over the delooping BG\mathbf{B}G:

() hG:GAct(H)H /BG BGH. (-)^{h G} \;\colon\; G Act(\mathbf{H}) \simeq \mathbf{H}_{/\mathbf{B}G} \overset {\;\;\;\prod_{\mathbf{B}G}\;\;\;} {\longrightarrow} \mathbf{H} \,.

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 XX-associated ∞-bundle to the universal principal ∞-bundle

X XG BG. \array{ X &\longrightarrow& X \sslash G \\ && \big\downarrow \\ && \mathbf{B}G \,. }

Such sections

are equivalently homotopy-equivariant function out of the point equipped with the trivial GG-action:

X hGGAct(*,X). X^{h G} \simeq G Act(\ast, X) \,.

For geometrically discrete ∞-group ∞-actions, hence for H=\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 EG=WG\mathbf{E}G = W G. Therefore in this case one finds

X hGRHom G(*,X)Hom G(EG,X). X^{h G} \simeq R Hom_G(\ast, X) \simeq Hom_G(E G, X) \,.

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).

Relation to periodicity theorems

In the context of complex oriented cohomology theory

… (Hill-Hopkins-Ravenel, theorem 8.4)…

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

homotopy type theoryrepresentation theory
pointed connected context BG\mathbf{B}G∞-group GG
dependent type on BG\mathbf{B}GGG-∞-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}Hrestricted representation
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)


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.