nLab
homotopy fixed point

Contents

Context

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

Introductions

Definitions

Paths and cylinders

Homotopy groups

Basic facts

Theorems

Representation theory

Contents

Idea

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

See at

Properties

Presentation

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} \stackrel{\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 X//G BG. \array{ X &\to& X//G \\ && \downarrow \\ && \mathbf{B}G } \,.

This is the same as maps 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 discusison there, there 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 (for instance in the historical discussion of the Sullivan conjecture ).

Homotopy fixed point and 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)

References

Last revised on September 17, 2018 at 06:40:47. See the history of this page for a list of all contributions to it.