# nLab homotopy fixed point

Contents

### Context

#### Representation theory

representation theory

geometric 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 $G$ an ∞-group and $X$ equipped with an ∞-action, then the homotopy fixed points is the base change along/dependent product over the delooping $\mathbf{B}G$:

$(-)^{h G} \;\colon\; G Act(\mathbf{H}) \simeq \mathbf{H}_{/\mathbf{B}G} \overset {\;\;\;\prod_{\mathbf{B}G}\;\;\;} {\longrightarrow} \mathbf{H} \,.$

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 $X$-associated ∞-bundle to the universal principal ∞-bundle

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

Such sections

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

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

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

$X^{h G} \simeq R Hom_G(\ast, X) \simeq Hom_G(E G, X) \,.$

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

homotopy type theoryrepresentation 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)