nLab global equivariant stable homotopy theory

Idea

What is called global equivariant stable homotopy theory is a variant of equivariant stable homotopy theory where spectra are equipped with $G$-infinity-actions “for all compact Lie groups $G$ at once”.

Often this is referred to just as “global stable homotopy theory” or even just “global homotopy theory”. But there is also unstable global equivariant homotopy theory.

More precisely, given an orthogonal spectrum $X$, then every representation $\rho \colon G \to O(n)$ of a compact Lie group on the Cartesian space $\mathbb{R}^n$ by orthogonal group actions induces a $G$-equivariant spectrum and hence a notion of $G$-equivariant homotopy groups.

One says that a morphism of orthogonal spectra is a global equivariant equivalence if it induces isomorphisms on all $G$- equivariant homotopy groups, for all $G$, this way. (This definition appears for instance as (Schwede 13, def. 2.9), there referred to just as “global equivalence”. See also at equivariant Whitehead theorem.)

The global equivariant stable homotopy category $\mathcal{GH}$ is the (simplicial) localization of the category of orthogonal spectra at these global equivariant equivalences (this is a stable (infinity,1)-category/triangulated category.)

Since a global equivariant equivalence is in particular an ordinary weak homotopy equivalence of spectra, there is a canonical functor

$$U \;\colon\; \mathcal{GH} \longrightarrow \mathcal{SH}$$

from the global equivariant to the ordinary stable homotopy category.

This functor has a (derived) left adjoint and right adjoint, which are both full and faithful functors, hence which exhibit discrete object/codiscrete object structure

$$\array{ \mathcal{GH} &\stackrel{\overset{L}{\leftarrow}}{\stackrel{\overset{U}{\longrightarrow}}{\underset{R}{\leftarrow}}}& \mathcal{SH} }$$

on stable/triangulated categories (Schwede 13, theorem IV 5.2)

Global Borel-type equivariant cohomology is in the image of the right adjoint $R$ (Schwede 13, example IV 5.12)

Properties

Relation to plain stable homotopy theory

The forgetful functor from global stable homotopy theory to plain stable homotopy theory exhibits a recollement.

(…)

Examples

• global equivariant sphere spectrum

Related concepts

• global family

References

A comprehensive textbook account is in

• Stefan Schwede, Global homotopy theory (arXiv:1802.09382)

Survey (with emphasis on global equivariant bordism homology theory):

• Stefan Schwede, Equivariant bordism from the global perspective, 2015 (pdf, pdf)

Original articles are

• L. Gaunce Lewis, Jr., Peter May, M. Steinberger, chapter II of Equivariant stable homotopy theory. Lecture Notes in Mathematics, 1213, Springer-Verlag, 1986

• John Greenlees, Peter May, section 5 of Localization and completion theorems for $MU$-module spectra Ann. of Math. (2) 146 (1997), 509-544.

Discussion specifically in terms of equivariant orthogonal spectra is in

• Anna Marie Bohmann, Global Orthogonal Spectra (arXiv:1208.4997)

Discussion for collections of finite subgroups includes

• Wolfgang Lueck, The Burnside Ring and Equivariant Stable Cohomotopy for Infinite Groups (arXiv:math/0504051)

• Irakli Patchkoria, Proper equivariant stable homotopy and virtual cohomological dimension, in Oberwolfach Report No. 14/2015 (pdf)

• Markus Hausmann, Symmetric spectra model global homotopy theory of finite groups (arXiv:1509.09270)

The example of algebraic K-theory:

• Stefan Schwede, Global algebraic K-theory (arXiv:1912.08872)