geometric representation theory
representation, 2-representation, ∞-representation
Grothendieck group, lambda-ring, symmetric function, formal group
principal bundle, torsor, vector bundle, Atiyah Lie algebroid
Eilenberg-Moore category, algebra over an operad, actegory, crossed module
Be?linson-Bernstein localization?
group cohomology, nonabelian group cohomology, Lie group cohomology
cohomology with constant coefficients / with a local system of coefficients
differential cohomology
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
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
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)
The forgetful functor from global stable homotopy theory to plain stable homotopy theory exhibits a recollement.
(…)
A comprehensive textbook account is in
Survey (with emphasis on global equivariant bordism homology theory):
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
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:
Last revised on November 21, 2020 at 11:34:23. See the history of this page for a list of all contributions to it.