Equivariant stable homotopy theory over some topological group is the stable homotopy theory of G-spectra. This includes the naive G-spectra which constitute the actual stabilization of equivariant homotopy theory, but is more general.
|Borel equivariant cohomology||general (Bredon) equivariant cohomology||non-equivariant cohomology with homotopy fixed point coefficients|
|trivial action on coefficients||trivial action on domain space|
A G-universe in this context is (e.g. Greenlees-May, p. 10) an infinite dimensional real inner product space equipped with a linear -action that is the direct sum of countably many copies of a given set of (finite dimensional? -DMR) representations of , at least containing the trivial representation on (so that contains at least a copy of ).
Each such subspace of (representation contained in ? -DMR) is called an indexing space . For indexing spaces, write for the orthogonal complement of in . Write for the one-point compactification of and for any (pointed) topological space write for the corresponding (based) sphere space.
A G-space in the following means a pointed topological space equipped with a continuous action of the topological group that fixes the base point. A morphism of -spaces is a continuous map that fixes the basepoints and is -equivariant.
A weak equivalence of -spaces is a morphism that induces isomorphism on all -fixed homotopy groups (…)
A -spectrum (indexed on the chosen universe ) is
for each indexing space a -space ;
for each pair of indexing spaces a -equivariant homeomorphism
A morphism of -spectra is for each indexing space a morphism of -spaces , such that this makes the obvious diagrams commute.
This becomes a category with weak equivalences by setting:
a morphism of -spectra is a weak equivalence of -spectra if for every indexing space the component is a weak equivalence of -spaces (as defined above).
This may be expressed directly in terms of the notion of homotopy group of a -spectrum: this is
A Mackey functor with values in spectra (“spectral Mackey functor”) is an (∞,1)-functor on a suitable (∞,1)-category of correspondences which sends coproducts to smash product. (This is similar to the concept of sheaf with transfer.)
For a finite group and its category of permutation representations, then is a genuine -equivariant spectrum (Guillou-May 11). So in this case the homotopy theory of spectral Mackey functors is a presentation for equivariant stable homotopy theory (Guillou-May 11, Barwick 14).
Introductions and surveys include
and a more modern version taking into account the theory of symmetric monoidal categories of spectra is in
An alternative perspective on this is in
The characterization of -equivariant functors in terms of topological Mackey functors is discussed in example 3.4 (i) of
A comprehensive construction of equivariant stable homotopy theory in terms of Mackey functors is in the series
Permutative -categories in equivariant infinite loop space theory (arXiv:1207.3459)
A fully (∞,1)-category theoretic formulation is i