If in the context of stable homotopy theory the topological spaces and spectra are equipped with an action of a topological group the theory refines to a -equivariant version. This is to equivariant homotopy theory (roughly) as stable homotopy theory is to homotopy theory.
The definition of -spectrum is typically given in generalization of the definition of coordinate-free spectrum.
A -universe in this context is a 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 -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
… see the references below, for the moment
The characterization of -equivariant functors in terms of topological Mackey-functors is discussed in example 3.4 (i) of
Something on modelling the equivariant stable category using functors on all (nice) -spaces (instead of on just the orbit category) is in
The May recognition theorem? for -spaces and genuine -spectra is discussed in