It is a classical observation that a based continuous functor
from the
category of finite CW–complexes to the category of based spaces that takes homotopy
pushouts to homotopy pullbacks “represents” a homology theory—the collection of spaces
obtained by evaluating
on spheres yields
an –prespectrum.
Such functors are sometimes referred to as linear or excisive. The
main theorem of this paper provides an equivariant analogue of
this result. We show that a based continuous functor from finite
–CW–complexes
to based –spaces
represents a genuine equivariant homology theory if and only if it takes
–homotopy pushouts
to –homotopy
pullbacks and satisfies an additional condition requiring compatibility with Atiyah duality for
orbit spaces .
Our motivation for this work is the development of a recognition principle for
equivariant infinite loop spaces. In order to make the connection to infinite loop space
theory precise, we reinterpret the main theorem as providing a fibrancy condition in
an appropriate model category of spectra. Specifically, we situate this result in the
context of the study of equivariant diagram spectra indexed on the category
of based
–spaces homeomorphic to
finite –CW–complexes
for a compact Lie group .
Using the machinery of Mandell–May–Schwede–Shipley, we show that there
is a stable model structure on this category of diagram spectra which
admits a monoidal Quillen equivalence to the category of orthogonal
–spectra.
We construct a second “absolute” stable model structure which is Quillen equivalent
to the stable model structure. There is a model-theoretic identification of the
fibrant continuous functors in the absolute stable model structure as functors
such that for
the collection
forms an
––prespectrum
as varies over
the universe .
Thus, our main result provides a concrete identification of the fibrant objects in the
absolute stable model structure.
This description of fibrant objects in the absolute stable model structure makes
it clear that in the equivariant setting we cannot hope for a comparison
between the category of equivariant continuous functors and equivariant
–spaces,
except when
is finite. We provide an explicit analysis of the failure of the category of equivariant
–spaces to model
connective –spectra,
even for .
Keywords
equivariant infinite loop space theory, excisive functors,
Atiyah duality
