Paths and cylinders
Stable Homotopy theory
Classes of bundles
Examples and Applications
Special and general types
A parameterized spectrum is a bundle of spectra (May-Sigurdsson 06), hence a stable homotopy type in parameterized homotopy theory.
Specifically, for a homotopy type thought of as an ∞-groupoid, then a spectrum parameterized over is equivalently an (∞,1)-functor from to the stable (∞,1)-category of spectra (Ando-Blumberg-Gepner 11): this assigns to each object of a spectrum, to each morphism an equivalence of spectra, to each 2-morphism a homotopy between such equivalences, and so forth.
More generally, given an (∞,1)-topos , then its tangent (∞,1)-topos is the (∞,1)-category of all spectrum objects in parameterized over any object of (an observation promoted by Joyal).
The intrinsic cohomology of such a tangent (∞,1)-topos of parameterized spectra is twisted generalized cohomology in , and generally is twisted bivariant cohomology in .
For more see also at tangent cohesive (∞,1)-topos.
As (co)module spectra
For a connected homotopy type, then the -parameterized spectra are equivalently the module spectra over the ∞-group ∞-ring of the ∞-group corresponding to the loop space.
To see this, use first that -parameterized spectra are equivalently ∞-functors of (∞,1)-categories of the form
from the delooping ∞-groupoid of to the (∞,1)-category of spectra, then use that these are equivalenty Spectrum enriched functors out of the one-object spectrum enriched -category with hom-spectrum .
Moreover -module spectra are equivalent to comodule spectra over the coalgebra induced from the coalgebra object? structure of in the Cartesian monoidal (∞,1)-category ∞Grpd via the diagonal (here), and using that is a strong monoidal functor:
(Hess-Shipley 14, theorem 1.2 with prop. 5.18)
See also at A-theory.
Six operations yoga
For any map between ∞-groupoids, the parameterized spectra form a Wirthmüller context version of the yoga of six functors, in that
in that is not only a strong monoidal functor but also a strong closed functor, hence that Frobenius reciprocity holds.
Moreover, along (co-)spans of morphisms pull-push satisfies the Beck-Chevalley condition (Hopkins-Lurie 14, prop. 4.3.3).
One way to summarize all this is to say that parameterized spectra over ∞Grpd constitute a linear homotopy type theory (Schreiber 14).
In twisted cohomology.
Parameterized spectra over a fixed base (in any suitable model category) are discussed in
- Stefan Schwede, section 3 of Spectra in model categories and applications to the algebraic cotangent complex, Journal of Pure and Applied Algebra 120 (1997) 104 (pdf)
A comprehensive textbook account on parameterized spectra in ∞Grpd Top is in
A formulation of aspects of this in (∞,1)-category theory is in
See also the further references at (∞,1)-module bundle.
Discussion of the Beck-Chevalley condition is in prop. 4.3.3 of
Discussion of the Kozul duality between -module spectra and -comodule spectra is in
Discussion as a linear homotopy type theory is in