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A parameterized spectrum is a bundle of spectra (May-Sigurdsson 06). Specifically, for 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.
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.
In twisted cohomology.
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 references at (∞,1)-module bundle.
Revised on October 14, 2013 04:30:30
by Urs Schreiber