nLab
parametrized spectrum

Context

Homotopy theory

Stable Homotopy theory

Bundles

Cohomology

cohomology

Special and general types

Special notions

Variants

Extra structure

Operations

Theorems

Contents

Idea

A parameterized spectrum is a bundle of spectra (May-Sigurdsson 06). Specifically, for XX an ∞-groupoid, then a spectrum parameterized over XX is equivalently an (∞,1)-functor XSpecX \longrightarrow Spec from XX to the stable (∞,1)-category of spectra (Ando-Blumberg-Gepner 11): this assigns to each object of XX 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 H\mathbf{H}, then its tangent (∞,1)-topos THT\mathbf{H} is the (∞,1)-category of all spectrum objects in H\mathbf{H} parameterized over any object of H\mathbf{H} (an observation promoted by Joyal).

The intrinsic cohomology of such a tangent (∞,1)-topos of parameterized spectra is twisted generalized cohomology in H\mathbf{H}, and generally is twisted bivariant cohomology in H\mathbf{H}.

For more see also at tangent cohesive (∞,1)-topos.

Applications

In twisted cohomology.

References

A comprehensive textbook account on parameterized spectra in ∞Grpd \simeq L wheL_{whe}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 (80.237.234.148)