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parametrized spectrum

Idea

A parameterized spectrum is a bundle of spectra (May-Sigurdsson 06), hence a stable object? in parameterized homotopy theory.

Specifically, for $X$ a homotopy type thought of as an ∞-groupoid, then a spectrum parameterized over $X$ is equivalently an (∞,1)-functor $X \longrightarrow Spec$ from $X$ to the stable (∞,1)-category of spectra (Ando-Blumberg-Gepner 11): this assigns to each object of $X$ 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 $\mathbf{H}$, then its tangent (∞,1)-topos $T\mathbf{H}$ is the (∞,1)-category of all spectrum objects in $\mathbf{H}$ parameterized over any object of $\mathbf{H}$ (an observation promoted by Joyal).

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

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

Properties

Six operations yoga

For any map $f\colon X\longrightarrow Y$ between ∞-groupoids, the parameterized spectra form a Wirthmüller context version of the yoga of six functors, in that

$$[X,Spectra] \stackrel{\overset{f_!}{\longrightarrow}}{\stackrel{\overset{f^\ast}{\longleftarrow}}{\underset{f_\ast}{\longrightarrow}}} [Y,Spectra]$$

in that $f^\ast$ 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 $(f_!\dashv f^\ast)$ 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).

Applications

In twisted cohomology.

Related concepts

• ex-space

• sheaf of spectra

References

A comprehensive textbook account on parameterized spectra in ∞Grpd $\simeq$ $L_{whe}$Top is in

• Peter May, J. Sigurdsson, Parametrized Homotopy Theory, 2006

A formulation of aspects of this in (∞,1)-category theory is in

• Matthew Ando, Andrew Blumberg, David Gepner, Parametrized spectra, multiplicative Thom spectra, and the twisted Umkehr map (arXiv:1112.2203)

See also the further references at (∞,1)-module bundle.

Discussion of the Beck-Chevalley condition is in prop. 4.3.3 of

• Michael Hopkins, Jacob Lurie, Ambidexterity in K(n)-Local Stable Homotopy Theory

Discussion as a linear homotopy type theory is in

• Urs Schreiber, Quantization via Linear homotopy types (arXiv:1402.7041)