nLab
parametrized spectrum

Idea

A parameterized spectrum is a bundle of spectra (May-Sigurdsson 06), hence a stable homotopy type 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

As (co)module spectra

For $X$ a connected homotopy type, then the $X$-parameterized spectra are equivalently the module spectra over the ∞-group ∞-ring $\mathbb{S}[\Omega X]$ of the ∞-group corresponding to the loop space.

To see this, use first that $X$-parameterized spectra are equivalently ∞-functors of (∞,1)-categories of the form

$$B \Omega X \longrightarrow Spectra$$

from the delooping ∞-groupoid of $\Omega X$ to the (∞,1)-category of spectra, then use that these are equivalenty Spectrum enriched functors out of the one-object spectrum enriched $\infty$-category with hom-spectrum $\mathbb{S}[\Omega X] \simeq \Sigma^\infty_+ \Omega X$.

Moreover $\mathbb{S}[\Omega X]$-module spectra are equivalent to comodule spectra over the coalgebra $\mathbb{S}[X] = \Sigma^\infty_+ X$ induced from the coalgebra object? structure of $X$ in the Cartesian monoidal (∞,1)-category ∞Grpd via the diagonal (here), and using that $\Sigma^\infty$ is a strong monoidal functor:

$$CoModSpectra_{\mathbb{S}[X]} \;\simeq\; ModSpectra_{\mathbb{S}[\Omega X]}$$

(Hess-Shipley 14, theorem 1.2 with prop. 5.18)

See also at A-theory.

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

• excisive functor

• tangent (infinity,1)-category

• sheaf of spectra

• rational parameterized stable homotopy theory

• ex-space

• the K-theory of spectra parameterized over a connected $X$ is the A-theory of $X$.

• twisted differential cohomology

References

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 $\simeq$ $L_{whe}$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, Geom. Topol. 22 (2018) 3761-3825 (arXiv:1112.2203)

Discussion of convenient model category presentations:

• Vincent Braunack-Mayer, Combinatorial parametrised spectra, based on the PhD thesis (arXiv:1907.08496)

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 of the Kozul duality between $\mathbb{S}[\Omega X]$-module spectra and $\mathbb{S}[X]$-comodule spectra is in

• Kathryn Hess, Brooke Shipley, Waldhausen K-theory of spaces via comodules, Advances in Mathematics 290 (2016): 1079-1137 (arXiv:1402.4719)

Discussion as a linear homotopy type theory is in

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