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 $\mathbf{H}$ (see also at Joyal locus).

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

• retractive space

• model structure for parameterized spectra

• excisive functor

• tangent (infinity,1)-category, Joyal locus

• cartesian doubly closed monoidal (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

Over a fixed base

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

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 of model structures for parameterized spectra in ∞Grpd $\simeq$ $L_{whe}$Top:

• Peter May, Johann Sigurdsson, Parametrized Homotopy Theory, Mathematical Surveys and Monographs 132, AMS (2006) [arXiv:math/0411656]

• Cary Malkiewich, Parametrized spectra, a low-tech approach [arXiv:1906.04773, user guide: pdf, pdf]

• Cary Malkiewich, A convenient category of parametrized spectra [arXiv:2305.15327]

Alternative formulation in (∞,1)-category theory:

• 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 the Beck-Chevalley condition:

• Michael Hopkins, Jacob Lurie, prop. 4.3.3 in: Ambidexterity in K(n)-Local Stable Homotopy Theory

Discussion of the Koszul 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)

Over varying bases

Discussion of parameterized spectra over varying bases, hence of tangent $\infty$-categories:

Discussion of suitable model structures for parameterized spectra:

• Vincent Braunack-Mayer, Combinatorial parametrised spectra, Algebr. Geom. Topol. 21 (2021) 801-891 [arXiv:1907.08496, doi:10.2140/agt.2021.21.801]

  (based on the PhD thesis, 2018)

• Fabian Hebestreit, Steffen Sagave, Christian Schlichtkrull, Multiplicative parametrized homotopy theory via symmetric spectra in retractive spaces, Forum of Mathematics, Sigma 8 (2020) e16 [arXiv:1904.01824, doi:10.1017/fms.2020.11]

• Eric Finster: A Tour of Parameterized Spectra, talk at Running HoTT 2024, CQTS@NYUAD (April 2024) [video:kt]

Discussion as a linear homotopy type theory:

• Urs Schreiber, Quantization via Linear homotopy types [arXiv:1402.7041]

  (intended categorical semantics)

• Mitchell Riley, Eric Finster, Daniel R. Licata, Synthetic Spectra via a Monadic and Comonadic Modality [arXiv:2102.04099]

  (inference rules for the classical modality $\natural$)

• Mitchell Riley, A Bunched Homotopy Type Theory for Synthetic Stable Homotopy Theory, PhD Thesis (2022) [doi:10.14418/wes01.3.139, ir:3269, pdf]

  (including inference rules for the multiplicative conjunction by bringing in the required bunched logic)