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parametrized spectrum

Context

Homotopy theory

Stable Homotopy theory

Bundles

Cohomology

cohomology

Special and general types

Special notions

Variants

Extra structure

Operations

Theorems

Goodwillie calculus

Contents

Idea

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

Specifically, for XX a homotopy type thought of as 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.

More 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.

Properties

As (co)module spectra

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

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

BΩXSpectra B \Omega X \longrightarrow Spectra

from the delooping ∞-groupoid of ΩX\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 𝕊[ΩX]Σ + ΩX\mathbb{S}[\Omega X] \simeq \Sigma^\infty_+ \Omega X.

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

CoModSpectra 𝕊[X]ModSpectra 𝕊[ΩX] 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:XYf\colon X\longrightarrow Y between ∞-groupoids, the parameterized spectra form a Wirthmüller context version of the yoga of six functors, in that

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

in that f *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 !f *)(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.

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 wheL_{whe}Top is in

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

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

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

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

Discussion as a linear homotopy type theory is in

Revised on March 7, 2017 12:38:59 by Urs Schreiber (192.41.135.245)