nLab parametrized spectrum

Contents

Context

Homotopy theory

homotopy theory, (∞,1)-category theory, homotopy type theory

flavors: stable, equivariant, rational, p-adic, proper, geometric, cohesive, directed

models: topological, simplicial, localic, …

see also algebraic topology

Introductions

Definitions

Paths and cylinders

Homotopy groups

Basic facts

Theorems

Stable Homotopy theory

Bundles

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} (see also at Joyal locus).

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

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 wheL_{whe}Top:

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

Discussion of the Beck-Chevalley condition:

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

Over varying bases

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

Discussion of suitable model structures for parameterized spectra:

Discussion as a linear homotopy type theory:

Last revised on May 25, 2023 at 10:35:50. See the history of this page for a list of all contributions to it.