homotopy theory, (∞,1)-category theory, homotopy type theory
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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.
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
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:
(Hess-Shipley 14, theorem 1.2 with prop. 5.18)
See also at A-theory.
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
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).
See also the further references at (∞,1)-module bundle.
Parameterized spectra over a fixed base (in any suitable model category) are discussed in
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:
Discussion of the Beck-Chevalley condition:
Discussion of the Koszul duality between $\mathbb{S}[\Omega X]$-module spectra and $\mathbb{S}[X]$-comodule spectra is in
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]
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)
Last revised on May 25, 2023 at 10:35:50. See the history of this page for a list of all contributions to it.