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\section*{parametrized spectrum}

\hypertarget{context}{}\subsubsection*{{Context}}\label{context}

\hypertarget{homotopy_theory}{}\paragraph*{{Homotopy theory}}\label{homotopy_theory}

[[!include homotopy - contents]]

\hypertarget{stable_homotopy_theory}{}\paragraph*{{Stable Homotopy theory}}\label{stable_homotopy_theory}

[[!include stable homotopy theory - contents]]

\hypertarget{bundles}{}\paragraph*{{Bundles}}\label{bundles}

[[!include bundles - contents]]

\hypertarget{cohomology}{}\paragraph*{{Cohomology}}\label{cohomology}

[[!include cohomology - contents]]

\hypertarget{goodwillie_calculus}{}\paragraph*{{Goodwillie calculus}}\label{goodwillie_calculus}

[[!include Goodwillie calculus - contents]]

\hypertarget{contents}{}\section*{{Contents}}\label{contents}

\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{properties}{Properties}\dotfill \pageref*{properties} \linebreak
\noindent\hyperlink{as_comodule_spectra}{As (co)module spectra}\dotfill \pageref*{as_comodule_spectra} \linebreak
\noindent\hyperlink{SixOperationsYoga}{Six operations yoga}\dotfill \pageref*{SixOperationsYoga} \linebreak
\noindent\hyperlink{applications}{Applications}\dotfill \pageref*{applications} \linebreak
\noindent\hyperlink{related_concepts}{Related concepts}\dotfill \pageref*{related_concepts} \linebreak
\noindent\hyperlink{References}{References}\dotfill \pageref*{References} \linebreak


\hypertarget{idea}{}\subsection*{{Idea}}\label{idea}

A \emph{parameterized spectrum} is a [[bundle]] of [[spectra]] (\hyperlink{MaySigurdsson06}{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]] (\hyperlink{AndoBlumbergGepner11}{Ando-Blumberg-Gepner 11}): this assigns to each [[object]] of $X$ a [[spectrum]], to each [[morphism]] an [[equivalence in an (infinity,1)-category|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 \emph{[[tangent cohesive (∞,1)-topos]]}.

\hypertarget{properties}{}\subsection*{{Properties}}\label{properties}

\hypertarget{as_comodule_spectra}{}\subsubsection*{{As (co)module spectra}}\label{as_comodule_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

\begin{displaymath}
B \Omega X \longrightarrow Spectra
\end{displaymath}
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]] (\href{cartesian+monoidal+infinity%2C1-category#CoalgebraObjects}{here}), and using that $\Sigma^\infty$ is a [[strong monoidal functor]]:

\begin{displaymath}
CoModSpectra_{\mathbb{S}[X]}
  \;\simeq\;
   ModSpectra_{\mathbb{S}[\Omega X]}
\end{displaymath}
(\hyperlink{HessShipley14}{Hess-Shipley 14, theorem 1.2 with prop. 5.18})

See also at \emph{[[A-theory]]}.

\hypertarget{SixOperationsYoga}{}\subsubsection*{{Six operations yoga}}\label{SixOperationsYoga}

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

\begin{displaymath}
[X,Spectra]
    \stackrel{\overset{f_!}{\longrightarrow}}{\stackrel{\overset{f^\ast}{\longleftarrow}}{\underset{f_\ast}{\longrightarrow}}}
  [Y,Spectra]
\end{displaymath}
in that $f^\ast$ is not only a [[strong monoidal functor]] but also a [[strong closed functor]], hence that [[Frobenius reciprocity]] holds.

Moreover, along ([[cospan|co-]])[[spans]] of morphisms pull-push $(f_!\dashv f^\ast)$ satisfies the [[Beck-Chevalley condition]] (\hyperlink{HopkinsLurie14}{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]] (\hyperlink{Schreiber14}{Schreiber 14}).

\hypertarget{applications}{}\subsection*{{Applications}}\label{applications}

In [[twisted cohomology]].

\hypertarget{related_concepts}{}\subsection*{{Related concepts}}\label{related_concepts}

\begin{itemize}%
\item [[excisive functor]]


\item [[tangent (infinity,1)-category]]


\item [[sheaf of spectra]]


\item [[rational parameterized stable homotopy theory]]


\item [[ex-space]]


\item the [[K-theory]] of spectra parameterized over a connected $X$ is the \emph{[[A-theory]]} of $X$.


\item [[twisted differential cohomology]]



\end{itemize}
\hypertarget{References}{}\subsection*{{References}}\label{References}

Parameterized spectra over a fixed base (in any suitable [[model category]]) are discussed in

\begin{itemize}%
\item [[Stefan Schwede]], section 3 of \emph{Spectra in model categories and applications to the algebraic cotangent complex}, Journal of Pure and Applied Algebra 120 (1997) 104 (\href{http://www.math.uni-bonn.de/people/schwede/modelspec.pdf}{pdf})

\end{itemize}
A comprehensive textbook account on parameterized spectra in [[∞Grpd]] $\simeq$ $L_{whe}$[[Top]] is in

\begin{itemize}%
\item [[Peter May]], J. Sigurdsson, \emph{[[Parametrized Homotopy Theory]]}, 2006

\end{itemize}
A formulation of aspects of this in [[(∞,1)-category theory]] is in

\begin{itemize}%
\item [[Matthew Ando]], [[Andrew Blumberg]], [[David Gepner]], \emph{Parametrized spectra, multiplicative Thom spectra, and the twisted Umkehr map}, Geom. Topol. 22 (2018) 3761-3825 (\href{http://arxiv.org/abs/1112.2203}{arXiv:1112.2203})

\end{itemize}
Discussion of convenient [[model category]] [[presentable (infinity,1)-category|presentations]]:

\begin{itemize}%
\item [[Vincent Braunack-Mayer]], \emph{Combinatorial parametrised spectra}, based on the [[schreiber:thesis Braunack-Mayer|PhD thesis]] (\href{https://arxiv.org/abs/1907.08496}{arXiv:1907.08496})

\end{itemize}
See also the further references at \emph{[[(∞,1)-module bundle]]}.

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

\begin{itemize}%
\item [[Michael Hopkins]], [[Jacob Lurie]], \emph{[[Ambidexterity in K(n)-Local Stable Homotopy Theory]]}

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

\begin{itemize}%
\item [[Kathryn Hess]], [[Brooke Shipley]], \emph{Waldhausen K-theory of spaces via comodules}, Advances in Mathematics 290 (2016): 1079-1137 (\href{https://arxiv.org/abs/1402.4719}{arXiv:1402.4719})

\end{itemize}
Discussion as a [[linear homotopy type theory]] is in

\begin{itemize}%
\item [[Urs Schreiber]], \emph{[[schreiber:Quantization via Linear homotopy types]]} (\href{http://arxiv.org/abs/1402.7041}{arXiv:1402.7041})

\end{itemize}
[[!redirects parametrized spectra]]

[[!redirects parameterized spectrum]] [[!redirects parameterized spectra]]

[[!redirects parametrised spectrum]] [[!redirects parametrised spectra]]

[[!redirects spectrum bundle]] [[!redirects spectrum bundles]]

[[!redirects bundle of spectra]] [[!redirects bundles of spectra]]



\end{document}