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\begin{document}

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\section*{smooth spectrum}

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

\hypertarget{cohesive_toposes}{}\paragraph*{{Cohesive $\infty$-Toposes}}\label{cohesive_toposes}

[[!include cohesive infinity-toposes - contents]]

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

[[!include stable homotopy theory - contents]]

\hypertarget{differential_cohomology}{}\paragraph*{{Differential cohomology}}\label{differential_cohomology}

[[!include differential cohomology - 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{FromChainComplexesOfSmoothModules}{From chain complexes of smooth modules}\dotfill \pageref*{FromChainComplexesOfSmoothModules} \linebreak
\noindent\hyperlink{examples}{Examples}\dotfill \pageref*{examples} \linebreak
\noindent\hyperlink{ExamplesDeRhamSpectra}{De Rham spectra}\dotfill \pageref*{ExamplesDeRhamSpectra} \linebreak
\noindent\hyperlink{algebraic_ktheory_of_smooth_manifolds}{Algebraic K-theory of smooth manifolds}\dotfill \pageref*{algebraic_ktheory_of_smooth_manifolds} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak


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

A [[sheaf of spectra]] on the [[site]] of all [[smooth manifolds]] may be thought of as a [[spectrum]] equipped with generalized [[smooth structure]], in just the same way as an [[(∞,1)-sheaf]] on this site may be thought of as a [[smooth ∞-groupoid]]. Therefore one might speak of the [[stable (∞,1)-category]]

\begin{displaymath}
Sh_\infty(SmoothMfd, Spectra)
  \simeq
  Stab(Sh_\infty(SmoothMfd))
  =
  Stab(Smooth \infty Grpd)
\end{displaymath}
which is the [[stabilization]] of that of [[smooth ∞-groupoids]] as being the $\infty$-category of \emph{smooth spectra}, just as the [[stable (∞,1)-category of spectra]] itself is the [[stabilization]] of that of bare [[∞-groupoids]].

Together with [[smooth ∞-groupoids]] smooth spectra sit inside the [[tangent cohesive (∞,1)-topos]] over smooth manifolds. By the discussion \href{tangent+cohesive+∞-topos#CohesiveAndDifferentialRefinement}{there}, every smooth spectrum sits in a hexagonal [[differential cohomology diagram]] which exhibits it (\hyperlink{BunkeNikolausVoelkl13}{Bunke-Nikolaus-V\"o{}lkl 13}) as the moduli of a generalized [[differential cohomology]] theory (in generalization of how every ordinary spectrum, via the [[Brown representability theorem]], corresponds to a bare [[generalized (Eilenberg-Steenrod) cohomology theory]]).

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

\hypertarget{FromChainComplexesOfSmoothModules}{}\subsubsection*{{From chain complexes of smooth modules}}\label{FromChainComplexesOfSmoothModules}

\begin{defn}
\label{}\hypertarget{}{}
Write

\begin{itemize}%
\item $Smooth0Type \coloneqq Sh(SmthMfd)$ for the [[topos]] of [[smooth spaces]];


\item $\mathbf{R} \in Smooth0Type$ for the sheaf of [[real number]]-valued [[smooth functions]] (the canonical [[line object]] in $Smooth0Type$);


\item $\mathbf{R} Mod$ for the category of [[abelian sheaves]] over smooth manifolds which are $\mathbf{R}$-[[modules]].



\end{itemize}
\end{defn}
\begin{defn}
\label{RModulesAsInfinityPresheaves}\hypertarget{RModulesAsInfinityPresheaves}{}
Let $C_\bullet \in Ch_\bullet(\mathbf{R}Mod)$ be a [[chain complex]] (unbounded) of [[abelian sheaves]] of $\mathbf{R}$-modules. Via the projective [[model structure on functors]] this defines an [[(∞,1)-presheaf]] of [[chain complexes]]

\begin{displaymath}
Ch_\bullet(\mathbf{R}Mod)
  \longrightarrow
  Sh(SmthMfd, Ch_{\bullet})
  \longrightarrow
  L_{qi} PSh(SmthMfd, Ch_\bullet)
  \simeq
  PSh_\infty(SmthMfs, Ch_\bullet)
  \,.
\end{displaymath}
We still write $C_\bullet\in PSh_\infty(SmthMfd, Ch_\bullet)$ for this [[(∞,1)-presheaf]] of chain complexes.

\end{defn}
\begin{prop}
\label{}\hypertarget{}{}
Under the [[stable Dold-Kan correspondence]]

\begin{displaymath}
DK \;\colon\; Ch_\bullet \longrightarrow Spectra
\end{displaymath}
a chain complex of $\mathbf{R}$-modules $C_\bullet \in Ch_\bullet(\mathbf{R}Mod)$, regarded as an [[(∞,1)-presheaf]] of spectra on $SmthMfd$ as in def. \ref{RModulesAsInfinityPresheaves}, is already an [[(∞,1)-sheaf]], hence a smooth spectrum (i.e. without further [[∞-stackification]]).

\end{prop}
This appears as (\hyperlink{BunkeNikolausVoelkl13}{Bunke-Nikolaus-V\"o{}lkl 13, lemma 7.12}).

\hypertarget{examples}{}\subsection*{{Examples}}\label{examples}

\hypertarget{ExamplesDeRhamSpectra}{}\subsubsection*{{De Rham spectra}}\label{ExamplesDeRhamSpectra}

Write $Ch_\bullet$ for the [[(∞,1)-category of chain complexes]] (of [[abelian groups]], hence over the [[ring]] $\mathbb{Z}$ of [[integers]]). It is convenient to choose for $A_\bullet \in Ch_\bullet$ the grading convention

\begin{displaymath}
\itexarray{
    \vdots
    \\
    \downarrow
    \\
    A_{-1}
    \\
    \downarrow
    \\
    A_0
    \\
    \downarrow
    \\
    A_1
    \\
    \downarrow
    \\
    \vdots
   }
\end{displaymath}
such that under the [[stable Dold-Kan correspondence]]

\begin{displaymath}
DK \;\colon\; Ch_\bullet \stackrel{}{\longrightarrow} Spectra
\end{displaymath}
the [[homotopy groups]] of spectra relate to the [[homology groups]] by

\begin{displaymath}
\pi_n(DK(A_\bullet)) \simeq  H_{-n}(A_\bullet)
  \,.
\end{displaymath}
In particular for $A \in$ [[Ab]] an [[abelian group]] then $A[n]$ denotes the chain complex concentrated on $A$ in degree $-n$ in this counting.

The grading is such as to harmonize well with the central example of a sheaf of chain complexes over the site of [[smooth manifolds]], which is the [[de Rham complex]], regarded as a [[smooth spectrum]] via the discussion at \emph{\href{smooth+spectrum#FromChainComplexesOfSmoothModules}{smooth spectrum -- from chain complexes of smooth modules}}

\begin{displaymath}
\Omega^\bullet \in Sh_\infty(SmthMfd, Ch_\bullet) \longrightarrow Sh_\infty(SmthMfd, Spectra) \hookrightarrow T \mathbf{H}
\end{displaymath}
\begin{displaymath}
\Omega^{\bullet} \;\colon\;  X\mapsto (\cdots \to 0 \to 0 \to \Omega^0(X) \stackrel{\mathbf{d}}{\to} \Omega^1(X)\stackrel{\mathbf{d}}{\to} \cdots)
\end{displaymath}
with $\Omega^0(X) = C^\infty(X, \mathbb{R})$ in degree 0.

We also need for $n \in \mathbb{N}$ the truncated sheaf of complexes

\begin{displaymath}
\Omega^{\bullet \geq n} \in Sh_\infty(SmthMfd, Ch_\bullet) \longrightarrow Sh_\infty(SmthMfd, Spectra) \hookrightarrow T \mathbf{H}
\end{displaymath}
\begin{displaymath}
\Omega^{\bullet \geq n} \;\colon\;  X\mapsto (\cdots \to 0 \to 0 \to \Omega^n(X) \stackrel{\mathbf{d}}{\to} \Omega^{n+1}(X)\stackrel{\mathbf{d}}{\to} \cdots)
\end{displaymath}
with $\Omega^n(X)$ in degree $n$.

More genereally, for $C \in Ch_\bullet$ any [[chain complex]], there is $(\Omega \otimes C)^{\bullet \geq n}$ given over each manifold $X$ by the [[tensor product of chain complexes]] followed by truncation.

Hence

\begin{displaymath}
(\Omega \otimes C)^{\bullet \geq n}
  = 
  (\cdots \to 0 \to 0 \to \oplus_{k \in \mathbb{N}} \Omega^{k}(X) \otimes C_{n-k} \stackrel{\mathbf{d} \pm d_{C}}{\to} \oplus_{k \in \mathbb{N}} \Omega^{k}(X) \otimes C_{n-k+1}\stackrel{\mathbf{d}\pm d_{C}}{\to} \cdots)
 \,.
\end{displaymath}
\hypertarget{algebraic_ktheory_of_smooth_manifolds}{}\subsubsection*{{Algebraic K-theory of smooth manifolds}}\label{algebraic_ktheory_of_smooth_manifolds}

see at \emph{[[algebraic K-theory of smooth manifolds]]}

\hypertarget{references}{}\subsection*{{References}}\label{references}

\begin{itemize}%
\item [[Ulrich Bunke]], [[Thomas Nikolaus]], [[Michael Völkl]], \emph{Differential cohomology theories as sheaves of spectra} (\href{http://arxiv.org/abs/1311.3188}{arXiv:1311.3188})

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



\end{document}