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\begin{document}

%-------------------------------------------------------------------


\section*{tangent cohesive (∞,1)-topos}

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

\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{properties}{Properties}\dotfill \pageref*{properties} \linebreak
\noindent\hyperlink{StableExtensionOfCohesion}{Stable extension of cohesion}\dotfill \pageref*{StableExtensionOfCohesion} \linebreak
\noindent\hyperlink{stable_homotopy_types}{Stable homotopy types}\dotfill \pageref*{stable_homotopy_types} \linebreak
\noindent\hyperlink{Cohomology}{Cohomology -- Twisted bivariant generalized geometric cohomology theory}\dotfill \pageref*{Cohomology} \linebreak
\noindent\hyperlink{CohesiveAndDifferentialRefinement}{Cohesive and differential refinement}\dotfill \pageref*{CohesiveAndDifferentialRefinement} \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}

The [[tangent (∞,1)-category]] $T\mathbf{H}$ to a [[cohesive (∞,1)-topos]] is itself cohesive: the \emph{tangent cohesive (∞,1)-topos}.

This $T \mathbf{H}$ is the $\infty$-topos of [[parameterized spectra]] in $\mathbf{H}$, hence is the context for cohesive [[stable homotopy theory]].

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

\hypertarget{StableExtensionOfCohesion}{}\subsubsection*{{Stable extension of cohesion}}\label{StableExtensionOfCohesion}

Let $\mathbf{H}$ be a [[cohesive (∞,1)-topos]].

By the discussion at \emph{\href{tangent+%28infinity%2C1%29-category#TangentTopos}{tangent ∞-category -- Examples -- Of an ∞-topos}} the tangent $\infty$-topos $T \mathbf{H}$ constitutes an [[extension]] of $\mathbf{H}$ by its [[stabilization]] $Stab(\mathbf{H})$:

\begin{displaymath}
\itexarray{
    && Stab(\mathbf{H})
    &
    \stackrel{\overset{L\Pi^{seq}}{\longrightarrow}}{\stackrel{\overset{Disc^{seq}}{\leftarrow}}{\stackrel{\overset{\Gamma^{seq}}{\longrightarrow}}{\underset{coDisc^{seq}}{\leftarrow}}}}
    &
    Stab(\infty Grpd) \simeq Spectra
    \\
    && \simeq && \simeq
    \\
    && T_\ast \mathbf{H} && T_\ast \infty Grpd
    \\
    && \downarrow^{\mathrlap{incl}} && \downarrow^{\mathrlap{incl}}
    \\
    \mathbf{H}
    &\stackrel{\overset{d}{\longrightarrow}}{\underset{\Omega^\infty \circ tot}{\leftarrow}}&
    T \mathbf{H}
    &
    \stackrel{\overset{L\Pi^{seq}}{\longrightarrow}}{\stackrel{\overset{Disc^{seq}}{\leftarrow}}{\stackrel{\overset{\Gamma^{seq}}{\longrightarrow}}{\underset{coDisc^{seq}}{\leftarrow}}}}
  &
  T \infty Grpd
  \\
  &&
  {}^{\mathllap{base}}\downarrow \uparrow^{\mathrlap{0}} 
  && 
  {}^{\mathllap{base}}\downarrow \uparrow^{\mathrlap{0}} 
  \\
  &&
    \mathbf{H}
    &
    \stackrel{\overset{\Pi}{\longrightarrow}}{\stackrel{\overset{Disc}{\leftarrow}}{\stackrel{\overset{\Gamma}{\longrightarrow}}{\underset{coDisc}{\leftarrow}}}}
  &
  \infty Grpd
  }
  \,.
\end{displaymath}
Here

\begin{itemize}%
\item $\Omega^\infty \circ tot \;\colon\; T \mathbf{H} \longrightarrow \mathbf{H}$ assigns the \emph{total space} of a spectrum bundle;

its [[left adjoint]] is the \href{tangent+%28infinity%2C1%29-category#CotangentComplex}{tangent complex functor};


\item $base \;\colon\; T \mathbf{H} \longrightarrow \mathbf{H}$ assigns the \emph{base space} of a spectrum bundle;

its [[left adjoint]] produces the 0-bundle;

together these exhibit $T \mathbf{H}$ as an [[infinitesimal cohesive (infinity,1)-topos]] over $\mathbf{H}$.



\end{itemize}
\hypertarget{stable_homotopy_types}{}\subsubsection*{{Stable homotopy types}}\label{stable_homotopy_types}

In a tangent cohesive $\infty$-topos $T \mathbf{H}$ all the [[homotopy types]] in $T_\ast \mathbf{H} \hookrightarrow T\mathbf{H}$ are [[stable homotopy types]].

\hypertarget{Cohomology}{}\subsubsection*{{Cohomology -- Twisted bivariant generalized geometric cohomology theory}}\label{Cohomology}

Where the [[(∞,1)-categorical hom-space]] in a general [[(∞,1)-topos]] constitute a notion of [[cohomology]], those of a [[tangent (∞,1)-topos]] specifically constitute [[twisted generalized cohomology]], in fact [[twisted bivariant cohomology]].

For consider a [[spectrum object]] $E \in T_\ast \mathbf{H}$ and write $GL_1(E) \in Grp(\mathbf{H})$ for its [[∞-group of units]]. Then the [[∞-action]] of this on $E$ is (by the discussion there) exhibited by an object

\begin{displaymath}
\left(
   \itexarray{
     E//GL_1(E)
     \\
     \downarrow
     \\
     \mathbf{B}GL_1(E)
   }
  \right)
  \;\;\;
   \in
  \;\;\;
  T_{\mathbf{B}GL_1(E)}\mathbf{H} \hookrightarrow T\mathbf{H}
  \,.
\end{displaymath}
More generally, for $Pic(E) \in \mathbf{H}$ the [[Picard ∞-groupoid]] of $E$ there is the universal [[(∞,1)-line bundle]]

\begin{displaymath}
(\widehat{Pic(E)} \to Pic(E)) \in T \mathbf{H}
  \,.
\end{displaymath}
Now for any object $X \in \mathbf{H}$ we have the trivial [[sphere spectrum]] [[spectrum bundle]] over $X$

\begin{displaymath}
X \times \mathbb{S}
  \simeq
  \left(
   \itexarray{
     X \times \mathbb{S}
     \\
     \downarrow
     \\
     X
   }
  \right)
  \;\;\;
   \in
  \;\;\;
  T_{X}\mathbf{H} \hookrightarrow T\mathbf{H}
  \,.
\end{displaymath}
then morphisms in $T \mathbf{H}$ from the latter to the former

\begin{displaymath}
\left(
   \itexarray{
     X \times \mathbb{S}
     \\
     \downarrow
     \\
     X
   }
  \right)
  \longrightarrow
  \left(
   \itexarray{
     E//GL_1(E)
     \\
     \downarrow
     \\
     \mathbf{B}GL_1(E)
   }
  \right)
\end{displaymath}
are equivalently [[homotopy]] [[commuting diagrams]] of the form

\begin{displaymath}
\itexarray{
    X \times \mathbb{S} &\stackrel{\sigma}{\longrightarrow}& E//GL_1(E)
    \\
    \downarrow && \downarrow
    \\
    X &\stackrel{\chi}{\longrightarrow}& \mathbf{B}GL_1(E)
  }
\end{displaymath}
and hence

\begin{enumerate}%
\item a choice of [[twisted cohomology|twist of E-cohomology]] $\chi \;\colon \; X  \longrightarrow \mathbf{B}GL_1(E)$, modulating a $GL_1(E)$-[[principal ∞-bundle]];


\item an element in the $\chi$-twisted $E$-cohomology of $X$, $\sigma \in E^{\bullet + \chi}(X,E)$, hence a [[section]] of the [[associated ∞-bundle|associated]] [[(∞,1)-line bundle]].



\end{enumerate}
If we consider the [[internal hom]] instead of the external [[(∞,1)-categorical hom space]] then things work even more nicely and we can use just $X$ instead of $X \times \mathbb{S}$:

\begin{prop}
\label{MappingSpectrumOutOfUnstableTypeAsCartesianInternalHom}\hypertarget{MappingSpectrumOutOfUnstableTypeAsCartesianInternalHom}{}
For $X \in \mathbf{H} \stackrel{0}{\hookrightarrow} T \mathbf{H}$ a [[geometric homotopy type]] and $E \in Stab(\mathbf{H}) \simeq T_\ast \mathbf{H} \hookrightarrow T \mathbf{H}$ a [[spectrum object]], then the [[internal hom]]/[[mapping stack]]

\begin{displaymath}
[X,E]_{T \mathbf{H}} \in T \mathbf{H}
\end{displaymath}
(with respect to the Cartesian [[closed monoidal (∞,1)-category]] structure on the [[(∞,1)-topos]] is equivalently the [[mapping spectrum]]

\begin{displaymath}
[\Sigma^\infty X, E]_{Stab(\mathbf{H})} \in Stab(\mathbf{H}) \hookrightarrow T \mathbf{H}
  \,,
\end{displaymath}
in that

\begin{displaymath}
[X,E]_{T \mathbf{H}} \simeq [\Sigma^\infty X,E]_{Stab(\mathbf{H})}
  \,.
\end{displaymath}
\end{prop}
\begin{proof}
Notice that as an object of $T \mathbf{H} \hookrightarrow \mathbf{H}^{seq}$, the object $X$ is the constant [[(∞,1)-presheaf]] on $seq$. By the formula for the [[internal hom]] in an [[(∞,1)-category of (∞,1)-presheaves]] we have

\begin{displaymath}
[X,E]_\bullet \simeq \mathbf{H}^{seq}(X \times \bullet, E)
  \,.
\end{displaymath}
But since $X$ is constant the object $X \times \bullet$ is for each object of $seq$ the [[representable functor|presheaf represented]] by that object. Therefore by the [[(∞,1)-Yoneda lemma]] it follows that

\begin{displaymath}
[X,E]_\bullet \simeq [X,E_\bullet]
  \,.
\end{displaymath}
This is manifestly the same formula as for the [[mapping spectrum]] out of $\Sigma^\infty X$.

\end{proof}
Similar kind of arguments give the following more general statement.

\begin{prop}
\label{}\hypertarget{}{}
For $X \in \mathbf{H} \stackrel{0}{\hookrightarrow} T \mathbf{H}$ a [[geometric homotopy type]], for $E \in E_\infty(\mathbf{H})$ an [[E-∞ ring]] with $(\widehat{Pic(E)} \to Pic(E)) \hookrightarrow T \mathbf{H}$ its universal [[(∞,1)-line bundle]] over its [[Picard ∞-groupoid]], then the [[internal hom]]/[[mapping stack]]

\begin{displaymath}
[X,\widehat{Pic(E)}]_{T \mathbf{H}} \in T \mathbf{H}
\end{displaymath}
is the object whose

\begin{itemize}%
\item base homotopy type is the [[E-∞ ring]] $[X, Pic(E)]$ of $E$-[[twisted cohomology|twist]] on $X$;


\item whose [[spectrum bundle]] is the collection of $\chi$-[[twisted cohomology|twisted E-cohomology spectra]] for all twists $\chi$.



\end{itemize}
\end{prop}
In full generality we may formulate the [[internal hom]] [[mapping space]] in $T \mathbf{H}$ in [[homotopy type theory]] notation as follows.

\begin{prop}
\label{InternalHomGeneral}\hypertarget{InternalHomGeneral}{}
For

\begin{displaymath}
(a \colon A) \;\vdash\; E_a \colon Spectra(\mathbf{H})
\end{displaymath}
and

\begin{displaymath}
(b \colon B) \;\vdash\; F_b \colon Spectra(\mathbf{H})
\end{displaymath}
two [[spectrum bundle]] [[dependent type|dependent types]] over base [[homotopy types]], $A,B \colon \mathbf{H}$, respectively, then the [[function type]] $(E \to F) \colon T\mathbf{H}$ between them (regarded as [[homotopy types]] in $T \mathbf{H}$) is

\begin{displaymath}
\chi \colon (A \to B); \sigma \colon \underset{a \colon A}{\prod} SpMap(E_a,F_{\chi(a)})
  \;\vdash\;
  \underset{a \colon A}{\prod} F_{\chi(a)}
  \,.
\end{displaymath}
\end{prop}
\begin{proof}
Let $(x:X)\vdash M_x : Spectra$ be another [[spectrum bundle]]. The [[cartesian product]] $M\times E$ in $T \mathbf{H}$ is then $(x:X),(a:A) \vdash M_x \oplus E_a$, with $\oplus$ also the [[coproduct]] (hence the [[direct sum]]), since [[spectra]] are [[stable (infinity,1)-category|stable]] and hence [[additive (infinity,1)-category|additive]]. We compute the [[mapping space]] $T\mathbf{H}(M\times E,F)$ as follows:

\begin{displaymath}
\begin{aligned}
  \sum_{(\phi:X\times A \to B)} \prod_{((x,a):X\times A)} SpMap(M_x\oplus E_a,F_{\phi(x,a)})
  &=& \sum_{(\phi:X \to A \to B)} \prod_{(x:X)} \prod_{(a:A)} SpMap(E_a,F_{\phi(x,a)}) \times SpMap(M_x,F_{\phi(x,a)})\\
  &=& \prod_{(x:X)} \sum_{(\chi:A \to B)} \left(\prod_{(a:A)} SpMap(E_a,F_{\chi(a)})\right) \times \left( \prod_{(a:A)} SpMap(M_x,F_{\chi(a)}) \right)\\
  &=& \prod_{(x:X)} \sum_{\psi : \sum_{(\chi:A \to B)} \prod_{(a:A)} SpMap(E_a,F_{\chi(a)})} SpMap\left(M_x, \prod_{(a:A)} F_{pr_1(\psi)(a)}\right)\\
  &=& \sum_{\rho : X \to \sum_{(\chi:A \to B)} \prod_{(a:A)} SpMap(E_a,F_{\chi(a)})} \prod_{(x:A)} SpMap\left(M_x, \prod_{(a:A)} F_{pr_1(\rho(x))(a)}\right)
\end{aligned}
\end{displaymath}
In the first line, we [[currying|curry]] $\phi$, apply the [[induction principle]] for [[dependent type|dependent]] maps out of $X\times A$, and also apply the [[universal property]] of the [[coproduct]] $M_x \oplus E_a$. In the second line, we apply the [[universal property]] for mapping into [[∞-types]] (the ``[[type-theoretic axiom of choice]]'') and also that for dependent functions into a [[Cartesian product|product]]. In the third line we apply the associativity of [[∞-types]], and also the [[universal property]] for mapping into the [[dependent product]] $\prod$ of [[spectra]]. Finally, in the fourth line, we apply the [[type-theoretic axiom of choice]] again in the other direction. The resulting [[type]] is the [[mapping space]] from $M$ to the claimed [[function type]] $(E\to F)$ defined above. (See also \href{http://nforum.mathforge.org/discussion/5321/parameterized-cohesive-spectra/?Focus=42394#Comment_42394}{this discussion}.)

\end{proof}
\begin{example}
\label{}\hypertarget{}{}
We have the following special cases of prop. \ref{InternalHomGeneral}.

\begin{enumerate}%
\item If $E_a = 0$ for all $a \colon A$, and if $B = \ast$, then the function type is

\begin{displaymath}
\vdash \; \underset{a \colon A}{\prod} F
\end{displaymath}
which reproduces the [[mapping spectrum]] $SpMap(\Sigma^\infty A, F)$ from prop. \ref{MappingSpectrumOutOfUnstableTypeAsCartesianInternalHom}.


\item If $A = B = \ast$ then the mapping type is

\begin{displaymath}
\sigma \colon SpMap(E,F) \;\vdash \; F \colon Spectra
\end{displaymath}

\item If $E_a = 0$ for all $a \colon A$ and $F_b = 0$ for all $b \colon B$ then the mapping type is

\begin{displaymath}
\chi \colon (A \to B)\;\vdash \; 0 \colon Spectra
  \,.
\end{displaymath}


\end{enumerate}
\end{example}
\hypertarget{CohesiveAndDifferentialRefinement}{}\subsubsection*{{Cohesive and differential refinement}}\label{CohesiveAndDifferentialRefinement}

Let $T\mathbf{H}$ be a tangent cohesive $(\infty,1)$-topos and write $T_\ast \mathbf{H}$ for the [[stable (∞,1)-category]] of [[spectrum objects]] inside it. We discuss how every [[stable homotopy type]] here canonically sits in the middle of a \emph{[[differential cohomology diagram]]}.

\begin{prop}
\label{}\hypertarget{}{}
For every $A \in T_\ast \mathbf{H}$ the [[natural transformation|naturality square]]

\begin{displaymath}
\itexarray{
    A &\stackrel{}{\longrightarrow}& A/\flat A
    \\
    \downarrow &{}^{(pb)}& \downarrow
    \\
    \Pi(A) &\stackrel{}{\longrightarrow}& \Pi(A/\flat A)
  }
\end{displaymath}
(of the [[shape modality]] applied to the [[homotopy cofiber]] of the [[counit of a comonad|counit]] of the [[flat modality]]) is an [[(∞,1)-pullback]] square.

\end{prop}
This was observed in (\hyperlink{BunkeNikolausVoelkl13}{Bunke-Nikolaus-V\"o{}lkl 13}). It is an incarnation of a [[fracture theorem]].

\begin{proof}
By [[cohesive (infinity,1)-topos|cohesion]] and [[stable (infinity,1)-category|stability]] we have the diagram

\begin{displaymath}
\itexarray{
    \flat A &\longrightarrow & A &\stackrel{}{\longrightarrow}& A/\flat A
    \\
    \downarrow^{\mathrlap{\simeq}} && \downarrow && \downarrow
    \\
    \Pi(\flat A) &\longrightarrow& \Pi(A) &\stackrel{}{\longrightarrow}& \Pi(A/\flat A)
  }
\end{displaymath}
where both rows are [[homotopy fiber sequences]]. By [[cohesive (infinity,1)-topos|cohesion]] the left vertical map is an [[equivalence in an (infinity,1)-category|equivalence]]. The claim now follows with the \href{homotopy+pullback#HomotopyFiberCharacterization}{homotopy fiber characterization} of [[homotopy pullbacks]].

\end{proof}
\begin{remark}
\label{}\hypertarget{}{}
This means that in stable cohesion every cohesive stable homotopy type is in controled sense a cohesive extension/refinement of its [[geometric realization of cohesive infinity-groupoids|geometric realization]] [[discrete infinity-groupoid|geometrically discrete]] (``bare'') stable [[homotopy type]] by the non-[[discrete object|discrete]] part of its cohesive structure;

In particular, $A/\flat A$ may be identified with differential cycle data. Indeed, by stability and cohesion it is the \href{cohesive+%28infinity,1%29-topos+--+structures#deRhamCohomology}{flat de Rham coefficient object}

\begin{displaymath}
A/\flat A = \flat_{dR}\Sigma A
\end{displaymath}
of the [[suspension]] of $A$, and the map to this quotient is thus the [[Maurer-Cartan form]] $\theta_A$. So

\begin{displaymath}
\itexarray{
    A &\stackrel{\theta_A}{\longrightarrow}& \flat_{dR}\Sigma A
    \\
    \downarrow &{}^{(pb)}& \downarrow
    \\
    \Pi(A) &\stackrel{}{\longrightarrow}& \Pi(A/\flat A)
  }
\end{displaymath}
exhibits $A$ as a [[differential cohomology]]-coefficient of the [[generalized cohomology theory]] $\Pi(A)$.

It follows by the discussion at [[schreiber:differential cohomology in a cohesive topos]] that the further differential refinement $\widehat{A}$ of $A$ should be given by a further [[homotopy pullback]]

\begin{displaymath}
\itexarray{
    \widehat{A} &\longrightarrow& \Omega^1(-,Lie(A))
    \\
    \downarrow &{}^{(pb)}& \downarrow
    \\
    A &\stackrel{\theta_A}{\longrightarrow}& \flat_{dR}\Sigma A
    \\
    \downarrow &{}^{(pb)}& \downarrow
    \\
    \Pi(A) &\stackrel{}{\longrightarrow}& \Pi(A/\flat A)
  }
  \,.
\end{displaymath}
But of course by the generality of the above proposition, such an $\widehat{A}$ sits itself again in its fracture-like pullback diagram.

\end{remark}
Dually:

\begin{prop}
\label{}\hypertarget{}{}
For every $A \in T_\ast \mathbf{H}$ the [[natural transformation|naturality square]]

\begin{displaymath}
\itexarray{
    \flat(\Pi_{dR} \Sigma^{-1} A) &\longrightarrow & \Pi_{dR}(\Sigma^{-1} A)
    \\
    \downarrow && \downarrow
    \\
    \flat A &\stackrel{}{\longrightarrow}& A 
  }
\end{displaymath}
(of the [[flat modality]] applied to the [[homotopy fiber]] of the [[unit of a monad|unit]] of the [[shape modality]]) is an [[(∞,1)-pullback]] square.

\end{prop}
\begin{proof}
As before but dually, the diagram extends to a morphism of [[homotopy cofiber]] diagrams of the form

\begin{displaymath}
\itexarray{
    \flat(\Pi_{dR} \Sigma^{-1} A) &\longrightarrow & \Pi_{dR}(\Omega A)
    \\
    \downarrow && \downarrow
    \\
    \flat A &\stackrel{}{\longrightarrow}& A 
    \\
    \downarrow && \downarrow 
    \\
    \flat \Pi(A) &\stackrel{\simeq}{\longrightarrow}& \Pi(A) 
  }
  \,,
\end{displaymath}
and by [[cohesion]] the bottom horizontal morphism is an [[equivalence in an (infinity,1)-category|equivalence]].

\end{proof}
Combining these two statements yields the following (\hyperlink{BunkeNikolausVoelkl13}{Bunke-Nikolaus-V\"o{}lkl 13}).

\begin{cor}
\label{TheDifferentialDiagram}\hypertarget{TheDifferentialDiagram}{}
For $\mathbf{H}$ a [[cohesive (∞,1)-topos]] every [[stable homotopy type]] $A \in Stab(\mathbf{H}) \hookrightarrow T \mathbf{H}$ sits inside a [[diagram]] of the form

\begin{displaymath}
\itexarray{
    &&  \Pi_{dR} \Omega A && \longrightarrow && \flat_{dR}\Sigma A
    \\
    & \nearrow & & \searrow & & \nearrow_{\mathrlap{\theta_A}} && \searrow
    \\
    \flat \Pi_{dR} \Omega A  && && A && && \Pi \flat_{dR}\Sigma A
    \\
    & \searrow &  & \nearrow & & \searrow && \nearrow_{\mathrlap{\Pi \theta_A}}
    \\
    && \flat A && \longrightarrow && \Pi A
  }
  \,,
\end{displaymath}
where the two squares are [[homotopy pullback]] squares and the two diagonals are the [[fiber sequences]] of the [[Maurer-Cartan form]] $\theta_A$ and its dual.

\end{cor}
\begin{remark}
\label{}\hypertarget{}{}
The bottom horizontal morphisms in the diagram in prop. \ref{TheDifferentialDiagram} are the canonical \href{cohesive%20topos#CanonicalComparison}{points-to-pieces transform}.

\end{remark}
\begin{remark}
\label{}\hypertarget{}{}
This kind of diagram under forming $\pi_0$ has been traditionally known from [[ordinary differential cohomology]] and from [[differential K-theory]], and had been used in proposals to axiomatize [[differential cohomology]], see for instance (\href{http://arxiv.org/abs/1208.3961}{Bunke 12, prop. 4.57}) and see at \emph{[[differential cohomology diagram]]}. Here we see that this holds fully generally for every stable cohesive homotopy type. If one still regards this diagram as characteristic of ``differential'' refinement it hence exhibits every cohesive stable type as a [[coefficients]] of \emph{some} [[differential cohomology]] theory. This is a strong version of the synthetic notion ``[[schreiber:differential cohomology in a cohesive topos]]'' . For more on this see also at \emph{[[smooth spectrum]]}.

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

\begin{itemize}%
\item [[sheaf of spectra]]


\item [[smooth spectrum]]


\item [[twisted generalized cohomology]]


\item [[twisted bivariant cohomology]]


\item [[twisted differential cohomology]]


\item [[twisted smooth cohomology in string theory]]



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

The idea of forming $T_\ast \mathbf{H}$ as a home for nontrivial [[stable homotopy types]] was originally suggested by [[Georg Biedermann]] and [[André Joyal]], see section 35 of

\begin{itemize}%
\item [[André Joyal]], \emph{Notes on Logoi}, 2008 (\href{http://www.math.uchicago.edu/~may/IMA/JOYAL/Joyal.pdf}{pdf})

\end{itemize}
and see the further references at \emph{[[tangent (infinity,1)-topos]]}.

Discussion of [[differential cohomology]] in $T_\ast Smooth \infty Grpd \simeq Stab(Smooth \infty Grpd)$ is in

\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}
For details see at \emph{[[differential cohomology hexagon]]}.

The above discussion of geometric twisted generalized cohomology as cohomology in the tangent cohesive $\infty$-topos was presented in

\begin{itemize}%
\item [[Urs Schreiber]], talk at \emph{\href{http://wwwmath.uni-muenster.de/sfb/sfb878/2013/twists.html}{Twists, generalised cohomology and applications}}, October 2013 (\href{twisted+smooth+cohomology+in+string+theory#LocalPrequantumFieldTheory}{talk notes})

\end{itemize}
Discussion in a comprehensive context of cohesion is in section 4.2.3 of

\begin{itemize}%
\item [[Urs Schreiber]], \emph{[[schreiber:differential cohomology in a cohesive topos]]} (\href{http://arxiv.org/abs/1310.7930}{arXiv:1310.7930})

\end{itemize}
[[!redirects cohesive tangent (∞,1)-topos]]

[[!redirects stable cohesive homotopy theory]] [[!redirects stable cohesive homotopy type theory]]

[[!redirects cohesive stable homotopy theory]] [[!redirects cohesive stable homotopy type theory]]

[[!redirects tangent cohesion]]

[[!redirects stable cohesion]]

[[!redirects tangent cohesive ∞-topos]] [[!redirects tangent cohesive ∞-toposes]] [[!redirects tangent cohesive infinity-topos]] [[!redirects tangent cohesive infinity-toposes]]



\end{document}