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\begin{document}

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\section*{cohesive (infinity,1)-topos -- structures II}

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

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

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

\begin{quote}%
This is a sub-section of the entry \emph{[[cohesive (∞,1)-topos]]} . See there for background and context


\end{quote}

\vspace{.5em} \hrule \vspace{.5em}
\hypertarget{contents}{}\section*{{Contents}}\label{contents}

\noindent\hyperlink{structures_in_a_cohesive_topos}{Structures in a cohesive $(\infty,1)$-topos}\dotfill \pageref*{structures_in_a_cohesive_topos} \linebreak
\noindent\hyperlink{Concordance}{Concordance}\dotfill \pageref*{Concordance} \linebreak
\noindent\hyperlink{Homotopy}{Geometric homotopy and Galois theory}\dotfill \pageref*{Homotopy} \linebreak
\noindent\hyperlink{vanKampenTheorem}{van Kampen theorem}\dotfill \pageref*{vanKampenTheorem} \linebreak
\noindent\hyperlink{Paths}{Paths and geometric Postnikov towers}\dotfill \pageref*{Paths} \linebreak
\noindent\hyperlink{Coverings}{Universal coverings and geometric Whitehead towers}\dotfill \pageref*{Coverings} \linebreak
\noindent\hyperlink{FlatDifferentialCohomology}{Flat $\infty$-connections and local systems}\dotfill \pageref*{FlatDifferentialCohomology} \linebreak
\noindent\hyperlink{deRhamCohomology}{de Rham cohomology}\dotfill \pageref*{deRhamCohomology} \linebreak
\noindent\hyperlink{LieAlgebras}{Exponentiated $\infty$-Lie algebras}\dotfill \pageref*{LieAlgebras} \linebreak
\noindent\hyperlink{CurvatureCharacteristics}{Maurer-Cartan forms and curvature characteristic forms}\dotfill \pageref*{CurvatureCharacteristics} \linebreak
\noindent\hyperlink{DifferentialCohomology}{Differential cohomology}\dotfill \pageref*{DifferentialCohomology} \linebreak
\noindent\hyperlink{ChernWeilTheory}{Chern-Weil homomorphism and $\infty$-connections}\dotfill \pageref*{ChernWeilTheory} \linebreak
\noindent\hyperlink{ChernSimonsTheory}{Higher holonomy and Chern-Simons functional}\dotfill \pageref*{ChernSimonsTheory} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak


\hypertarget{structures_in_a_cohesive_topos}{}\subsection*{{Structures in a cohesive $(\infty,1)$-topos}}\label{structures_in_a_cohesive_topos}

This continues the list of structures whose first part is at \emph{[[cohesive (infinity,1)-topos -- structures]]} .

\hypertarget{Concordance}{}\subsubsection*{{Concordance}}\label{Concordance}

Since $\mathbf{H}$ is an [[(∞,1)-topos]] it carries canonically the structure of a [[cartesian closed (∞,1)-category]]. For\newline $X, Y \in \mathbf{H}$, write $Y^X \in \mathbf{H}$ for the corresponding [[internal hom]].

Since $\Pi : \mathbf{H} \to$ [[∞Grpd]] preserves products, we have for all $X,Y, Z \in \mathbf{H}$ canonically induced a morphism

\begin{displaymath}
\Pi(Y^X) \times \Pi(Z^Y)
   \stackrel{\simeq}{\to}
  \Pi(Y^X \times Z^Y)
  \stackrel{\Pi(comp_{X,Y,Z})}{\to}
  \Pi(Z^X)
  \,.
\end{displaymath}
This should yield an [[(∞,1)-category]] $\tilde \mathbf{H}$ with the same objects as $\mathbf{H}$ and hom-$\infty$-groupoids defined by

\begin{displaymath}
\tilde \mathbf{H}(X,Y) := \Pi(Y^X)
  \,.
\end{displaymath}
We have that

\begin{displaymath}
\tilde \mathbf{H}(X,\mathbf{B}G) = \Pi(\mathbf{B}G^X)
\end{displaymath}
is the $\infty$-groupoid whose objects are $G$-[[principal ∞-bundle]]s on $X$ and whose morphisms have the interpretaton of $G$-principal bundles on the cylinder $X \times I$. These are \emph{[[concordance]]s of $\infty$-bundles.}

\hypertarget{Homotopy}{}\subsubsection*{{Geometric homotopy and Galois theory}}\label{Homotopy}

We discuss canonical internal realizations of the notions of [[homotopy group]], [[local system]] and [[Galois theory]] in $\mathbf{H}$.

\begin{defn}
\label{}\hypertarget{}{}
For $\mathbf{H}$ a [[locally ∞-connected (∞,1)-topos]] and $X \in \mathbf{H}$ an [[object]], we call $\Pi X \in$ [[∞Grpd]] the \textbf{[[fundamental ∞-groupoid in a locally ∞-connected (∞,1)-topos]]} of $X$.

The ([[categorical homotopy groups in an (∞,1)-topos|categorical]]) [[homotopy group]]s of $\Pi(X)$ we call the [[geometric homotopy groups in an (∞,1)-topos|geometric homotopy groups]] of $X$

\begin{displaymath}
\pi_\bullet^{geom}(X) := \pi_\bullet(\Pi (X))
  \,.
\end{displaymath}
\end{defn}
\begin{defn}
\label{GeometricRealization}\hypertarget{GeometricRealization}{}
For $\vert - \vert :$ [[∞Grpd]] $\stackrel{\simeq}{\to}$ [[Top]] the [[homotopy hypothesis]]-[[equivalence of (∞,1)-categories|equivalence]] we write

\begin{displaymath}
\vert X \vert := \vert \Pi X \vert \in Top
\end{displaymath}
and call this the \textbf{topological [[geometric realization]]} of $X$, or just the \emph{geometric realization} for short.

\end{defn}
\begin{note}
\label{GeometricRealizationInTheLiterature}\hypertarget{GeometricRealizationInTheLiterature}{}
In presentations of $\mathbf{H}$ by a [[model structure on simplicial presheaves]] as in prop. \ref{SimplicialPresheavesOverInfinityCohesviveSite} this abstract notion reproduces the notion of geometric realization of [[∞-stack]]s in (\hyperlink{Simpson}{Simpson}). See remark 2.22 in (\hyperlink{SimpsonTeleman}{SimpsonTeleman}).

\end{note}
\begin{udef}
We say a \textbf{geometric [[homotopy]]} between two morphism $f,g : X \to Y$ in $\mathbf{H}$ is a diagram

\begin{displaymath}
\itexarray{
    X 
    \\
    \downarrow^{\mathrlap{(Id,i)}} & \searrow^{\mathrlap{f}}
    \\
    X \times I &\stackrel{\eta}{\to}& Y
    \\
    \uparrow^{\mathrlap{(Id,o)}} & \nearrow_{\mathrlap{g}}
    \\
    X
  }
\end{displaymath}
such that $I$ is geometrically connected, $\pi_0^{geom}(I) = *$.

\end{udef}
\begin{uprop}
If $f,g : X\to Y$ are geometrically homotopic in $\mathbf{H}$, then their images $\Pi(f), \Pi(g)$ are equivalent in $\infty Grpd$.

\end{uprop}
\begin{proof}
By the condition that $\Pi$ preserves products in a cohesive $(\infty,1)$-topos we have that the image of the geometric homotopy in $\infty Grpd$ is a diagram of the form

\begin{displaymath}
\itexarray{
    \Pi(X) 
    \\
    \downarrow^{\mathrlap{(Id,\Pi(i))}} & \searrow^{\mathrlap{\Pi(f)}}
    \\
    \Pi(X) \times \Pi(I) &\stackrel{\Pi(\eta)}{\to}& \Pi(Y)
    \\
    \uparrow^{\mathrlap{(Id,\Pi(o))}} & \nearrow_{\mathrlap{\Pi(g)}}
    \\
    \Pi(X)
  }
  \,.
\end{displaymath}
Now since $\Pi(I)$ is connected by assumption, there is a diagram

\begin{displaymath}
\itexarray{
     && *
     \\
     & {}^{\mathllap{Id}}\nearrow & \downarrow^{\mathrlap{\Pi(i)}}
     \\
     * &\to& \Pi(I)
     \\
     & {}_{\mathllap{Id}}\searrow & \uparrow^{\mathrlap{\Pi(o)}}
     \\
     && *
  }
\end{displaymath}
in [[∞Grpd]].

Taking the product of this diagram with $\Pi(X)$ and pasting the result to the above image $\Pi(\eta)$ of the geometric homotopy constructs the equivalence $\Pi(f) \Rightarrow \Pi(g)$ in $\infty Grpd$.

\end{proof}
\begin{uprop}
For $\mathbf{H}$ a [[locally ∞-connected (∞,1)-topos]], also all its objects $X \in \mathbf{H}$ are locally $\infty$-connected, in the sense their [[petit topos|petit]] [[over-(∞,1)-toposes]] $\mathbf{H}/X$ are locally $\infty$-connected.

The two notions of fundamental $\infty$-groupoids of $X$ induced this way do agree, in that there is a natural equivalence

\begin{displaymath}
\Pi_X(X \in \mathbf{H}/X) \simeq \Pi(X \in \mathbf{H})
  \,.
\end{displaymath}
\end{uprop}
\begin{proof}
By the general facts recalled at [[étale geometric morphism]] we have a composite [[essential geometric morphism]]

\begin{displaymath}
(\Pi_X \dashv \Delta_X \dashv \Gamma_X) : 
  \mathbf{H}_{/X}
   \stackrel{\overset{X_!}{\to}}{\stackrel{\overset{X^*}{\leftarrow}}{\underset{\X_*}{\to}}}
  \mathbf{H}
   \stackrel{\overset{\Pi}{\to}}{\stackrel{\overset{\Delta}{\leftarrow}}{\underset{\Gamma}{\to}}}  
   \infty Grpd
\end{displaymath}
and $X_!$ is given by sending $(Y \to X) \in \mathbf{H}/X$ to $Y \in \mathbf{H}$.

\end{proof}
\begin{udef}
For $\kappa$ a [[regular cardinal]] write

\begin{displaymath}
Core \infty Grpd_\kappa \in \infty Grpd
\end{displaymath}
for the [[∞-groupoid]] of $\kappa$-[[small (∞,1)-category|small ∞-groupoid]]s: the [[core]] of the full [[sub-(∞,1)-category]] of [[∞Grpd]] on the $\kappa$-small ones.

\end{udef}
\begin{uremark}
We have

\begin{displaymath}
Core \infty Grpd_\kappa \simeq
  \coprod_i \mathbf{B} Aut(F_i)
  \,,
\end{displaymath}
where the coproduct ranges over all $\kappa$-small [[homotopy type]]s $[F_i]$ and $Aut(F_i)$ is the [[automorphism ∞-group]] of any representative $F_i$ of $[F_i]$.

\end{uremark}
\begin{udef}
For $X \in \mathbf{H}$ write

\begin{displaymath}
LConst(X) := \mathbf{H}(X, Disc Core \infty Grpd_\kappa)
  \,.
\end{displaymath}
We call this the $\infty$-groupoid of \textbf{[[locally constant ∞-stack]]s} on $X$.

\end{udef}
\begin{uobservation}
Since $Disc$ is [[left adjoint]] and [[right adjoint]] it commutes with [[coproduct]]s and with [[delooping]] and therefore

\begin{displaymath}
Disc Core \infty Grpd_\kappa 
    \simeq 
  \coprod_i \mathbf{B} Disc Aut(F_i)
  \,.
\end{displaymath}
Therefore a cocycle $P \in LConst(X)$ may be identified on each geometric connected component of $X$ as a $Disc Aut(F_i)$-[[principal ∞-bundle]] $P \to X$ over $X$ for the [[∞-group]] object $Disc Aut(F_i) \in \mathbf{H}$. We may think of this as an object $P \in \mathbf{H}/X$ in the [[little topos]] over $X$. This way the objects of $LConst(X)$ are indeed identified $\infty$-stacks over $X$.

\end{uobservation}
The following proposition says that the central statements of [[Galois theory]] hold for these canonical notions of geometric homotopy groups and locally constant $\infty$-stacks.

\begin{uprop}
For $\mathbf{H}$ [[locally ∞-connected (∞,1)-topos|locally ∞-connected]] and [[∞-connected (∞,1)-topos|∞-connected]], we have

\begin{itemize}%
\item a natural equivalence

\begin{displaymath}
LConst(X) \simeq \infty \mathrm{Grpd}(\Pi(X), \infty Grpd_\kappa)
\end{displaymath}
of locally constant $\infty$-stacks on $X$ with $\infty$-[[permutation representation]]s of the [[fundamental ∞-groupoid in a locally ∞-connected (∞,1)-topos|fundamental ∞-groupoid]] of $X$ ([[local system]]s on $X$);


\item for every point $x : * \to X$ a natural equivalence of the endomorphisms of the fiber functor $x^*$ and the [[loop space]] of $\Pi(X)$ at $x$

\begin{displaymath}
End( x^* : LConst(X) \to \infty Grpd ) \simeq \Omega_x \Pi(X)
  \,.
\end{displaymath}


\end{itemize}
\end{uprop}
\begin{proof}
The first statement is just the adjunction $(\Pi \dashv Disc)$.

\begin{displaymath}
\begin{aligned}
     LConst(X) & := \mathbf{H}(X, Disc Core \infty Grpd_\kappa)
     \\
     &  \simeq \infty Grpd(\Pi(X), Core \infty Grpd_\kappa)
     \\
     &  \simeq \infty Grpd(\Pi(X), \infty Grpd_\kappa)     
  \end{aligned}
  \,.
\end{displaymath}
Using this and that $\Pi$ preserves the [[terminal object in an (∞,1)-category|terminal object]], so that the [[adjunct]] of $(* \to X \to Disc Core \infty Grpd_\kappa)$ is $(* \to \Pi(X) \to \infty Grpd_\kappa)$ the second statement follows with an iterated application of the [[(∞,1)-Yoneda lemma]] (this is pure [[Tannaka duality]] as discussed there):

The fiber functor $x^* : Func(\Pi(X), \infty Grpd) \to \infty Grpd$ evaluates an $(\infty,1)$-presheaf on $\Pi(X)^{op}$ at $x \in \Pi(X)$. By the [[(∞,1)-Yoneda lemma]] this is the same as homming out of $j(x)$, where $j : \Pi(X)^{op} \to Func(\Pi(X), \infty Grpd)$ is the [[(∞,1)-Yoneda embedding]]:

\begin{displaymath}
x^* \simeq Hom_{PSh(\Pi(X)^{op})}(j(x), -)
  \,.
\end{displaymath}
This means that $x^*$ itself is a representable object in $PSh(PSh(\Pi(X)^{op})^{op})$. If we denote by $\tilde j : PSh(\Pi(X)^{op})^{op} \to PSh(PSh(\Pi(X)^{op})^{op})$ the corresponding Yoneda embedding, then

\begin{displaymath}
x^* \simeq \tilde j (j (x))
  \,.
\end{displaymath}
With this, we compute the endomorphisms of $x^*$ by applying the [[(∞,1)-Yoneda lemma]] two more times:

\begin{displaymath}
\begin{aligned}
    End x^*
     & \simeq
    End_{PSh(PSh(\Pi(X)^{op})^{op})} (\tilde j(j (x)))
     \\
     & \simeq
     End(PSh(\Pi(X))^{op}) (j(x))
     \\
     & \simeq
     End_{\Pi(X)^{op}}(x,x)
    \\
     & \simeq
     Aut_x \Pi(X)
    \\
     & =: \Omega_x \Pi(X)
  \end{aligned}
   \,.
\end{displaymath}
\end{proof}
\hypertarget{vanKampenTheorem}{}\subsubsection*{{van Kampen theorem}}\label{vanKampenTheorem}

A [[higher homotopy van Kampen theorem|higher]] [[van Kampen theorem]] asserts that passing to [[fundamental ∞-groupoid]]s preserves certain colimits.

On a cohesive $(\infty,1)$-topos $\mathbf{H}$ the fundamental $\infty$-groupoid functor $\Pi : \mathbf{H} \to \infty Grpd$ is a [[left adjoint]] [[(∞,1)-functor]] and hence preserves all [[(∞,1)-colimit]]s.

More interesting is the question which $(\infty,1)$-colimits of \hyperlink{ConcreteObjects}{concrete spaces} in

\begin{displaymath}
Conc(\mathbf{H}) 
    \stackrel{\overset{conc}{\leftarrow}}{\underset{inj}{\hookrightarrow}} 
  \mathbf{H}
\end{displaymath}
are preserved by $\Pi \circ inj : Conc(\mathbf{H}) \to \infty Grpd$. These colimits are computed by first computing them in $\mathbf{H}$ and then applying the concretization functor. So we have

\begin{ulemma}
Let $U_\bullet : K \to Conc(\mathbf{H})$ be a [[diagram]] such that the [[(∞,1)-colimit]] $\lim_\to inj \circ U_\bullet$ is concrete, $\cdots \simeq inj(X)$.

Then the [[fundamental ∞-groupoid]] of $X$ is computed as the $(\infty,1)$-colimit

\begin{displaymath}
\Pi(X) \simeq {\lim_\to} \Pi(U_\bullet)
  \,.
\end{displaymath}
\end{ulemma}
In the \hyperlink{Examples}{Examples} we discuss the cohesive $(\infty,1)$-topos $\mathbf{H} = (\infty,1)Sh(TopBall)$ of [[topological ∞-groupoid]]s For that case we recover the ordinary [[higher van Kampen theorem]]:

\begin{uprop}
Let $X$ be a [[paracompact space|paracompact]] or [[locally contractible space|locally contractible]] [[topological space]]s and $U_1 \hookrightarrow X$, $U_2 \hookrightarrow X$ a [[covering]] by two [[open subsets]].

Then under the [[singular simplicial complex]] functor $Sing : Top \to$ [[sSet]] we have a [[homotopy pushout]]

\begin{displaymath}
\itexarray{
    Sing(U_1) \cap Sing(U_2) &\to& Sing(U_2)
    \\
    \downarrow && \downarrow
    \\
    Sing(U_1) &\to& Sing(X)
  }
  \,.
\end{displaymath}
\end{uprop}
\begin{proof}
We inject the topological space via the external [[Yoneda embedding]]

\begin{displaymath}
Top \hookrightarrow Sh(TopBalls) \hookrightarrow
  \mathbf{H} := (\infty,1)Sh(OpenBalls)
\end{displaymath}
as a 0-[[truncated]] [[topological ∞-groupoid]] in the cohesive $(\infty,1)$-topos $\mathbf{H}$. Being an [[(∞,1)-category of (∞,1)-sheaves]] this is [[locally presentable (∞,1)-category|presented]] by the [[Bousfield localization of model categories|left Bousfield localization]] $Sh(TopBalls, sSet)_{inj,loc}$ of the injective [[model structure on simplicial sheaves]] on $TopBalls$ (as described at [[models for ∞-stack (∞,1)-toposes]]).

Notice that the injection $Top \hookrightarrow Sh(TopBalls)$ of topological spaces as [[concrete sheaves]] on the site of open balls preserves the pushout $X = U_1 \coprod_{U_1 \cap U_2} U_2$. (This is effectively the statement that $X$ as a [[representable functor|representable]] on [[Diff]] is a [[sheaf]].) Accordingly so does the further inclusion into $Sh(TopBall,sSet) \simeq Sh(TopBalls)^{\Delta^{op}}$ as simplicially constant simplicial sheaves.

Since cofibrations in that model structure are objectwise and degreewise injective maps, it follows that the ordinary [[pushout]] diagram

\begin{displaymath}
\itexarray{
    U_1 \cap U_2 &\to& U_2
    \\
    \downarrow && \downarrow
    \\
    U_1 &\to& X
  }
\end{displaymath}
in $Sh(TopBalls, sSet)_{inj,loc}$ has all objects cofibrant and is the pushout along a cofibration, hence is a [[homotopy pushout]] (as described there). By the general theorem at [[(∞,1)-colimit]] homotopy pushouts model $(\infty,1)$-pushouts, so that indeed $X$ is the $(\infty,1)$-pushout

\begin{displaymath}
X \simeq U_1 \coprod_{U_1 \cap U_2} U_2 \in \mathbf{H}
  \,.
\end{displaymath}
The proposition now follows with the above observation that $\Pi$ preserves all $(\infty,1)$-colimits and with the statement (from [[topological ∞-groupoid]]) that for a topological space (locally contractible or paracompact) we have $\Pi X \simeq Sing X$.

\end{proof}
\hypertarget{Paths}{}\subsubsection*{{Paths and geometric Postnikov towers}}\label{Paths}

The \hyperlink{Homotopy}{above} construction of the [[fundamental ∞-groupoid in a locally ∞-connected (∞,1)-topos|fundamental ∞-groupoid]] of objects in $\mathbf{H}$ as an object in [[∞Grpd]] may be reflected back into $\mathbf{H}$, where it gives a notion of homotopy [[path n-groupoid]]s and a geometric notion of [[Postnikov tower]]s of objects in $\mathbf{H}$.

\begin{udef}
For $\mathbf{H}$ a [[locally ∞-connected (∞,1)-topos]] define the composite [[adjoint (∞,1)-functor]]s

\begin{displaymath}
(\mathbf{\Pi} \dashv \mathbf{\flat}) 
  :=
  (Disc \Pi \dashv Disc \Gamma)
  : 
  \mathbf{H}
   \to 
  \mathbf{H}
  \,.
\end{displaymath}
\end{udef}
We say

\begin{itemize}%
\item $\mathbf{\Pi}(X)$ is the \textbf{path $\infty$-groupoid} of $X$ -- the reflection of the [[fundamental ∞-groupoid in a locally ∞-connected (∞,1)-topos]] back into the cohesive context of $\mathbf{H}$;


\item $\mathbf{\flat} A$ (``flat $A$'') is the coefficient object for \textbf{\hyperlink{FlatDifferentialCohomology}{flat differential A-cohomology}} or for $A$-[[local system]]s



\end{itemize}
Write

\begin{displaymath}
(\tau_n \dashv i_n)
  :
  \mathbf{H}_{\leq n}
  \stackrel{\overset{\tau_{n}}{\leftarrow}}{\underset{i}{\hookrightarrow}}
  \mathbf{H}
\end{displaymath}
for the [[reflective sub-(∞,1)-category]] of [[truncated|n-truncated object]]s and

\begin{displaymath}
\mathbf{\tau}_n : 
    \mathbf{H} \stackrel{\tau_n}{\to} \mathbf{H}_{\leq n}
  \hookrightarrow
  \mathbf{H}
\end{displaymath}
for the truncation-[[localization of an (∞,1)-category|localization]] funtor.

We say

\begin{displaymath}
\mathbf{\Pi}_n : \mathbf{H} \stackrel{\mathbf{\Pi}_n}{\to}
   \mathbf{H} \stackrel{\mathbf{\tau}_n}{\to}
  \mathbf{H}
\end{displaymath}
is the \textbf{homotopy [[path n-groupoid]]} functor.

We say that the (truncated) components of the $(\Pi \dashv Disc)$-[[unit of an adjunction|unit]]

\begin{displaymath}
X \to \mathbf{\Pi}(X)
\end{displaymath}
are the \textbf{constant path inclusions}. Dually we have canonical morphism

\begin{displaymath}
\mathbf{\flat}A \to A
  \,.
\end{displaymath}
\begin{ulemma}
If $\mathbf{H}$ is cohesive, then $\mathbf{\flat}$ has a [[right adjoint]] $\mathbf{\Gamma}$

\begin{displaymath}
(\mathbf{\Pi} \dashv \mathbf{\flat} \dashv \mathbf{\Gamma}) 
  :=
  (Disc \Pi \dashv Disc \Gamma \dashv coDisc \Gamma)
  : 
  \mathbf{H}
     \stackrel{\overset{\mathbf{\Pi}}{\to}}{\stackrel{\overset{\mathbf{\flat}}{\leftarrow}}{\underset{\mathbf{\Gamma}}{\to}}}
  \mathbf{H}
  \,.
\end{displaymath}
and this makes $\mathbf{H}$ be $\infty$-connected and locally $\infty$-connected over itself.

\end{ulemma}
\begin{udef}
Let $\mathbf{H}$ be a [[locally ∞-connected (∞,1)-topos]]. If $X \in \mathbf{H}$ is [[small-projective]] then the [[over-(∞,1)-topos]] $\mathbf{H}/X$ is

\begin{enumerate}%
\item [[locally ∞-connected (∞,1)-topos|locally ∞-connected]];


\item [[local (∞,1)-topos|local]].



\end{enumerate}
\end{udef}
\begin{proof}
The first statement is proven at [[locally ∞-connected (∞,1)-topos]], the second at [[local (∞,1)-topos]].

\end{proof}
\begin{udef}
In a cohesive $(\infty,1)$-topos $\mathbf{H}$, if $X$ is [[small-projective]] then so is its path ∞-groupoid $\mathbf{\Pi}(X)$.

\end{udef}
\begin{proof}
Because of the [[adjoint triple]] of [[adjoint (∞,1)-functor]]s $(\mathbf{\Pi} \dashv \mathbf{\flat} \dashv \mathbf{\Gamma})$ we have for [[diagram]] $A : I \to \mathbf{H}$ that

\begin{displaymath}
\begin{aligned}
     \mathbf{H}(\mathbf{\Pi}(X), {\lim_\to}_i A_i)
     & \simeq 
     \mathbf{H}(X, \mathbf{\flat}{\lim_\to}_i A_i)
    \\
     & \simeq 
     \mathbf{H}(X, {\lim_\to}_i \mathbf{\flat} A_i)
     \\
     & \simeq 
     {\lim_\to}_i \mathbf{H}(X,  \mathbf{\flat} A_i)
  \end{aligned}
  \,,
\end{displaymath}
where in the last step we used that $X$ is [[small-projective]] by assumption.

\end{proof}
\begin{udef}
For $X \in \mathbf{H}$ we say that the \textbf{geometric Postnikov tower} of $X$ is the [[Postnikov tower in an (∞,1)-category]] of $\mathbf{\Pi}(X)$:

\begin{displaymath}
\mathbf{\Pi}(X) \to \cdots
   \to 
   \mathbf{\Pi}_2(X) \to \mathbf{\Pi}_1(X) \to \mathbf{\Pi}_0(X)
  \,.
\end{displaymath}
\end{udef}
\hypertarget{Coverings}{}\subsubsection*{{Universal coverings and geometric Whitehead towers}}\label{Coverings}

We discuss an intrinsic notion of [[Whitehead tower]]s in a [[locally ∞-connected (∞,1)-topos|locally ∞-connected]] [[∞-connected (∞,1)-topos]] $\mathbf{H}$.

\begin{udef}
For $X \in \mathbf{H}$ a [[pointed object]], the \textbf{[[Whitehead tower in an (∞,1)-topos|geometric Whitehead tower]]} of $X$ is the sequence of objects

\begin{displaymath}
X^{\mathbf{(\infty)}} \to \cdots \to X^{\mathbf{(2)}} \to X^{\mathbf{(1)}} \to X^{\mathbf{(0)}} \simeq X
\end{displaymath}
in $\mathbf{H}$, where for each $n \in \mathbb{N}$ the object $X^{(n+1)}$ is the [[homotopy fiber]] of the canonical morphism $X \to \mathbf{\Pi}_{n+1} X$ to the \hyperlink{Paths}{path n+1-groupoid} of $X$.

We call $X^{\mathbf{(n+1)}}$ the $(n+1)$-fold \textbf{[[universal covering space]]} of $X$.

We write $X^{\mathbf{(\infty)}}$ for the homotopy fiber of the untruncated constant path inclusion.

\begin{displaymath}
X^{\mathbf{(\infty)}} \to X \to \mathbf{\Pi}(X)
  \,.
\end{displaymath}
Here the morphisms $X^{\mathbf{(n+1)}} \to X^{\mathbf{n}}$ are those induced from this 

\begin{displaymath}
\itexarray{
    X^{\mathbf{(n)}} &\to& *
    \\
    \downarrow && \downarrow
    \\
    X^{\mathbf{(n-1)}} & \to & \mathbf{B}^n \mathbf{\pi}_n(X) &\to& *
    \\
    \downarrow && \downarrow && \downarrow
    \\
    X &\to& \mathbf{\Pi}_n(X) &\to& \mathbf{\Pi}_{(n-1)}(X)
  }
  \,,
\end{displaymath}
where the object $\mathbf{B}^n \mathbf{\pi}_n(X)$ is defined as the [[homotopy fiber]] of the bottom right morphism.

\end{udef}
\begin{uprop}
Every object $X \in \mathbf{H}$ is covered by objects of the form $X^{\mathbf{(\infty)}}$ for different choices of base points in $X$, in the sense that every $X$ is the [[(∞,1)-colimit]] over a [[diagram]] whose vertices are of this form.

\end{uprop}
\begin{proof}
Consider the diagram

\begin{displaymath}
\itexarray{
    {\lim_\to}_{s \in \Pi(X)} (i^* *)
      &\to& {\lim_\to}_{s \in \Pi(X)} *
    \\
    {}^{\mathllap{\simeq}}\downarrow && \downarrow^{\mathrlap{\simeq}}
    \\
    X &\stackrel{i}{\to}& \mathbf{\Pi}(X)
  }
  \,.
\end{displaymath}
The bottom morphism is the constant path inclusion, the $(\Pi \dashv Disc)$-[[unit of an adjunction|unit]]. The right morphism is the [[equivalence in an (∞,1)-category]] that is the image under $Disc$ of the decomposition ${\lim_\to}_S * \stackrel{\simeq}{\to} S$ of every [[∞-groupoid]] as the [[(∞,1)-colimit]] (see there) over itself of the [[(∞,1)-functor]] constant on the point.

The left morphism is the [[(∞,1)-pullback]] along $i$ of this equivalence, hence itself an equivalence. By [[universal colimits]] in the [[(∞,1)-topos]] $\mathbf{H}$ the top left object is the [[(∞,1)-colimit]] over the single [[homotopy fiber]]s $i^* *_s$ of the form $X^{\mathbf{(\infty)}}$ as indicated.

\end{proof}
\begin{uprop}
The inclusion $\Pi(i^* *) \to \Pi(X)$ of the [[fundamental ∞-groupoid in a locally ∞-connected (∞,1)-topos|fundamental ∞-groupoid]] $\Pi(i^* *)$ of each of these objects into $\Pi(X)$ is homotopic to the point.

\end{uprop}
\begin{proof}
We apply $\Pi(-)$ to the above diagram over a single vertex $s$ and attach the $(\Pi \dashv Disc)$-[[unit of an adjunction|counit]] to get

\begin{displaymath}
\itexarray{
    \Pi(i^* *)
      &\to& 
      &\to&
      *
    \\
    \downarrow && && \downarrow 
    \\
    \Pi X &\stackrel{\Pi(i)}{\to}& \Pi Disc \Pi(X)
     &\to& \Pi(X)
  }
  \,.
\end{displaymath}
Then the bottom morphism is an equivalence by the $(\Pi \dashv Disc)$-[[zig-zag-identity]].

\end{proof}
\hypertarget{FlatDifferentialCohomology}{}\subsubsection*{{Flat $\infty$-connections and local systems}}\label{FlatDifferentialCohomology}

We describe for a [[locally ∞-connected (∞,1)-topos]] $\mathbf{H}$ a canonical intrinsic notion of \emph{flat} [[connections on ∞-bundles]], \emph{flat} [[higher parallel transport]] and higher [[local system]]s.

Write $(\mathbf{\Pi} \dashv\mathbf{\flat}) := (Disc \Pi \dashv Disc \Gamma) : \mathbf{H} \to \mathbf{H}$ for the adjunction given by the \hyperlink{Paths}{path ∞-groupoid}. Notice that this comes with the canonical $(\Pi \dashv Disc)$-[[unit of an adjunction|unit]] with components

\begin{displaymath}
X \to \mathbf{\Pi}(X)
\end{displaymath}
and the $(Disc \dashv \Gamma)$-counit with components

\begin{displaymath}
\mathbf{\flat} A \to A
  \,.
\end{displaymath}
\begin{udef}
For $X, A \in \mathbf{H}$ we write

\begin{displaymath}
\mathbf{H}_{flat}(X,A) := \mathbf{H}(\mathbf{\Pi}X, A)
\end{displaymath}
and call $H_{flat}(X,A) := \pi_0 \mathbf{H}_{flat}(X,A)$ the \textbf{flat (nonabelian) differential cohomology} of $X$ with coefficients in $A$.

We say a morphism $\nabla : \mathbf{\Pi}(X) \to A$ is a \textbf{flat [[connection on an ∞-bundle|∞-connnection]] on the [[principal ∞-bundle]] corresponding to $X \to \mathbf{\Pi}(X) \stackrel{\nabla}{\to} A$, or an}$A$-[[local system]]** on $X$.

The induced morphism

\begin{displaymath}
\mathbf{H}_{flat}(X,A) \to \mathbf{H}(X,A)
\end{displaymath}
we say is the [[forgetful functor]] that forgets flat connections.

\end{udef}
\begin{uremark}
The object $\mathbf{\Pi}(X)$ has the interpretation of the \hyperlink{Paths}{path ∞-groupoid} of $X$: it is a cohesive $\infty$-groupoid whose [[k-morphism]]s may be thought of as generated from the $k$-morphisms in $X$ and $k$-dimensional cohesive paths in $X$.

Accordingly a mophism $\mathbf{Pi}(X) \to A$ may be thought of as assigning

\begin{itemize}%
\item to each point of $X$ a fiber in $A$;


\item to each path in $X$ an equivalence between these fibers;


\item to each disk in $X$ a 2-equivalalence between these equivaleces associated to its boundary


\item and so on.



\end{itemize}
This we think of as encoding a flat [[higher parallel transport]] on $X$, coming from some flat $\infty$-connection and \emph{defining} this flat $\infty$-connection.

\end{uremark}
\begin{ulemma}
By the $(\mathbf{\Pi} \dashv \mathbf{\flat})$-adjunction we have a [[natural transformation|natural]] equivalence

\begin{displaymath}
\mathbf{H}_{flat}(X,A) \simeq \mathbf{H}(X,\mathbf{\flat}A)
  \,.
\end{displaymath}
A [[cocycle]] $g : X \to A$ for a [[principal ∞-bundle]] on $X$ is in the image of

\begin{displaymath}
\mathbf{H}_{flat}(X,A) \to \mathbf{H}(X,A)
\end{displaymath}
precisely if there is a lift $\nabla$ in the [[diagram]]

\begin{displaymath}
\itexarray{
     && \mathbf{\flat}A 
     \\
     & {}^{\nabla}\nearrow& \downarrow
    \\
    X &\stackrel{g}{\to}& A
  }
  \,.
\end{displaymath}
\end{ulemma}
We call $\mathbf{\flat}A$ the \textbf{coefficient object for flat $A$-connections}.

\begin{uprop}
For $G := Disc G_0 \in \mathbf{H}$ a [[discrete ∞-groupoid|discrete ∞-group]] the canonical morphism $\mathbf{H}_{flat}(X,\mathbf{B}G) \to \mathbf{H}(X,\mathbf{B}G)$ is an [[equivalence in an (∞,1)-category|equivalence]].

\end{uprop}
\begin{proof}
Since $Disc$ is a [[full and faithful (∞,1)-functor]] we have that the [[unit of an adjunction|unit]] $Id \to \Gamma Disc$ is a natural equivalence. It follows that on $Disc G_0$ also the counit $Disc \Gamma Disc G_0 \to Disc G_0$ is a weak equivalence (since by the [[triangle identity]] we have that $Disc G_0 \stackrel{\simeq}{\to} Disc \Gamma Disc G_0 \to Disc G_0$ is the identity).

\end{proof}
\begin{uremark}
This says that for discrete structure [[∞-group]]s $G$ there is an essentially unique flat $\infty$-connection on any $G$-[[principal ∞-bundle]]. Moreover, the further equivalence

\begin{displaymath}
\mathbf{H}(\mathbf{\Pi}(X), \mathbf{B}G)
  \simeq
  \mathbf{H}_{flat}(X, \mathbf{B}G)
  \simeq
  \mathbf{H}(X, \mathbf{B}G)
\end{displaymath}
may be read as saying that the $G$-principal $\infty$-bundle is entirely characterized by the flat [[higher parallel transport]] of this unique $\infty$-connection.

\end{uremark}
\hypertarget{deRhamCohomology}{}\subsubsection*{{de Rham cohomology}}\label{deRhamCohomology}

In every [[locally ∞-connected (∞,1)-topos]] $\mathbf{H}$ there is an intrinsic notion of [[nonabelian cohomology|nonabelian]] [[de Rham cohomology]].

\begin{udef}
For $X \in \mathbf{H}$ an object, write $\mathbf{\Pi}_{dR}X := * \coprod_X \mathbf{\Pi} X$ for the [[(∞,1)-colimit|(∞,1)-pushout]]

\begin{displaymath}
\itexarray{
    X &\to& *
    \\
    \downarrow && \downarrow
    \\
    \mathbf{\Pi}(X) &\to& \mathbf{\Pi}_{dR}X
  }
  \,.
\end{displaymath}
For $* \to A$ any [[pointed object]] in $\mathbf{H}$, write $\mathbf{\flat}_{dR} A : * \prod_A \mathbf{\flat}A$ for the [[(∞,1)-pullback]]

\begin{displaymath}
\itexarray{
    \mathbf{\flat}_{dR} A &\to& \mathbf{\flat} A 
    \\
    \downarrow && \downarrow
    \\
    * &\to& A
  }
  \,.
\end{displaymath}
\end{udef}
\begin{uprop}
This construction yields a pair of [[adjoint (∞,1)-functor]]s

\begin{displaymath}
(\mathbf{\Pi}_{dR} \dashv \mathbf{\flat}_{dR} )
  : 
  */\mathbf{H}
  \stackrel{
     \overset{\mathbf{\Pi}_{dR}}{\leftarrow}
   }{
     \underset{\mathbf{\flat}_{dR}}{\to}
   }
  \mathbf{H}
  \,.
\end{displaymath}
\end{uprop}
\begin{proof}
We check the defining natural hom-equivalence

\begin{displaymath}
{*}/\mathbf{H}(\mathbf{\Pi}_{dR}X,A)
  \simeq
  \mathbf{H}(X, \mathbf{\flat}_{dR}A)
  \,.
\end{displaymath}
The hom-space in the [[over-(∞,1)-category|under-(∞,1)-category]] $*/\mathbf{H}$ is (as discussed there), computed by the [[(∞,1)-pullback]]

\begin{displaymath}
\itexarray{
     */\mathbf{H}(\mathbf{\Pi}_{dR}X, A)
      &\to&
     \mathbf{H}(\mathbf{\Pi}_{dR}X, A)
     \\
     \downarrow && \downarrow
     \\
     * &\stackrel{pt_A}{\to}& \mathbf{H}(*,A)
  }
  \,.
\end{displaymath}
By the fact that the [[hom-functor]] $\mathbf{H}(-,-) : \mathbf{H}^{op} \times \mathbf{H} \to \infty Grpd$ preserves limits in both arguments we have a natural equivalence

\begin{displaymath}
\begin{aligned}
     \mathbf{H}(\mathbf{\Pi}_{dR} X, A)
     & :=
     \mathbf{H}( *\coprod_{X} \mathbf{\Pi}(X), A  )     
     \\
     & \simeq 
     \mathbf{H}(*,A) \prod_{\mathbf{H}(X,A)} 
      \mathbf{H}(\mathbf{\Pi}(X),A)
  \end{aligned}
  \,.
\end{displaymath}
We paste this pullback to the above pullback diagram to obtain

\begin{displaymath}
\itexarray{
     */\mathbf{H}(\mathbf{\Pi}_{dR}X, A)
      &\to&
     \mathbf{H}(\mathbf{\Pi}_{dR}X, A) &\to&
      \mathbf{H}(\mathbf{\Pi}(X),A)
     \\
     \downarrow && \downarrow  && \downarrow
     \\
     * &\stackrel{pt_A}{\to}& \mathbf{H}(*,A)
     &\to& 
      \mathbf{H}(X,A)
  }
  \,.
\end{displaymath}
By the pasting law for [[(∞,1)-pullback]]s the outer diagram is still a pullback. We may evidently rewrite the bottom composite as in

\begin{displaymath}
\itexarray{
     */\mathbf{H}(\mathbf{\Pi}_{dR}X, A)
      &\to&
      &\to&
      \mathbf{H}(\mathbf{\Pi}(X),A)
     \\
     \downarrow && && \downarrow
     \\
     * &\stackrel{\simeq}{\to}& \mathbf{H}(X,*)
     &\stackrel{(pt_A)_*}{\to}& 
      \mathbf{H}(X,A)
  }
  \,.
\end{displaymath}
This exhibits the hom-space as the pullback

\begin{displaymath}
\begin{aligned}
    */\mathbf{H}(\mathbf{\Pi}_{dR}(X),A)
    \simeq
     \mathbf{H}(X,*) \prod_{\mathbf{H}(X,A)} \mathbf{H}(X,\mathbf{\flat} A)
   \end{aligned}
  \,,
\end{displaymath}
where we used the $(\mathbf{\Pi} \dashv \mathbf{\flat})$-adjunction. Now using again that $\mathbf{H}(X,-)$ preserves pullbacks, this is

\begin{displaymath}
\cdots \simeq \mathbf{H}(X, * \prod_A \mathbf{\flat}A )
  \simeq \mathbf{H}(X , \mathbf{\flat}_{dR}A)
  \,.
\end{displaymath}
\end{proof}
\begin{uprop}
If $\mathbf{H}$ is also [[local (∞,1)-topos|local]], then there is a further [[right adjoint]] $\mathbf{\Gamma}_{dR}$

\begin{displaymath}
(\mathbf{\Pi}_{dR} \dashv \mathbf{\flat}_{dR} \dashv \mathbf{\Gamma}_{dR})
  :
  \mathbf{H}
    \stackrel{\overset{\mathbf{\Pi}_{dR}}{\to}}{\stackrel{\stackrel{\mathbf{\flat}_{dR}}{\leftarrow}}{\underset{\mathbf{\Gamma}_{dR}}{\to}}}
  */\mathbf{H}
\end{displaymath}
given by

\begin{displaymath}
\mathbf{\Gamma}_{dR} X {:=}
    * \coprod_{X} \mathbf{\Gamma}(X)
  \,,
\end{displaymath}
where $(\mathbf{\Pi} \dashv \mathbf{\flat} \dashv \mathbf{\Gamma}) : \mathbf{H} \to \mathbf{H}$ is the triple of adjunctions discussed at \hyperlink{Paths}{Paths}.

\end{uprop}
\begin{proof}
This follows by the same kind of argument as above.

\end{proof}
\begin{udef}
For $X, A \in \mathbf{H}$ we write

\begin{displaymath}
\mathbf{H}_{dR}(X,A)
  :=
  \mathbf{H}(\mathbf{\Pi}_{dR}X, A)
  \simeq
  \mathbf{H}(X, \mathbf{\flat}_{dR} A)
  \,.
\end{displaymath}
A [[cocycle]] $\omega : X \to \mathbf{\flat}_{dR}A$ we call an \textbf{flat $A$-valued differential form} on $X$.

We say that $H_{dR}(X,A) {:=} \pi_0 \mathbf{H}_{dR}(X,A)$ is the \textbf{de Rham cohomology} of $X$ with coefficients in $A$.

\end{udef}
\begin{uremark}
A [[cocycle]] in de Rham cohomology

\begin{displaymath}
\omega : \mathbf{\Pi}_{dR}X \to A
\end{displaymath}
is precisely a \hyperlink{FlatDifferentialCohomology}{flat ∞-connetion} on a \emph{trivializable} $A$-principal $\infty$-bundle. More precisely, $\mathbf{H}_{dR}(X,A)$ is the [[homotopy fiber]] of the [[forgetful functor]] from $\infty$-bundles with flat $\infty$-connection to $\infty$-bundles: we have an [[(∞,1)-pullback]]

\begin{displaymath}
\itexarray{
    \mathbf{H}_{dR}(X,A) &\to& *
    \\
    \downarrow && \downarrow
    \\
    \mathbf{H}_{flat}(X,A) &\to& \mathbf{H}(X,A)
  }  
  \,.
\end{displaymath}
\end{uremark}
\begin{proof}
This follows by the fact that the [[hom-functor]] $\mathbf{H}(X,-)$ preserves the defining [[(∞,1)-pullback]] for $\mathbf{\flat}_{dR} A$.

\end{proof}
Just for emphasis, notice the dual description of this situation: by the [[universal property]] of the [[(∞,1)-colimit]] that defines $\mathbf{\Pi}_{dR} X$ we have that $\omega$ corresponds to a diagram

\begin{displaymath}
\itexarray{
    X &\to& *
    \\
    \downarrow &\swArrow_{\simeq}& \downarrow
    \\
    \mathbf{\Pi}(X) &\stackrel{\omega}{\to}& A
  }
  \,.
\end{displaymath}
The bottom horizontal morphism is a flat connection on the $\infty$-bundle given by the cocycle $X \to \mathbf{\Pi}(X) \stackrel{\omega}{\to} A$. The diagram says that this is equivalent to the trivial bundle given by the trivial cocycle $X \to * \to A$.

\begin{uprop}
The de Rham cohomology with coefficients in discrete objects is trivial: for all $S \in \infty Grpd$ we have

\begin{displaymath}
\mathbf{\flat}_{dR} Disc S \simeq *
  \,.
\end{displaymath}
\end{uprop}
\begin{proof}
Using that in a [[∞-connected (∞,1)-topos]] the functor $Disc$ is a [[full and faithful (∞,1)-functor]] so that the [[unit of an adjunction|unit]] $Id \to \Gamma Disc$ is an [[equivalence in an (∞,1)-category|equivalence]] and using that by the [[zig-zag identity]] we have then that the [[unit of an adjunction|counit]] component $\mathbf{\flat} Disc S := Disc \Gamma Disc S \to Disc S$ is also an equivalence, we have

\begin{displaymath}
\begin{aligned}
     \mathbf{\flat}_{dR} Disc S 
     & {:=}
     * \prod_{Disc S} \mathbf{\flat} Disc S
     \\
     & \simeq * \prod_{Disc S} Disc S
     \\
     & \simeq *
  \end{aligned}
  \,,
\end{displaymath}
since the pullback of an equivalence is an equivalence.

\end{proof}
\begin{uprop}
In a cohesive $\mathbf{H}$ \emph{\hyperlink{PiecesHavePoints}{pieces have points}} precisely if for all $X \in \mathbf{H}$, the de Rham coefficient object $\mathbf{\Pi}_{dR} X$ is \emph{globally [[connected]]} in that $\pi_0 \mathbf{H}(*, \mathbf{\Pi}_{dR}X) = *$.

If $X$ has at least one point ($\pi_0(\Gamma X) \neq \emptyset$) and is geometrically connected ($\pi_0 (\Pi X) = {*}$) then $\mathbf{\Pi}_{\mathrm{dR}}(X)$ is also locally connected: $\tau_0 \mathbf{\Pi}_{\mathrm{dR}}X \simeq {*} \in \mathbf{H}$.

\end{uprop}
\begin{proof}
Since $\Gamma$ preserves [[(∞,1)-colimit]]s in a cohesive $(\infty,1)$-topos we have

\begin{displaymath}
\begin{aligned}
    \mathbf{H}(*, \mathbf{\Pi}_{dR}X)
    & \simeq
    \Gamma \mathbf{\Pi}_{dR} X
    \\  
    & \simeq
    * \coprod_{\Gamma X} \Gamma \mathbf{\Pi}X  
    \\
    & \simeq
    * \coprod_{\Gamma X} \Pi X  
  \end{aligned}
  \,,
\end{displaymath}
where in the last step we used that $Disc$ is a [[full and faithful (∞,1)-functor|full and faithful]], so that there is an equivalence $\Gamma \mathbf{\Pi}X := \Gamma Disc \Pi X \simeq \Pi X$.

To analyse this [[(∞,1)-colimit|(∞,1)pushout]] we present it by a [[homotopy pushout]] in the standard [[model structure on simplicial sets]] $\mathrm{sSet}_{\mathrm{Quillen}}$. Denoting by $\Gamma X$ and $\Pi X$ any representatives in $\mathrm{sSet}_{\mathrm{Quillen}}$ of the objects of the same name in $\infty \mathrm{Grpd}$, this may be computed by the ordinary [[pushout]] in [[sSet]]

\begin{displaymath}
\itexarray{
    \Gamma X  &\to& (\Gamma X) \times \Delta[1] \coprod_{\Gamma X} {*}
    \\
    \downarrow && \downarrow
    \\
    \Pi X &\to & Q
  }
  \,,
\end{displaymath}
where on the right we have inserted the [[cone]] on $\Gamma X$ in order to turn the top morphism into a cofibration. From this ordinary pushout it is clear that the connected components of $Q$ are obtained from those of $\Pi X$ by identifying all those in the image of a connected component of $\Gamma X$. So if the left morphism is surjective on $\pi_0$ then $\pi_0(Q) = *$. This is precisely the condition that \hyperlink{PiecesHavePoints}{pieces have points} in $\mathbf{H}$.

For the local analysis we consider the same setup objectwise in the injective [[model structure on simplicial presheaves]] $[C^{\mathrm{op}}, \mathrm{sSet}]_{\mathrm{inj},\mathrm{loc}}$. For any $U \in C$ we then have the pushout $Q_U$ in

\begin{displaymath}
\itexarray{
    X(U) &\to & (X(U)) \times \Delta[1] \coprod_{X(U)} {*}
    \\
    \downarrow && \downarrow
    \\
    \mathrm{sSet}(\Gamma(U), \Pi X) & \to & Q_U
  }
  \,,
\end{displaymath}
as a model for the value of the simplicial presheaf presenting $\mathbf{\Pi}_{\mathrm{dR}}(X)$. If $X$ is geometrically connected then $\pi_0 \mathrm{sSet}(\Gamma(U), \Pi(X)) = *$ and hence for the left morphism to be surjective on $\pi_0$ it suffices that the top left object is not empty. Since the simplicial set $X(U)$ contains at least the vertices $U \to * \to X$ of which there is by assumption at least one, this is the case.

\end{proof}
\begin{uremark}
In summary this means that in a cohesive $(\infty,1)$-topos the objects $\mathbf{\Pi}_{dR} X$ have the abstract properties of [[de Rham schematic homotopy type|pointed geometric de Rham homotopy types]].

In the \hyperlink{Examples}{Examples} we will see that, indeed, the intrinsic de Rham cohomology $H_{dR}(X, A) {:=} \pi_0 \mathbf{H}(\mathbf{\Pi}_{dR} X, A)$ reproduces ordinary de Rham cohomology in degree $d\gt 1$.

In degree 0 the intrinsic de Rham cohomology is necessrily trivial, while in degree 1 we find that it reproduces closed 1-forms, not divided out by exact forms. This difference to ordinary de Rham cohomology in the lowest two degrees may be interpreted in terms of the obstruction-theoretic meaning of de Rham cohomology by which we essentially characterized it above: we have that the intrinsic $H_{dR}^n(X,K)$ is the home for the obstructions to flatness of $\mathbf{B}^{n-2}K$-[[principal ∞-bundle]]s. For $n = 1$ this are groupoid-principal bundles over the \emph{groupoid} with $K$ as its space of objects. But the 1-form curvatures of groupoid bundles are not to be regarded modulo exact forms. More details on this are at [[circle n-bundle with connection]].

\end{uremark}
\hypertarget{LieAlgebras}{}\subsubsection*{{Exponentiated $\infty$-Lie algebras}}\label{LieAlgebras}

\begin{udef}
For a [[connected]] object $\mathbf{B}\exp(\mathfrak{g})$ in $\mathbf{H}$ that is \emph{geometrically contractible}

\begin{displaymath}
\Pi (\mathbf{B}\exp(\mathfrak{g})) \simeq *
\end{displaymath}
we call its [[loop space object]] $\exp(\mathfrak{g}) := \Omega_* \mathbf{B}\exp(\mathfrak{g})$ the \textbf{[[Lie integration]] of an [[∞-Lie algebra]]} in $\mathbf{H}$.

\end{udef}
\begin{udef}
Set

\begin{displaymath}
\exp Lie
  := 
  \mathbf{\Pi}_{dR} \circ \mathbf{\flat}_{dR} 
  : */\mathbf{H} \to */\mathbf{H}
  \,.
\end{displaymath}
\end{udef}
\begin{uprop}
If $\mathbf{H}$ is cohesive, then $\exp Lie$ is a [[left adjoint]].

\end{uprop}
\begin{proof}
When $\mathbf{H}$ is cohesive we have the \hyperlink{TripleOfDeRhamAdjunctions}{de Rham triple of adjunction} $(\mathbf{\Pi}_{dR} \dashv \mathbf{\flat}_{dR} \dashv \mathbf{\Gamma}_{dR})$. Accordingly then $Lie$ is part of an [[adjunction]]

\begin{displaymath}
(\exp Lie \dashv \mathbf{\Gamma}_{dR}\mathbf{\flat}_{dR})
  \,.
\end{displaymath}
\end{proof}
\begin{uexample}
For all $X$ the object $\mathbf{\Pi}_{dR}(X)$ is geometrically contractible.

\end{uexample}
\begin{proof}
Since on the [[locally ∞-connected (∞,1)-topos]] and [[∞-connected (∞,1)-topos|∞-connected]] $\mathbf{H}$ the functor $\Pi$ preserves [[(∞,1)-colimit]]s and the [[terminal object in an (∞,1)-category|terminal object]], we have

\begin{displaymath}
\begin{aligned}
    \Pi \mathbf{\Pi}_{dR} X
    & {:=}
     \Pi (*) \coprod_{\Pi X} \Pi \mathbf{\Pi} X
    \\
    & \simeq
      * \coprod_{\Pi X} \Pi Disc \Pi X   
    \\
    & \simeq 
      * \coprod_{\Pi X} \Pi X
    & \simeq
      *
  \end{aligned}
  \,,
\end{displaymath}
where we used that in the [[∞-connected (∞,1)-topos|∞-connected]] $\mathbf{H}$ the functor $Disc$ is[[full and faithful (∞,1)-functor|full and faithful]].

\end{proof}
\begin{ucorollary}
We have for every $\mathbf{B}G$ that $\exp Lie \mathbf{B}G$ is geometrically contractible.

\end{ucorollary}
We shall write $\mathbf{B}\exp(\mathfrak{g})$ for $\exp Lie \mathbf{B}G$, when the context is clear.

\begin{uprop}
Every \hyperlink{deRhamCohomology}{de Rham cocycle} $\omega : \mathbf{\Pi}_{dR} X \to \mathbf{B}G$ factors through the ∞-Lie algebra of $G$

\begin{displaymath}
\itexarray{
     && \mathbf{B}\exp(\mathfrak{g})
     \\
     & \nearrow & \downarrow
     \\
     \mathbf{\Pi}_{dR}X 
     &\stackrel{\omega}{\to}&
     \mathbf{B}G
  }
  \,.
\end{displaymath}
\end{uprop}
\begin{proof}
By the  of $(\mathbf{\Pi}_{dR} \dashv \mathbf{\flat}_{dR})$ we have that $\omega$ factors through the [[unit of an adjunction|counit]9 $\exp Lie \mathbf{B}G \to \mathbf{B}G$.

\end{proof}
Therefore instead of speaking of a $G$-valued de Rham cocycle, it is less redundant to speak of an $\exp(\mathfrak{g})$-valued de Rham cocycle. In particular we have the following.

\begin{uprop}
Every morphism $\exp Lie \mathbf{B}H \to \mathbf{B}G$ from an exponentiated $\infty$-Lie algebra to an $\infty$-group factors through the exponentiated $\infty$-Lie algebra of that $\infty$-group

\begin{displaymath}
\itexarray{
    \mathbf{B}\exp(\mathfrak{h}) &\to& \mathbf{B}\exp(\mathfrak{g})
    \\
    & \searrow& \downarrow
    \\
    && \mathbf{B}G
  }
  \,.
\end{displaymath}
\end{uprop}
\begin{uprop}
If $\mathbf{H}$ is cohesive then we have

\begin{displaymath}
\exp Lie \circ \exp Lie \simeq \exp Lie \circ \Sigma \circ \Omega
  \,.
\end{displaymath}
\end{uprop}
\begin{proof}
First observe that for all $A \in */\mathbf{H}$ we have

\begin{displaymath}
\mathbf{\flat} \mathbf{\flat}_{dR} A \simeq *
\end{displaymath}
This follows using

\begin{itemize}%
\item $\mathbf{\flat}$ is a [[right adjoint]] and hence preserves [[(∞,1)-pullback]]s;


\item $\mathbf{\flat} \mathbf{\flat} := Disc \Gamma Disc \Gamma \simeq Disc \Gamma =: \mathbf{\flat}$ by the fact that $Disc$ is a [[full and faithful (∞,1)-functor]];


\item the counit $\mathbf{\flat} \mathbf{\flat} A \to \mathbf{\flat} A$ is equivalent to the identity, by the [[zig-zag-identity]] of the [[adjunction]] and using that equivalences satisfy [[category with weak equivalences|2-out-of-3]].



\end{itemize}
by computing

\begin{displaymath}
\begin{aligned}
    \mathbf{\flat} \mathbf{\flat}_{dR} A
    & 
    * \times_{\mathbf{\flat}A} \mathbf{\flat}\mathbf{\flat}A
    \\
    & \simeq 
    * \times_{\mathbf{\flat}A} \mathbf{\flat}A
    \\
    & \simeq *
 \end{aligned} 
 \,,
\end{displaymath}
using that the [[(∞,1)-pullback]] of an [[equivalence in an (∞,1)-category|equivalence]] is an equivalence.

From this we deduce that

\begin{displaymath}
\mathbf{\flat}_{dR} \circ \mathbf{\flat}_{dR} 
  \simeq \mathbf{\flat}_{dR} \circ \Omega
  \,.
\end{displaymath}
by computing for all $A \in \mathbf{H}$

\begin{displaymath}
\begin{aligned}
    \mathbf{\flat}_{dR} \circ \mathbf{\flat}_{dR} A
    & \simeq 
    * \times_{\mathbf{\flat}_{dR} A} \mathbf{\flat}\mathbf{\flat}_{dR} A
    \\
    & \simeq
    * \times_{\mathbf{\flat}_{dR} A} *
    \\
    & \simeq
    \mathbf{\flat}_{dR}( * \times_A * )
    \\
    & \simeq \mathbf{\flat}_{dR} \Omega A
  \end{aligned}
  \,.
\end{displaymath}
Also observe that by a \hyperlink{deRhamWithDiscCoeffsIsTrivial}{proposition above} we have

\begin{displaymath}
\mathbf{\flat}_{dR} \mathbf{\Pi} X \simeq *
\end{displaymath}
for all $X \in \mathbf{H}$.

Finally to obtain $\exp Lie \circ \exp Lie$ we do one more computation of this sort, using that

\begin{itemize}%
\item $\exp Lie := \mathbf{\Pi}_{dR} \mathbf{\flat}_{dR}$ preserves the terminal object (since $\mathbf{H}$ is [[locally ∞-connected (∞,1)-topos|locally ∞-connected]] and [[∞-connected (∞,1)-topos|∞-connected]])


\item and that it is a [[left adjoint]] by \hyperlink{LieAsALeftAdjoint}{the above}, since $\mathbf{H}$ is assumed to be cohesive.



\end{itemize}
We compute:

\begin{displaymath}
\begin{aligned}
    \exp Lie \exp Lie A 
    & \simeq 
    \exp Lie \mathbf{\Pi}_{dR} \mathbf{\flat}_{dR} A
    \\
    & \simeq
      * 
        \coprod_{\exp Lie \mathbf{\flat}_{dR} A} 
      \exp Lie \mathbf{\Pi} \mathbf{\flat}_{dR} A
    \\
    & \simeq
      * 
        \coprod_{\exp Lie \mathbf{\flat}_{dR} A} 
      \mathbf{\Pi}_{dR} \mathbf{\flat}_{dR} \mathbf{\Pi} \mathbf{\flat}_{dR} A
    \\
    & \simeq 
    * \coprod_{\exp Lie \mathbf{\flat}_{dR} A} *    
    \\
    & \simeq 
    * \coprod_{\mathbf{\Pi}_{dR} \mathbf{\flat}_{dR} \mathbf{\flat}_{dR} A} *
    \\
    & \simeq 
    * \coprod_{\mathbf{\Pi}_{dR} \mathbf{\flat}_{dR} \Omega A} *   
    \\
    & \simeq
    * \coprod_{\exp Lie \Omega A} *   
    \\
    & \simeq
    \exp Lie ( * \coprod_{\Omega A} * )
    \\
    & \simeq
    \exp Lie \Sigma \Omega A  
  \end{aligned} 
  \,.
\end{displaymath}
\end{proof}
\hypertarget{CurvatureCharacteristics}{}\subsubsection*{{Maurer-Cartan forms and curvature characteristic forms}}\label{CurvatureCharacteristics}

In the \hyperlink{deRhamCohomology}{intrinsic de Rham cohomology} of a [[locally ∞-connected (∞,1)-topos|locally ∞-connected]] [[∞-connected (∞,1)-topos|∞-connected]] there exist canonical cocycles that we may identify with [[Maurer-Cartan form]]s and with universal [[curvature characteristic form]]s.

\begin{udef}
For $G \in \mathbf{H}$ an [[∞-group]], write

\begin{displaymath}
\theta : G \to \mathbf{\flat}_{dR} \mathbf{B}G
\end{displaymath}
for the $\mathfrak{g}$-\hyperlink{LieAlgebras}{valued} de Rham cocycle on $G$ which is induced by the 

\begin{displaymath}
\itexarray{
     G &\to& *
     \\
     {}^{\mathllap{\theta}}\downarrow && \downarrow
     \\
     \mathbf{\flat}_{dR} \mathbf{B}G &\to& \mathbf{\flat}\mathbf{B}G
     \\
     \downarrow && \downarrow
     \\
     * &\to& \mathbf{B}G 
  }
\end{displaymath}
and the \hyperlink{LieValuesofDeRham}{above proposition}.

We call $\theta$ the \textbf{[[Maurer-Cartan form]]} on $G$.

\end{udef}
\begin{uremark}
By postcomposition the Maurer-Cartan form sends $G$-valued functions on $X$ to $\mathfrak{g}$-valued forms on $X$

\begin{displaymath}
\theta_* : \mathbf{H}(X,G) \to 
    \mathbf{H}^1_{dR}(X,G)
  \,.
\end{displaymath}
\end{uremark}
\begin{udefn}
For $G = \mathbf{B}^n A$ an [[Eilenberg-MacLane object]], we also write

\begin{displaymath}
curv : \mathbf{B}^n A \to \mathbf{\flat}_{dR} \mathbf{B}^{n+1} A
\end{displaymath}
for the intrinsic Maurer-Cartan form and call this the intrinsic \textbf{universal [[curvature characteristic form]]} on $\mathbf{B}^n A$.

\end{udefn}
\hypertarget{DifferentialCohomology}{}\subsubsection*{{Differential cohomology}}\label{DifferentialCohomology}

In every [[locally ∞-connected (∞,1)-topos|locally ∞-connected]] [[∞-connected (∞,1)-topos]] there is an intrinsic notion of [[ordinary differential cohomology]].

Fix a 0-[[truncated]] [[abelian group|abelian]] [[group object]] $A \in \tau_{\leq 0} \mathbf{H} \hookrightarrow \mathbf{H}$. For all $n \in \mathbf{N}$ we have then the [[Eilenberg-MacLane object]] $\mathbf{B}^n A$.

\begin{udef}
For $X \in \mathbf{H}$ any object and $n \geq 1$ write

\begin{displaymath}
\mathbf{H}_{diff}(X,\mathbf{B}^n A) := 
    \mathbf{H}(X,\mathbf{B}^n A) \prod_{\mathbf{H}_{dR}(X,\mathbf{B}^n A)}
    H_{dR}^{n+1}(X,A)
\end{displaymath}
for the cocycle $\infty$-groupoid of [[twisted cohomology]], def. \ref{TwistedCohomologyInOvertopos}, of $X$ with coefficients in $A$ and with twist given by the canonical \hyperlink{CurvatureCharacteristics}{curvature characteristic morphism} $curv : \mathbf{B}^n A \to \mathbf{\flat}_{dR}\mathbf{B}^{n+1} A$. This is the [[(∞,1)-pullback]]

\begin{displaymath}
\itexarray{
    \mathbf{H}_{diff}(X,\mathbf{B}^n A) 
      &\stackrel{[F]}{\to}& 
     H_{dR}^{n+1}(X,A)
    \\
    {}^{\mathllap{\eta}}\downarrow && \downarrow
    \\
    \mathbf{H}(X,\mathbf{B}^n A) 
        &\stackrel{curv_*}{\to}& 
     \mathbf{H}_{dR}(X,\mathbf{B}^{n+1} A)
  }
  \,,
\end{displaymath}
where the right vertical morphism $H^{n+1}_{dR}(X) = \pi_0 \mathbf{H}_{dR}(X,\mathbf{B}^{n+1}A)
 \to \mathbf{H}_{dR}(X,\mathbf{B}^{n+1}A)$ is any choice of cocycle representative for each cohomology class: a choice of point in every connected component.

We call

\begin{displaymath}
H_{diff}^n(X,A) {:=} \pi_0 \mathbf{H}_{diff}(X, \mathbf{B}^{n} A)
\end{displaymath}
the degree-$n$ \textbf{[[differential cohomology]]} of $X$ with coefficient in $A$.

For $\nabla \in \mathbf{H}_{diff}(X,\mathbf{B}^n A)$ a [[cocycle]], we call

\begin{itemize}%
\item $[\eta(\nabla)] \in H^n(X,A)$ the class of the \textbf{underlying $\mathbf{B}^{n-1} A$-[[principal ∞-bundle]]};


\item $F(\nabla) \in H_{dR}^{n+1}(X,A)$ the \textbf{[[curvature]]} class of $c$.



\end{itemize}
We also say $\nabla$ is an \textbf{$\infty$-connection} on $\eta(\nabla)$ (see \hyperlink{ChernWeilTheory}{below}).

\end{udef}
\begin{uprop}
The differential cohomology $H_{diff}^n(X,A)$ does not depend on the choice of morphism $H_{dR}^{n+1}(X,A) \to \mathbf{H}_{dR}(X, \mathbf{B}^{n+1}A)$ (as long as it is an isomorphism on $\pi_0$, as required). In fact, for different choices the corresponding cocycle [[∞-groupoid]]s $\mathbf{H}_{diff}(X,\mathbf{B}^n A)$ are equivalent.

\end{uprop}
\begin{proof}
The set

\begin{displaymath}
H_{dR}^{n+1}(X,A) = \coprod_{H_{dR}^{n+1}(X,A)} {*}
\end{displaymath}
is, as a 0-[[truncated]] [[∞-groupoid]], an [[(∞,1)-coproduct]] of the [[terminal object in an (∞,1)-category|terminal object]] in [[∞Grpd]]. By [[universal colimits]] in this [[(∞,1)-topos]] we have that [[(∞,1)-colimit]]s are preserved by [[(∞,1)-pullback]]s, so that $\mathbf{H}_{diff}(X, \mathbf{B}^n A)$ is the coproduct

\begin{displaymath}
\mathbf{H}_{diff}(X,\mathbf{B}^n A) \simeq
  \coprod_{H_{dR}^{n+1}(X,A)} 
   \left(
   \mathbf{H}(X,\mathbf{B}^n A) 
      \prod_{\mathbf{H}_{dR}(X,\mathbf{B}^{n+1}A)} 
    {*}
  \right)
\end{displaymath}
of the [[homotopy fiber]]s of $curv_*$ over each of the chosen points $* \to \mathbf{H}_{dR}(X,\mathbf{B}^{n+1}A)$. These homotopy fibers only depend, up to equivalence, on the connected component over which they are taken.

\end{proof}
\begin{uprop}
When restricted to vanishing curvature, differential cohomology coincides with \hyperlink{FlatDifferentialCohomology}{flat differential cohomology}:

\begin{displaymath}
H_{diff}^n (X,A)|_{[F] = 0} \simeq H_{flat}(X,\mathbf{B}^n A)
  \,.
\end{displaymath}
Moreover this is true at the level of [[cocycle]] [[∞-groupoid]]s

\begin{displaymath}
\left(
  \mathbf{H}_{diff}(X, \mathbf{B}^n A) 
    \prod_{H_{dR}^{n+1}(X,A)} \{[F] = 0\}
  \right)
  \simeq
  \mathbf{H}_{flat}(X,\mathbf{B}^n A)
  \,.
\end{displaymath}
\end{uprop}
\begin{proof}
By the  the claim is equivalently that we have a an $(\infty,1)$-pullback diagram

\begin{displaymath}
\itexarray{
    \mathbf{H}_{flat}(X, \mathbf{B}^n A) &\to& {*}
    \\
    \downarrow && \downarrow^{\mathrlap{\{[F] = 0\}}}
    \\
    \mathbf{H}_{diff}(X,\mathbf{B}^n A) &\stackrel{[F]}{\to}& 
     H_{dR}^{n+1}(X,A)
    \\
    {}^{\mathllap{\eta}}\downarrow && \downarrow
    \\
    \mathbf{H}(X,\mathbf{B}^n A) 
        &\stackrel{curv_*}{\to}& 
     \mathbf{H}_{dR}(X,\mathbf{B}^{n+1} A)
  }
  \,.
\end{displaymath}
By definition of flat cohomology and of intrinsic de Rham cohomology in $\mathbf{H}$, the outer rectangle is

\begin{displaymath}
\itexarray{
    \mathbf{H}(X,\mathbf{\flat}\mathbf{B}^n A) &\to& {*}
    \\ 
    \downarrow && \downarrow 
    \\
    \mathbf{H}(X, \mathbf{B}^n A) 
      &\stackrel{curv_*}{\to}&
    \mathbf{H}(X, \mathbf{\flat}_{dR}\mathbf{B}^{n+1} A)
  }
  \,.
\end{displaymath}
Since the [[hom-functor]] $\mathbf{H}(X,-)$ preserves [[(∞,1)-limit]]s this is a pullback if

\begin{displaymath}
\itexarray{
    \mathbf{\flat} \mathbf{B}^n A &\to& *
    \\
    \downarrow && \downarrow
    \\
    \mathbf{B}^n A 
      &\stackrel{curv}{\to}& 
    \mathbf{\flat}_{dR} \mathbf{B}^{n+1} A
  }
\end{displaymath}
is. Indeed, this is one step in the [[fiber sequence]]

\begin{displaymath}
\cdots
   \to 
   \mathbf{\flat} \mathbf{B}^n A
   \to
   \mathbf{B}^n A
   \stackrel{curv}{\to}
   \mathbf{\flat}_{dR} \mathbf{B}^{n+1}A
  \to
  \mathbf{\flat} \mathbf{B}^{n+1} A
   \to
  \mathbf{B}^{n+1} A
\end{displaymath}
that \hyperlink{CurvatureCharacteristics}{defines} $curv$ (using that $\mathbf{\flat}$ preserves limits and hence looping and delooping)

\end{proof}
\begin{uprop}
The differential cohomology group $H_{diff}^n(X,A)$ fits into a [[short exact sequence]] of [[abelian group]]s

\begin{displaymath}
0 
   \to 
  H_{dR}^n(X,A)/H^{n-1}(X,A) 
   \to 
  H_{diff}^n(X,A) 
   \to 
  H^n(X,A)
   \to 
  0
  \,.
\end{displaymath}
\end{uprop}
\begin{proof}
This is a general statement about the definition of [[twisted cohomology]]. We claim that for all $n \geq 1$ we have a [[fiber sequence]]

\begin{displaymath}
\mathbf{H}(X, \mathbf{B}^{n-1}A)
  \to 
  \mathbf{H}_{dR}(X, \mathbf{B}^n A)
  \to 
  \mathbf{H}_{diff}(X, \mathbf{B}^n A)
  \to
  \mathbf{H}(X, \mathbf{B}^n A)
\end{displaymath}
in [[∞Grpd]]. This implies the [[short exact sequence]] using that by construction the last morphism is surjective on connected components (because in the defining $(\infty,1)$-pullback for $\mathbf{H}_{diff}$ the right vertical morphism is by assumption surjective on connected components).

To see that we do have the fiber sequence as claimed consider the pasting composite of [[(∞,1)-pullback]]s

\begin{displaymath}
\itexarray{
    \mathbf{H}_{dR}(X,\mathbf{B}^{n-1} A) 
    &\to& 
      \mathbf{H}_{diff}(X,\mathbf{B}^n A)  
    &\to& 
     H_{dR}(X, \mathbf{B}^{n+1} A)
    \\
    \downarrow && \downarrow && \downarrow
    \\
    {*} 
      &\to& 
    \mathbf{H}(X, \mathbf{B}^n A) 
      &\stackrel{curv}{\to}&
    \mathbf{H}_{dR}(X, \mathbf{B}^{n+1} A)
  }
  \,.
\end{displaymath}
The square on the right is a pullback by the above definition. Since also the square on the left is assumed to be an $(\infty,1)$-pullback it follows by the  that the top left object is the $(\infty,1)$-pullback of the total rectangle diagram. That total diagram is

\begin{displaymath}
\itexarray{
    \Omega \mathbf{H}(X, \mathbf{\flat}_{dR} \mathbf{B}^{n+1}A) 
    &\to& 
     H(X,\mathbf{\flat}_{dR} \mathbf{B}^{n+1} A)
    \\
    \downarrow && \downarrow
    \\
    * &\to& \mathbf{H}(X,\mathbf{\flat}_{dR} \mathbf{B}^{n+1} A)
  }
  \,,
\end{displaymath}
because, as before, this $(\infty,1)$-pullback is the coproduct of the homotopy fibers, and they are empty over the connected components not in the image of the bottom morphism and are the [[loop space object]] over the single connected component that is in the image.

Finally using that (as discussed at [[cohomology]] and at [[fiber sequence]])

\begin{displaymath}
\Omega \mathbf{H}(X,\mathbf{\flat}_{dR} \mathbf{B}^{n+1}A) 
    \simeq 
   \mathbf{H}(X,\Omega \mathbf{\flat}_{dR} \mathbf{B}^{n+1}A)
\end{displaymath}
and

\begin{displaymath}
\Omega \mathbf{\flat}_{dR} \mathbf{B}^{n+1}A 
    \simeq 
   \mathbf{\flat}_{dR} \Omega \mathbf{B}^{n+1}A
\end{displaymath}
since both $\mathbf{H}(X,-)$ as well as $\mathbf{\flat}_{dR}$ preserve [[(∞,1)-limit]]s and hence formation of [[loop space object]]s, the claim follows.

\end{proof}
\begin{uremark}
This is essentially the short exact sequence whose form is familiar from the traditional definition of [[ordinary differential cohomology]] only up to the following slight nuances in notation:

\begin{enumerate}%
\item The cohomology groups of the short exact sequence above denote the groups obtained in the given [[(∞,1)-topos]] $\mathbf{H}$, not in [[Top]]. Notably for $\mathbf{H} =$ [[?LieGrpd]], $A = U(1) =\mathbb{R}/\mathbb{Z}$ the [[circle group]] and $|X| \in Top$ the [[geometric realization]] of a paracompact manifold $X$, we have that $H^n(X,\mathbb{R}/\mathbb{Z})$ above is $H^{n+1}_{sing}({|\Pi X|},\mathbb{Z})$.


\item The fact that on the left of the short exact sequence for differential cohomology we have the de Rham cohomology set $H_{dR}^n(X,A)$ instead of something like the set of all flat forms as familiar from [[ordinary differential cohomology]] is because the latter has no intrinsic meaning but depends on a choice of model. After fixing a specific [[presentable (∞,1)-category|presentation]] of $\mathbf{H}$ by a [[model category]] $C$ we can consider instead of $H_{dR}^{n+1}(X,A) \to \mathbf{H}_{dR}(X, \mathbf{B}^{n+1}A)$ the inclusion of the set of objects $\Omega_{cl}^{n+1}(X,A) {:=} \mathbb{R}Hom_C(X, \mathbf{B}^{n+1}A )_0
\hookrightarrow \mathbb{R}Hom_C(X, \mathbf{B}^{n+1}A )$. However, by the \hyperlink{DiffCohIsWellDefined}{above observation} this only adds multiple copies of the homotopy types of the connected components of $\mathbf{H}_{diff}(X, \mathbf{B}^n A)$.



\end{enumerate}
\end{uremark}
\hypertarget{ChernWeilTheory}{}\subsubsection*{{Chern-Weil homomorphism and $\infty$-connections}}\label{ChernWeilTheory}

Induced by the \hyperlink{DifferentialCohomology}{intrinsic differential cohomology} in any [[∞-connected (∞,1)-topos|∞-connected]] and [[locally ∞-connected (∞,1)-topos]] is an intrinsic notion of [[Chern-Weil homomorphism]].

Let $A$ be the chosen abelian [[∞-group]] as \hyperlink{DifferentialCohomology}{above}. Recall the \hyperlink{CurvatureCharacteristics}{universal curvature characteristic class}

\begin{displaymath}
curv : \mathbf{B}^n A \to \mathbf{\flat}_{dR}\mathbf{B}^{n+1}A
\end{displaymath}
for all $n \geq 1$.

\begin{udef}
For $G$ an [[∞-group]] and

\begin{displaymath}
\mathbf{c} : \mathbf{B}G \to \mathbf{B}^n A
\end{displaymath}
a representative of a [[characteristic class]] $[\mathbf{c}] \in H^n(\mathbf{B}G, A)$ we say that the composite

\begin{displaymath}
\mathbf{c}_{dR} : \mathbf{B}G \stackrel{\mathbf{c}}{\to}
    \mathbf{B}^n A 
     \stackrel{curv}{\to}
    \mathbf{\flat}_{dR} \mathbf{B}^{n+1} A
\end{displaymath}
represents the corresponding [[differential characteristic class]] or \textbf{[[curvature characteristic class]]} $[\mathbf{c}_{dR}] \in H_{dR}^{n+1}(\mathbf{B}G, A)$.

The induced map on cohomology

\begin{displaymath}
(\mathbf{c}_{dR})_* 
   : 
  H^1(-,G)
  \to
  H^{n+1}_{dR}(-,A)
\end{displaymath}
we call the (unrefined) \textbf{[[∞-Chern-Weil homomorphism]]} induced by $\mathbf{c}$.

\end{udef}
The following construction universally lifts the $\infty$-Chern-Weil homomorphism from taking values in \hyperlink{deRhamCohomology}{intrinsic de Rham cohomology} to values in \hyperlink{DifferentialCohomology}{intrinsic differential cohomology}.

\begin{udef}
For $X \in \mathbf{H}$ any object, define the [[∞-groupoid]] $\mathbf{H}_{conn}(X,\mathbf{B}G)$ as the [[(∞,1)-pullback]]

\begin{displaymath}
\itexarray{
    \mathbf{H}_{conn}(X, \mathbf{B}G) 
     &\stackrel{(\hat \mathbf{c}_i)_i}{\to}& 
     \prod_{[\mathbf{c}_i] \in H^{n_i}(\mathbf{B}G,A); i \geq 1} 
     \mathbf{H}_{diff}(X,\mathbf{B}^{n_i} A)
   \\
   {}^{\mathllap{\eta}}\downarrow && \downarrow
   \\
   \mathbf{H}(X, \mathbf{B}G) 
     &\stackrel{( \mathbf{c}_i)_i}{\to}& 
     \prod_{[\mathbf{c}_i] \in H^{n_i}(\mathbf{B}G,A); i \geq 1} 
   \mathbf{H}(X,\mathbf{B}^{n_i} A)
  }
 \,.
\end{displaymath}
We say

\begin{itemize}%
\item a cocycle in $\nabla \in \mathbf{H}_{conn}(X, \mathbf{B}G)$ is an [[connection on an ∞-bundle|∞-connection]]


\item on the [[principal ∞-bundle]] $\eta(\nabla)$;


\item a morphism in $\mathbf{H}_{conn}(X, \mathbf{B}G)$ is a [[gauge transformation]] of connections;


\item for each $[\mathbf{c}] \n H^n(\mathbf{B}G, A)$ the morphism

\begin{displaymath}
[\hat \mathbf{c}] :  H_{conn}(X,\mathbf{B}G) \to H_{diff}^n(X, A)
\end{displaymath}
is the (full/refined) \textbf{[[∞-Chern-Weil homomorphism]]} induced by the [[characteristic class]] $[\mathbf{c}]$.



\end{itemize}
\end{udef}
\begin{uprop}
Under the [[curvature]] projection $[F] : H_{diff}^n (X,A) \to H_{dR}^{n+1}(X,A)$ the refined Chern-Weil homomorphism for $\mathbf{c}$ projects to the unrefined Chern-Weil homomorphism.

\end{uprop}
\begin{proof}
This is due to the existence of the pasting composite

\begin{displaymath}
\itexarray{
    \mathbf{H}_{conn}(X, \mathbf{B}G) 
     &\stackrel{(\hat \mathbf{c}_i)_i}{\to}& 
     \prod_{[\mathbf{c}_i] \in H^{n_i}(\mathbf{B}G,A); i \geq 1} 
     \mathbf{H}_{diff}(X,\mathbf{B}^{n_i} A)
    &\stackrel{[F]}{\to}&
     \prod_{[\mathbf{c}_i] \in H^{n_i}(\mathbf{B}G,A); i \geq 1} 
     H_{dR}^{n_i+1}(X,A)
   \\
   {}^{\mathllap{\eta}}\downarrow && \downarrow && \downarrow
   \\
   \mathbf{H}(X, \mathbf{B}G) 
     &\stackrel{(\mathbf{c}_i)_i}{\to}& 
     \prod_{[\mathbf{c}_i] \in H^{n_i}(\mathbf{B}G,A); i \geq 1} 
   \mathbf{H}(X,\mathbf{B}^{n_i} A)
    &\stackrel{curv_*}{\to}&
     \prod_{[\mathbf{c}_i] \in H^{n_i}(\mathbf{B}G,A); i \geq 1} 
    \mathbf{H}_{dR}(X, \mathbf{B}^{n_i+1},A)
  }
\end{displaymath}
of the defining $(\infty,1)$-pullback for $\mathbf{H}_{conn}(X,\mathbf{B}G)$ with the products of the defining $(\infty,1)$-pullbacks for the $\mathbf{H}_{diff}(X, \mathbf{B}^{n_i}A)$.

\end{proof}
\hypertarget{ChernSimonsTheory}{}\subsubsection*{{Higher holonomy and Chern-Simons functional}}\label{ChernSimonsTheory}

The notion of \hyperlink{ChernWeilTheory}{intrinsic ∞-connections} in a cohesive $(\infty,1)$-topos induces a notion of [[higher parallel transport|higher holonomy]] and [[Chern-Simons theory|Chern-Simons functionals]].

\begin{uprop}
We say an object $\Sigma \in \mathbf{H}$ has \textbf{[[cohomological dimension]]} $\leq n \in \mathbb{N}$ if for all $n$-[[connected]] coefficient objects and $(n++1)$-[[truncated]] objects $\mathbf{B}^{n+1}A$ the corresponding cohomology on $\Sigma$ is trivial

\begin{displaymath}
H(\Sigma, \mathbf{B}^{n+1}A ) \simeq *
  \,.
\end{displaymath}
Let $dim(\Sigma)$ be the maximum $n$ for which this is true.

\end{uprop}
\begin{uprop}
If $\Sigma$ has [[cohomological dimension]] $\leq n$ then its \hyperlink{deRhamCohomology}{intrinsic de Rham cohomology} vanishes in degree $k \gt n$

\begin{displaymath}
H_{dR}^{k \gt n}(\Sigma, A) \simeq *
  \,.
\end{displaymath}
\end{uprop}
\begin{proof}
Since $\mathbf{\flat}$ is a [[right adjoint]] it preserves [[delooping]] and hence $\mathbf{\flat} \mathbf{B}^k A \simeq \mathbf{B}^k \mathbf{\flat}A$. It follows that

\begin{displaymath}
\begin{aligned}
     H_{dR}^{k}(\Sigma,A)
     & :=
     \pi_0 \mathbf{H}(\Sigma, \mathbf{\flat}_{dR} \mathbf{B}^k A)
    \\
    & \simeq
     \pi_0 \mathbf{H}(\Sigma, * \prod_{\mathbf{B}^k A} \mathbf{B}^k \mathbf{\flat}A)
    \\
    & \simeq
    \pi_0
     \left( 
     \mathbf{H}(\Sigma,*) \prod_{\mathbf{H}(\Sigma, \mathbf{B}^k A)}
      \mathbf{H}(\Sigma, \mathbf{B}^k \mathbf{\flat}A)
     \right)
   \\
   & \simeq
    \pi_0 (*)
  \end{aligned}
  \,.
\end{displaymath}
\end{proof}
Let now again $A$ be fixed as \hyperlink{DifferentialCohomology}{above}.

\begin{udef}
Let $\Sigma \in \mathbf{H}$, $n \in \mathbf{N}$ with $dim \Sigma \leq n$.

We say that the composite

\begin{displaymath}
\int_\Sigma 
  : 
  \mathbf{H}_{flat}(\Sigma, \mathbf{B}^n A)
   \stackrel{\simeq}{\to}
  \infty Gprd(\Pi(\Sigma), \Pi(\mathbf{B}^n A))
   \stackrel{\tau_{\leq n-dim(\Sigma)}}{\to}
  \tau_{n-dim(\Sigma)}
    \infty Gprd(\Pi(\Sigma), \Pi(\mathbf{B}^n A))
\end{displaymath}
of the [[adjunction]] equivalence followed by [[truncated|truncation]] is the \textbf{flat holonomy} operation on flat $\infty$-connections.

More generally, let

\begin{itemize}%
\item $\nabla \in \mathbf{H}_{diff}(X, \mathbf{B}^n A)$ be a differential coycle on some $X \in \mathbf{H}$


\item $\phi : \Sigma \to X$ a morphism.



\end{itemize}
Write

\begin{displaymath}
\phi^* : \mathbf{H}_{diff}(X, \mathbf{B}^{n+1} A) 
    \to 
  \mathbf{H}_{diff}(\Sigma, \mathbf{B}^n A)
   \simeq
  \mathbf{H}_{flat}(\Sigma, \mathbf{B}^n A)
\end{displaymath}
(using the \hyperlink{DiffCohomologyRestrictedToVanishingCurvature}{above proposition}) for the morphism on $(\infty,1)$-pullbacks induced by the morphism of diagrams

\begin{displaymath}
\itexarray{  
     \mathbf{H}(X, \mathbf{B}^n A) &\to&
     \mathbf{H}_{dR}(X, \mathbf{B}^{n+1} A)
     &\leftarrow&
     H_{dR}^{n+1}(X, A)
     \\
     \downarrow^{\mathrlap{\phi^*}}
     &&
     \downarrow^{\mathrlap{\phi^*}}
     &&
     \downarrow
     \\
     \mathbf{H}(\Sigma, \mathbf{B}^n A) &\to&
     \mathbf{H}_{dR}(X, \mathbf{B}^{n+1} A)
     &\leftarrow&
     *     
  }
\end{displaymath}
The \textbf{[[higher parallel transport|holonomomy]]} of $\nabla$ over $\sigma$ is the flat holonomy of $\phi^* \nabla$

\begin{displaymath}
\int_\phi \nabla := \int_{\Sigma} \phi^* \nabla
  \,.
\end{displaymath}
\end{udef}
\begin{udef}
Let $\Sigma \in \mathbf{H}$ be of [[cohomological dimension]] $dim\Sigma = n \in \mathbb{N}$ and let $\mathbf{c} : X \to \mathbf{B}^n A$ a representative of a [[characteristic class]] $[\mathbf{c}] \in H^n(X, A)$ for some object $X$. We say that the composite

\begin{displaymath}
\exp(S_{\mathbf{c}}(-))
  : 
  \mathbf{H}(\Sigma, X)
   \stackrel{\hat \mathbf{c}}{\to}
  \mathbf{H}_{diff}(\Sigma, \mathbf{B}^n A)
   \stackrel{\simeq}{\to}
  \mathbf{H}_{flat}(\Sigma, \mathbf{B}^n A)
   \stackrel{\int_\Sigma}{\to}
  \tau_{\leq 0}
    \infty Grpd(\Pi(\Sigma), \Pi \mathbf{B}^n A)
\end{displaymath}
where $\hat \mathbf{c}$ denotes the \hyperlink{ChernWeilTheory}{refined Chern-Weil homomorphism} induced by $\mathbf{c}$, is the \textbf{[[schreiber:∞-Chern-Simons theory|extended Chern-Simons functional]]} induced by $\mathbf{c}$ on $\Sigma$.

\end{udef}
\begin{udef}
In the language of [[sigma-model]] [[quantum field theory]] the ingredients of this definition have the following interpretation

\begin{itemize}%
\item $\Sigma$ is the \emph{worldvolume of a fundamental $(dim\Sigma-1)$-[[brane]]} ;


\item $X$ is the target space;


\item $\hat \mathbf{c}$ is the background [[gauge field]] on $X$;


\item $\mathbf{H}_{conn}(\Sigma,X)$ is the space of \emph{worldvolume field configurations} $\phi : \Sigma \to X$ or \emph{trajectories} of the brane in $X$;


\item $\exp(S_{\mathbf{c}}(\phi)) =  \int_\Sigma \phi^* \hat \mathbf{c}$ is the value of the [[action functional]] on the field configuration $\phi$.



\end{itemize}
\end{udef}
In suitable situations this construction refines to an internal construction.

Assume that $\mathbf{H}$ has a canonical [[line object]] $\mathbb{A}^1$ and a [[natural numbers object]] $\mathbb{Z}$. Then the [[action functional]] $\exp(i S(-))$ may lift to the [[internal hom]] with respect to the canonical  on any [[(∞,1)-topos]] to a morphism of the form

\begin{displaymath}
\exp(i S_{\mathbf{c}}(-)) : [\Sigma,\mathbf{B}G_{conn}] \to 
  \mathbf{B}^{n-dim \Sigma}\mathbb{A}^1/\mathbb{Z}
  \,.
\end{displaymath}
We call $[\Sigma, \mathbf{B}G_{conn}]$ the [[configuration space]] of the [[schreiber:∞-Chern-Simons theory]] defined by $\mathbf{c}$ and $\exp(i S_\mathbf{c}(-))$ the [[action functional]] in [[codimension]] $(n-dim\Sigma)$ defined on it.

See [[schreiber:∞-Chern-Simons theory]] for more discussion.

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

The [[category theory|category-theoretic]] definition of [[cohesive topos]] was proposed by [[Bill Lawvere]]. See the references at [[cohesive topos]].

The observation that the further left adjoint $\Pi$ in a [[locally ∞-connected (∞,1)-topos]] defines an intrinsic notion of paths and [[geometric homotopy groups in an (∞,1)-topos]] was suggested by [[Richard Williamson]].

The observation that the further right adjoint $coDisc$ in a [[local (∞,1)-topos]] serves to characterize [[concrete sheaf|concrete (∞,1)-sheaves]] was amplified by [[David Carchedi]].

Several aspects of the discussion here are, more or less explicitly, in

\begin{itemize}%
\item [[Carlos Simpson]], [[Constantin Teleman]], \emph{deRham theorem for $\infty$-stacks} (\href{http://math.berkeley.edu/~teleman/math/simpson.pdf}{pdf})

\end{itemize}
For instance something similar to the notion of [[infinity-connected (infinity,1)-site|∞-connected site]] and the [[fundamental ∞-groupoid in a locally ∞-connected (∞,1)-topos]] is the content of section 2.16. The \hyperlink{LieTheory}{infinitesimal path ∞-groupoid adjunction} $(\mathbf{Red} \dashv \mathbf{\Pi}_{inf} \dashv \mathbf{\flat}_{inf})$ is essentially discussed in section 3. The notion of geometric realization, \ref{GeometricRealization}, is touched on around remark 2.22, referring to

\begin{itemize}%
\item [[Carlos Simpson]], \emph{The topological realization of a simplicial presheaf} , \href{http://arxiv.org/abs/q-alg/9609004}{arXiv:q-alg/9609004}.

\end{itemize}
But, more or less explicitly, the presentation of geometric realization of simplicial presheaves is much older, going back to Artin-Mazur. See [[geometric homotopy groups in an (∞,1)-topos]] for a detailed commented list of literature.

A characterization of infinitesimal extensions and formal smoothness by adjoint functors (discussed at [[infinitesimal cohesion]]) is considered in

\begin{itemize}%
\item [[Maxim Kontsevich]], [[Alexander Rosenberg]], \emph{Noncommutative spaces}, preprint MPI-2004-35 (\href{http://www.mpim-bonn.mpg.de/preprints/send?bid=2331}{ps}, \href{http://www.mpim-bonn.mpg.de/preprints/send?bid=2303}{dvi})

\end{itemize}
in the context of \emph{[[Q-categories]]} .

The material presented here is also in section 2 of

\begin{itemize}%
\item [[Urs Schreiber]], \emph{[[schreiber:differential cohomology in a cohesive topos]]}

\end{itemize}
A commented list of further related references is at

\begin{itemize}%
\item [[schreiber:differential cohomology in an (∞,1)-topos -- references |differential cohomology in a cohesive topos -- references]]

\end{itemize}


\end{document}