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\begin{document}

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\section*{concretification}

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

\hypertarget{discrete_and_concrete_objects}{}\paragraph*{{Discrete and concrete objects}}\label{discrete_and_concrete_objects}

[[!include discrete and concrete objects - contents]]

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

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

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

\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{definition}{Definition}\dotfill \pageref*{definition} \linebreak
\noindent\hyperlink{InALocalTopos}{In a local topos}\dotfill \pageref*{InALocalTopos} \linebreak
\noindent\hyperlink{InAHigherCohesiveTopos}{In a cohesive $(\infty,1)$-topos}\dotfill \pageref*{InAHigherCohesiveTopos} \linebreak
\noindent\hyperlink{examples}{Examples}\dotfill \pageref*{examples} \linebreak
\noindent\hyperlink{ConcretificationOfDifferentialModuli}{Concretification of differential moduli}\dotfill \pageref*{ConcretificationOfDifferentialModuli} \linebreak
\noindent\hyperlink{models_for_image_factorization}{Models for $n$-image factorization}\dotfill \pageref*{models_for_image_factorization} \linebreak
\noindent\hyperlink{moduli_of_circle_connections}{Moduli of circle $n$-connections}\dotfill \pageref*{moduli_of_circle_connections} \linebreak
\noindent\hyperlink{differential_concretification_on_contractibles}{Differential concretification on contractibles}\dotfill \pageref*{differential_concretification_on_contractibles} \linebreak
\noindent\hyperlink{GeneralDifferentialConcretification}{General differential concretification}\dotfill \pageref*{GeneralDifferentialConcretification} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak


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

In a [[local topos]] there is a notion of \emph{[[concrete objects]]}. These form a [[reflective subcategory]]. The corresponding reflector is the \emph{concretification map} which universally approximates any object by a concrete object.

\hypertarget{definition}{}\subsection*{{Definition}}\label{definition}

There is a unique evident notion of concretification in any [[local topos]], this we discuss first in

\begin{itemize}%
\item \emph{\hyperlink{InALocalTopos}{In a local topos}}

\end{itemize}
This involves an [[image]] factorization. Since in [[higher category theory]]/[[homotopy theory]] image factorization refines to a tower of notions of [[n-image]] factorization in a [[local (∞,1)-topos]] there are different constructions that one may all think of as concretification. This we discuss in

\begin{itemize}%
\item \emph{\hyperlink{InAHigherCohesiveTopos}{In a higher cohesive topos}}

\end{itemize}
\hypertarget{InALocalTopos}{}\subsubsection*{{In a local topos}}\label{InALocalTopos}

A [[local topos]] is a [[topos]] equipped with a [[sharp modality]] $\sharp$.

\begin{defn}
\label{}\hypertarget{}{}
For $X$ any object of the topos, the [[image]] projection of the [[unit of a monad|unit]] $\iota_X \colon X \to \sharp X$ is the \emph{concretification} of $X$

\begin{displaymath}
(X \to Conc X) 
  \coloneqq
  (X \to im(\iota_X))
  \,.
\end{displaymath}
\end{defn}
\hypertarget{InAHigherCohesiveTopos}{}\subsubsection*{{In a cohesive $(\infty,1)$-topos}}\label{InAHigherCohesiveTopos}

In a [[local (∞,1)-topos]] there is still the [[sharp modality]], but here the [[1-image]]-factorization of its unit is rarely of interest, for this concretifies an object in degree 0 but makes it [[codiscrete object|codiscrete]] in all higher degrees. Typically one is interested in concretifying in all degrees. One needs to specify extra data to say what this means.

One case where this is arises is the \emph{differential concretification} of [[moduli ∞-stacks]] of [[principal ∞-connections]].

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

\hypertarget{ConcretificationOfDifferentialModuli}{}\subsubsection*{{Concretification of differential moduli}}\label{ConcretificationOfDifferentialModuli}

\hypertarget{models_for_image_factorization}{}\paragraph*{{Models for $n$-image factorization}}\label{models_for_image_factorization}

The following gives a sufficient condition for modeling [[n-image]] factorizations in some [[(∞,1)-toposes]] with particularly convenient presentation.

\begin{prop}
\label{nImageFactroizationModeledOnSimplicialPrsheaves}\hypertarget{nImageFactroizationModeledOnSimplicialPrsheaves}{}
Let $C$ be a site with [[point of a topos|enough points]], so that the weak equivalences in $sPSh(C)_{\mathrm{loc}}$ are detected on [[stalks]] (\href{model+structure+on+simplicial+presheaves#OverSiteWithEnoughPointsWeakEquivalencesDetectedOnStalks}{this prop.}). Then given a morphism of [[Kan complex]]-valued [[simplicial presheaves]]

\begin{displaymath}
f \colon X \longrightarrow Y$ in $sPSh(C)
\end{displaymath}
such that both $X$ and $Y$ are [[homotopy n-types|homotopy k-types]] for some finite $k \in \mathbb{N}$, then its [[n-image]] factorization in the [[(∞,1)-topos]] $L_{lwhe} sPSh(C)_{loc}$ for any $n \in \mathbb{N}$ is presented by any factorization $X \longrightarrow im_{n}(f) \longrightarrow Y$ in $sPSh(C)$ through some Kan-complex valued simplicial presheaf $im_n(f)$ such that for each object $U \in C$ the [[simplicial homotopy groups]] satisfy the following conditions:

\begin{enumerate}%
\item $\pi_{\bullet \lt n}\left(X(U) \to (im_{n}(f))(U)\right)$ are [[isomorphisms]];


\item $\pi_n\left(X(U) \to (im_{n}(f))(U)\to Y(U)\right)$ is the [[(epi,mono) factorization]] of $\pi_n(f(U))$;


\item $\pi_{\bullet \gt n}\left((im_{n}(f))(U) \to Y(U)\right)$ are [[isomorphisms]].



\end{enumerate}
\end{prop}
\begin{proof}
Evalutation on [[stalks]] is a [[filtered colimit]] which preserves the [[finite limits]] and [[finite colimits]] that go into the definition of [[simplicial homotopy groups]]. Therefore the global conditions assumed on the simplicial homotopy groups imply that the same kind of conditions holds for the stalkwise homotopy groups. These are the [[categorical homotopy groups in an infinity-topos|categorical homotopy groups]] in $L_{lwhe} sPSh(C)_{loc}$. By \href{n-truncated+object+of+an+infinity-category#RecognizngnTuncationOnSimplicialHomotopyGroups}{this prop.} and \href{n-connected+object+of+an+infinity-topos#Connectedness}{this def.} we may recognize $n$-truncation of morphisms on categorical homotopy groups (using the assumption that $X$ and $Y$ are $k$-truncated for some $k$). Therefore the claim now follows from the stalkwise [[long exact sequence of homotopy groups]].

\end{proof}
In order to appeal to prop. \ref{nImageFactroizationModeledOnSimplicialPrsheaves} we are interested in explicit models for $n$-image factorization of morphisms of [[Kan complexes]]. The following gives such for the special case that the the morphism of Kan complexes is the image under the [[Dold-Kan correspondence]] of a [[chain map]] between [[chain complexes]].

\begin{remark}
\label{ChainComplexnPlusOneImageInDegreen}\hypertarget{ChainComplexnPlusOneImageInDegreen}{}
Let $f_\bullet \colon V_\bullet \longrightarrow W_\bullet$ be a [[chain map]] between [[chain complexes]]

For $n \in \mathbb{N}$, consider the abelian group

\begin{displaymath}
(im_{n+1}(f))_n
    \;\coloneqq\;
  coker(\,
     ker(\partial_V) \cap ker(f_n)
     \to
     V_n
  \,)
\end{displaymath}
For the following it is helpful to think of this abelian group in the following equivalent ways.

Define an [[equivalence relation]] on $V_n$ by

\begin{displaymath}
\left(
     v_n \sim v'_n
  \right)
  \;\Leftrightarrow\;
  \left(
    (\partial_V v_n = \partial_V v'_n)
    \;\text{and}\;
    (f_n(v_n) = f_n(v'_n))
  \right)
  \,.
\end{displaymath}
Then

\begin{displaymath}
(im_{n+1}(f))_n \simeq V_n/_\sim
\end{displaymath}
is equivalently the set of [[equivalence classes]] of this equivalence relation, which inherits abelian group structure since the eqivalence relation is linear.

This is because the equivalence relation says equivalently that

\begin{displaymath}
\left(
     v_n \sim v'_n
  \right)
  \;\Leftrightarrow\;
  \left(
    v_n - v'_n \;\in\; ker(\partial_V) \cap ker(f_n)
  \right)
\end{displaymath}
and hence is generated under linearity by

\begin{displaymath}
\left(
     v_n \sim 0
  \right)
  \;\Leftrightarrow\;
  \left(
    v_n  \in ker(\partial_V) \cap ker(f_n)
  \right)
  \,.
\end{displaymath}
Moreover, notice that the [[Dold-Kan correspondence]]

\begin{displaymath}
DK
  \;\colon\;
  Ch_{\bullet \geq 0}
    \longrightarrow
  KanCplx
\end{displaymath}
factors through [[globe|globular]] [[strict omega-groupoids]] (\href{Dold-Kan+correspondence#GlobularAndCubical}{here}). An [[n-morphism]] in the [[strict omega-groupoid]] $DK(V_\bullet)$ is of the form

\begin{displaymath}
(v_{n-1})
    \overset{\phantom{AA}v_n\phantom{AA}}{\longrightarrow}
  (v_{n-1} + \partial v_n)
  \,.
\end{displaymath}
In terms of these morphisms the [[equivalence relation]] above says that two of them are equivalent precisely if

\begin{enumerate}%
\item they are ``[[parallel morphisms]]'' in that they have the same [[source]] and [[target]];


\item they have the same image under $f$ in the [[n-morphisms]] of $DK(W_\bullet)$.



\end{enumerate}
This suggests yet another equivalent way to think of $(im_{n+1}(f))_n$: it is the [[disjoint union]] over the [[target]] $(n-1)$-cells in $V_{n-1}$ of the images under $f$ of the sets of $n$-cells from zero to that target:

\begin{displaymath}
(im_{n+1}(f))_n
   \simeq
   \underset{v_{n-1} \in V_{n-1}}{\sqcup}
    \left\{
      f_n(v_n) \vert v_n \in V_n \,\text{and}\,\partial v_n = v_{n-1}
    \right\}
  \,.
\end{displaymath}
\end{remark}
\begin{prop}
\label{ImageFactorizationForChainComplexes}\hypertarget{ImageFactorizationForChainComplexes}{}
Let $f_\bullet \colon V_\bullet \longrightarrow W_\bullet$ be a [[chain map]] between [[chain complexes]] and let $n \in \mathbb{N}$. Recall the abelian group $\underset{v_{n-1}}{\sqcup}\{f_n(v_n) \vert \partial v_n = v_{n-1}\}$ from remark \ref{ChainComplexnPlusOneImageInDegreen}.

The following [[diagram]] of [[abelian groups]] [[commuting diagram|commutes]]:

\begin{displaymath}
\itexarray{
    \vdots && \vdots && \vdots
    \\
    \downarrow^{\mathrlap{\partial_{V}}}
     &&
    \downarrow^{\mathrlap{\partial_{W}}}
     &&
    \downarrow^{\mathrlap{\partial_{W}}}
    \\
    V_{n+3}
      &\overset{f_{n+3}}{\longrightarrow}&
    W_{n+3}
      &\overset{=}{\longrightarrow}&
    W_{n+3}
    \\
    \downarrow^{\mathrlap{\partial_{V}}}
     &&
    \downarrow^{\mathrlap{\partial_{W}}}
     &&
    \downarrow^{\mathrlap{\partial_{W}}}
    \\
    V_{n+2}
      &\overset{f_{n+2}}{\longrightarrow}&
    W_{n+2}
      &\overset{=}{\longrightarrow}&
    W_{n+2}
    \\
    \downarrow^{\mathrlap{\partial_{V}}}
     &&
    \downarrow^{\mathrlap{ \partial_W  } }
     &&
    \downarrow^{\mathrlap{\partial_{W}}}
    \\
    V_{n+1}
      &\overset{f_{n+1}}{\longrightarrow}&
    \left\{
      w_{n+1} | \exists v_n :  \partial_W w_{n+1} = f_n(v_n), \partial_V v_n = 0,
    \right\}
      &\overset{}{\longrightarrow}&
    W_{n+1}
    \\
    \downarrow^{\mathrlap{\partial_{V}}}
     &&
    \downarrow^{\partial_W}
     &&
    \downarrow^{\mathrlap{\partial_{W}}}
    \\
    V_n
      &\overset{ (f_n, \partial_V) }{\longrightarrow}&
    \underset{v_{n-1}}{\sqcup}
    \left\{
      f_n(v_n) \vert \partial_V v_n = v_{n-1}
    \right\}
     &\overset{  }{\longrightarrow}&
    W_n
    \\
    \downarrow^{\mathrlap{\partial_V}}
    &&
    \downarrow^{\mathrlap{(f_n(v_n),\partial_V v_n) \mapsto \partial_V v_n}}
    &&
    \downarrow^{\mathrlap{\partial_W}}
    \\
    V_{n-1}
      &\overset{=}{\longrightarrow}&
    V_{n-1}
      &\overset{f_{n-1}}{\longrightarrow}&
    W_{n-1}
    \\
    \downarrow^{\mathrlap{\partial_{V}}}
    &&
    \downarrow^{\mathrlap{\partial_{V}}}
    &&
    \downarrow^{\mathrlap{\partial_W}}
    \\
    V_{n-2}
      &\overset{=}{\longrightarrow}&
    V_{n-2}
      &\overset{f_{n-2}}{\longrightarrow}&
    W_{n-2}
    \\
    \\
    \downarrow^{\mathrlap{\partial_{V}}}
    &&
    \downarrow^{\mathrlap{\partial_{V}}}
    &&
    \downarrow^{\mathrlap{\partial_W}}
    \\
    \vdots && \vdots && \vdots
  }
\end{displaymath}
Moreover, the middle vertical sequence is a chain complex $im_{n+1}(f)_\bullet$, and hence the diagram gives a factorization of $f_\bullet$ into two chain maps

\begin{displaymath}
f_\bullet
   \;\colon\;
  V_\bullet \longrightarrow im_{n+1}(f)_\bullet \longrightarrow W_\bullet
  \,.
\end{displaymath}
Finally, this is a model for the [[n-image|(n+1)-image factorization]] of $f$ in that on [[homology groups]] the following holds:

\begin{enumerate}%
\item $H_{\bullet \lt n}(V) \overset{\simeq}{\to} H_{\bullet \lt n}(im_{n+1}(f))$ are [[isomorphisms]];


\item $H_n(V) \to H_n(im_{n+1}(f)) \hookrightarrow H_n(W)$ is the [[image|image factorization]] of $H_n(f)$;


\item $H_{\bullet \gt n}(im_{n+1}(f)) \overset{\simeq}{\to} H_{\bullet \gt n}(W)$ are [[isomorphisms]].



\end{enumerate}
\end{prop}
\begin{proof}
This follows by elementary and straightforward direct inspection.

\end{proof}
\hypertarget{moduli_of_circle_connections}{}\paragraph*{{Moduli of circle $n$-connections}}\label{moduli_of_circle_connections}

\begin{defn}
\label{TruncatedDelgneComplexes}\hypertarget{TruncatedDelgneComplexes}{}
For $p \in \mathbb{N}$ and $k \leq p+1$ write

\begin{displaymath}
\mathbf{B}^{p+1}U(1)_{conn^k}
  \coloneqq
  DK
  \left(
    U(1) \to \Omega^1 \to \Omega^2 \to \cdots \to \Omega^k \to 0 \to \cdots \to 0
  \right)
  \;\in\;
  sPSh(CartSp)
\end{displaymath}
for the [[simplicial presheaf]] which is the image under the [[Dold-Kan correspondence]] of the presheaf of chain complexes which is the [[Deligne complex]] starting with the presheaf represented by $U(1)$ in degree $p+1$ and truncated to the differential $k$-forms, as shown.

Since the $DK$ map sends surjections of chain complexes to [[Kan fibrations]], the canonical projection maps yield a tower of objectwise [[Kan fibrations]] of the following form:

\begin{displaymath}
\mathbf{B}^{p+1}U(1)_{conn}
  =
  \mathbf{B}^{p+1}U(1)_{conn^{p+1}}
    \longrightarrow
  \mathbf{B}^{p+1}U(1)_{conn^{p}}
    \longrightarrow
  \mathbf{B}^{p+1}U(1)_{conn^{p-1}}
    \longrightarrow
  \mathbf{B}^{p+1}U(1)_{conn^1}
    \longrightarrow
  \mathbf{B}^{p+1}U(1)_{conn^0}
    =
  \mathbf{B}^{p+1}U(1)
  \,.
\end{displaymath}
\end{defn}
\begin{defn}
\label{ModuliOfnFormConnection}\hypertarget{ModuliOfnFormConnection}{}
For $\Sigma$ a [[smooth manifold]], write

\begin{displaymath}
(\mathbf{B}^p U(1)) \mathbf{Conn}(\Sigma)
  \in
  sPSh(CartSp)
\end{displaymath}
for the image under the [[Dold-Kan correspondence]] of the presheaf of chain complexes which to $U \in CartSp$ assigns the \emph{vertical} [[Cech cohomology|Cech]]-[[Deligne complex]] on $\Sigma \times U \to U$ in the given degree, i.e. the Cech-Deligne complex involving differential forms on $\Sigma \times U$ that have no leg along $U$, i.e. those in $\Omega^{\bullet,0}(\Sigma \times U)$.

\end{defn}
\hypertarget{differential_concretification_on_contractibles}{}\paragraph*{{Differential concretification on contractibles}}\label{differential_concretification_on_contractibles}

We first consider differential concretification on geometrically contractible base spaces. Once this is established, then the general differential concretification follows simply by stackifying along the base space.

\begin{defn}
\label{DifferentialConcretificationForCirclenConnectionsOnContractible}\hypertarget{DifferentialConcretificationForCirclenConnectionsOnContractible}{}
\textbf{(differential concretification for higher circle connections on contractibles)}

Let $\Sigma$ be a [[contractible topological space|contractible]] [[smooth manifold]]. For $p \in \mathbb{N}$ write

\begin{displaymath}
(\mathbf{B}^p U(1))\mathbf{Conn}_0(\Sigma)
    \coloneqq
  [\Sigma, \mathbf{B}^{p+1}U(1)]
\end{displaymath}
and then for $0 \leq k \leq p$ define [[induction|inductively]]

\begin{displaymath}
(\mathbf{B}^p U(1))\mathbf{Conn}_{k+1}(\Sigma)
   \coloneqq
  im_{p+1-k}
  \left(
    [\Sigma, \mathbf{B}(\mathbf{B}^p U(1))_{conn^{k+1}}]
      \longrightarrow
    \sharp [ \Sigma, \mathbf{B}(\mathbf{B}^p U(1))_{conn^{k+1}} ]
      \underset{\sharp[\Sigma, \mathbf{B}(\mathbf{B}^p U(1))_{conn^k}]}{\times^h}
    (\mathbf{B}^p U(1))\mathbf{Conn}_k(\Sigma)
  \right)
 \,.
\end{displaymath}
\end{defn}
\begin{lemma}
\label{DifferentialConcretificationFornConnectionsOnContractibles}\hypertarget{DifferentialConcretificationFornConnectionsOnContractibles}{}
Let $\Sigma$ be a [[contractible topological space|contractible]] [[smooth manifold]]. Then there is a weak equivalence

\begin{displaymath}
(\mathbf{B}^p U(1)) \mathbf{Conn}_{p+1}(\Sigma)
    \simeq
  (\mathbf{B}^p U(1)) \mathbf{Conn}(\Sigma)
  \,,
\end{displaymath}
from the inductively defined object from def. \ref{DifferentialConcretificationForCirclenConnectionsOnContractible} to the moduli object from def. \ref{ModuliOfnFormConnection}.

\end{lemma}
\begin{proof}
By the assumption that $\Sigma$ is contractible, the Cech-direction of the Cech-Deligne double complex is trivial and so we have for all $U \in CartSp$ and $0 \leq k \leq p$ weak equivalences of the form

\begin{displaymath}
[\Sigma, \mathbf{B}^{p+1}U(1)_{conn^k}](U)
  \;\simeq\;
  DK\left(
    C^\infty(\Sigma \times U, U(1))
     \to
    \Omega^1(\Sigma \times U)
      \to
    \Omega^2(\Sigma \times U)
     \to
    \cdots
      \to
    \Omega^{p+1}(\Sigma \times U)
  \right)
\end{displaymath}
and

\begin{displaymath}
(\mathbf{B}^p U(1))\mathbf{Conn}(\Sigma)
    \simeq
  DK\left(
    C^\infty(\Sigma \times U, U(1))
     \to
    \Omega^{1,0}(\Sigma \times U)
     \to
    \Omega^{2,0}(\Sigma \times U)
      \to
    \cdots
      \to
    \Omega^{p+1,0}(\Sigma \times U)
  \right)
  \,.
\end{displaymath}
We claim now for all $k \leq p$ that

\begin{displaymath}
(\mathbf{B}^p U(1))\mathbf{Conn}_k(\Sigma)
    \simeq
  DK\left(
    C^\infty(\Sigma \times U, U(1))
     \to
    \Omega^{1,0}(\Sigma \times U)
     \to
    \cdots
     \to
    \Omega^{k,0}(\Sigma \times U)
     \to
    0
      \to
    \cdots
      \to
    0
  \right)
  \,.
\end{displaymath}
For $k = p$ this is the statement to be shown. Hence we may now prove this by [[induction]].

It is manifestly true for $k = 0$. Hence suppose it is true for some $k \lt p$. Observe then that

\begin{displaymath}
\sharp [\Sigma, \mathbf{B}^{p+1}U(1)_{conn^{k+1}}]
    \longrightarrow
  \sharp [\Sigma, \mathbf{B}^{p+1}U(1)_{conn^k}]
\end{displaymath}
is an objectwise Kan fibration, because so is $\mathbf{B}^{p+1}U(1)_{conn^{k+1}} \to \mathbf{B}^{p+1}U(1)_{conn^k}$ by def. \ref{TruncatedDelgneComplexes}, and both $[\Sigma,-]$ as well as $\sharp$ are right Quillen functors from $sPSh(C)$ with its global projective model structre to itself.

It follows (\href{homotopy+pullback#HomotopyPullbackByOrdinaryPullback}{this prop.}) that the homotopy fiber product in question is presented by the plain 1-categorical fiber product. Since $DK$ is right adjoint, this in turn is given by the degreewise fiber product of the corresponding chain complexes. By direct inspection this means that

\begin{displaymath}
\begin{aligned}
    &
    \sharp [ \Sigma, \mathbf{B}(\mathbf{B}^p U(1))_{conn_{k+1}} ]
      \underset{\sharp[\Sigma, \mathbf{B}(\mathbf{B}^p U(1))_{conn_k}]}{\times^h}
    (\mathbf{B}^p U(1))\mathbf{Conn}_k(\Sigma)
   \\
   & \simeq
   DK
   \left(
      C^\infty(\Sigma \times U, U(1))
        \to
      \Omega^{1,0}(\Sigma \times U)
        \to
      \cdots
        \to
      \Omega^{k,0}(\Sigma \times U)
        \to
      (\sharp \Omega^{k+1}(\Sigma \times -))(U)
        \to
      0
       \to
      \cdots
        \to
      0
   \right)
  \end{aligned}
\end{displaymath}
Hence we are now reduced to computing the $(p+1-k)$ image of

\begin{displaymath}
\itexarray{
  DK
  (
    C^\infty(\Sigma \times U)
     &\to&
    \Omega^1(\Sigma \times U)
      &\to&
    \cdots
       &\to&
    \Omega^{k}(\Sigma \times U)
      &\to&
    \Omega^{k+1}(\Sigma \times U)
      &\to&
    0
      &\to&
    \cdots
      &\to&
    0
  )
  \\
    \downarrow
    &&
    \downarrow
    &&
    &&
    \downarrow
    &&
    \downarrow
    &&
    \downarrow
    &&
    &&
    \downarrow
  \\
   DK
   (
      C^\infty(\Sigma \times U, U(1))
        &\to&
      \Omega^{1,0}(\Sigma \times U)
        &\to&
      \cdots
        &\to&
      \Omega^{k,0}(\Sigma \times U)
        &\to&
      (\sharp \Omega^{k+1}(\Sigma \times -))(U)
        &\to&
      0
       &\to&
      \cdots
        &\to&
      0
   )
  }
\end{displaymath}
Observe that in degree $(p+1)-(k+1)$ the ordinary image is

\begin{displaymath}
im\left(
    \Omega^{k+1}(\Sigma \times U)
      \to
     (\sharp \Omega^{k+1}(\Sigma \times -))(U)
  \right)
   \simeq
  \Omega^{k+1,0}(\Sigma \times U)
\end{displaymath}
With this the induction follows by prop. \ref{nImageFactroizationModeledOnSimplicialPrsheaves} and prop. \ref{ImageFactorizationForChainComplexes}.

\end{proof}
\hypertarget{GeneralDifferentialConcretification}{}\paragraph*{{General differential concretification}}\label{GeneralDifferentialConcretification}

\begin{defn}
\label{DifferentialConcretificationOfCirclenConnections}\hypertarget{DifferentialConcretificationOfCirclenConnections}{}
\textbf{(differential concretification of moduli for higher connection)}

For $\Sigma$ a [[smooth manifold]], define for $p \in \mathbb{N}$

\begin{displaymath}
(\mathbf{B}^{p}U(1))
  \mathbf{Conn}_{p+1}(\Sigma)
    \;\simeq\;
  \underset{\longleftarrow}{\lim}^h_i
  \;
  (\mathbf{B}^p U(1)) \mathbf{Conn}_{p+1}(U_i)
\end{displaymath}
to be the [[homotopy limit]] over the differential concretifications from def. \ref{DifferentialConcretificationForCirclenConnectionsOnContractible} of contractibles $U_i$, for

\begin{displaymath}
\Sigma \simeq \underset{\longrightarrow}{\lim}_i^h U_i
\end{displaymath}
a presentation of $\Sigma$ as a homotopy colimit of contractible manifolds (e.g. the realization of the [[Cech nerve]] of a [[good open cover]]).

\end{defn}
\begin{prop}
\label{}\hypertarget{}{}
For $\Sigma$ a [[smooth manifold]], then the differential concretifiction of def. \ref{DifferentialConcretificationOfCirclenConnections} is equivalent to the moduli object from def. \ref{ModuliOfnFormConnection}:

\begin{displaymath}
(\mathbf{B}^p U(1)) \mathbf{Conn}_{p+1}(\Sigma)
    \simeq
  (\mathbf{B}^{p}U(1)) \mathbf{Conn}(\Sigma)
  \,.
\end{displaymath}
\end{prop}
\begin{proof}
Let $\Sigma \simeq \underset{\longrightarrow}{\lim}_i^h U_i$ be the realization of the Cech nerve of a good open cover. Then

\begin{displaymath}
\underset{\longleftarrow}{\lim}_i (\mathbf{B}^p  U(1))\mathbf{Conn}_{p+1}(U_i)
\end{displaymath}
is equivalently the image under DK of the corresponding Cech hypercomplex with coefficients in the presheaf of chain complexes $(\mathbf{B}^p  U(1))\mathbf{Conn}_{p+1}(-)$. By lemma \ref{DifferentialConcretificationFornConnectionsOnContractibles} this is the vertical Deligne complex, and hence the claim follows.

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

Introductory lecture notes with an eye towards applications in fundamental [[physics]] are at

\begin{itemize}%
\item \emph{[[geometry of physics]]}

\end{itemize}
The differential concretification of differential moduli is discussed in

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

\end{itemize}
The full proof of example \ref{DifferentialConcretificationOfCirclenconnections} is due to Joost Nuiten\ldots{}.

[[!redirects concretifications]]

[[!redirects differential concretification]] [[!redirects differential concretifications]]



\end{document}