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\begin{document}

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\section*{differential function complex}

\begin{quote}%
under construction


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

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

[[!include differential cohomology - contents]]

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

\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{definitions}{Definitions}\dotfill \pageref*{definitions} \linebreak
\noindent\hyperlink{cocycles_with_values_in_graded_vector_spaces}{Cocycles with values in graded vector spaces}\dotfill \pageref*{cocycles_with_values_in_graded_vector_spaces} \linebreak
\noindent\hyperlink{differential_functions}{Differential functions}\dotfill \pageref*{differential_functions} \linebreak
\noindent\hyperlink{differential_function_complexes}{Differential function complexes}\dotfill \pageref*{differential_function_complexes} \linebreak
\noindent\hyperlink{differential_cohomology_2}{Differential $E$-cohomology}\dotfill \pageref*{differential_cohomology_2} \linebreak
\noindent\hyperlink{properties}{Properties}\dotfill \pageref*{properties} \linebreak
\noindent\hyperlink{homotopy_groups}{Homotopy groups}\dotfill \pageref*{homotopy_groups} \linebreak
\noindent\hyperlink{relation_to_differential_cohomology_in_cohesive_toposes}{Relation to differential cohomology in cohesive $(\infty,1)$-toposes}\dotfill \pageref*{relation_to_differential_cohomology_in_cohesive_toposes} \linebreak
\noindent\hyperlink{examples}{Examples}\dotfill \pageref*{examples} \linebreak
\noindent\hyperlink{line_bundles_with_connection}{Line bundles with connection}\dotfill \pageref*{line_bundles_with_connection} \linebreak
\noindent\hyperlink{differential_kcocycles}{Differential K-cocycles}\dotfill \pageref*{differential_kcocycles} \linebreak
\noindent\hyperlink{higher_filtration_degree}{Higher filtration degree}\dotfill \pageref*{higher_filtration_degree} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak


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

A \emph{differential function complex} (\hyperlink{HopkinsSinger}{HopkinsSinger}) is a [[Kan complex]] of [[cocycle]] s for \emph{generalized} [[differential cohomology]], hence for differential refinements of [[generalized (Eilenberg-Steenrod) cohomology]] theories:

roughly, given a [[spectrum]] $E$ [[Brown representability|representing]] a given [[cohomology theory]], its \emph{differential function complex} over any given [[smooth manifold]] $U$ is the [[simplicial set]] whose $k$-[[simplices]] are triples consisting of

\begin{itemize}%
\item a [[continuous function]] $f : U \times \Delta^k \to E_{n}$;


\item a smooth [[differential form]] $\omega$ on $U \times \Delta^k$ whose corresponding real cohomology class (under the [[de Rham theorem]]) is that of the pullback of the real cohomology classes of $E$ along $f$;


\item an explicit [[coboundary]] in real cohomology exhibiting this fact.



\end{itemize}
(More precisely, in order for this construction to yield not just a single [[simplicial set]] (which will be a [[Kan complex]]) but a suitable [[spectrum]] object, there are conditons on the dependency of $\omega$ on the [[tangent vector]]s to the [[simplex]].)

When applied to the [[Eilenberg-MacLane spectrum]] $K\mathbb{Z}$ this construction reproduces, on cohomology classes, [[ordinary differential cohomology]]. Applied to the [[classifying space]] $B U$ of [[topological K-theory]] it gives [[differential K-theory]].

See also at \emph{\href{differential+cohomology+diagram#HopkinsSingerCoefficients}{differential cohomology diagram --Hopkins-Singer coefficients}}.

\hypertarget{definitions}{}\subsection*{{Definitions}}\label{definitions}

\hypertarget{cocycles_with_values_in_graded_vector_spaces}{}\subsubsection*{{Cocycles with values in graded vector spaces}}\label{cocycles_with_values_in_graded_vector_spaces}

For the present purposes it will be convenient to collect [[cocycle]]s of various degrees together to a single cocycle. For that purpose we make the following simple definition.

\begin{defn}
\label{}\hypertarget{}{}
For $V = V^\bullet$ a [[graded vector space]] over the [[real number]]s set

\begin{itemize}%
\item for $E$ a [[topological space]]:

\begin{displaymath}
C^\bullet(E, V)^n := \oplus_{i + j = n} C^i(E, V^j)
\end{displaymath}

\item and so on



\end{itemize}
\end{defn}
(\ldots{})

\hypertarget{differential_functions}{}\subsubsection*{{Differential functions}}\label{differential_functions}

\begin{defn}
\label{DifferentialFunctions}\hypertarget{DifferentialFunctions}{}
For

\begin{itemize}%
\item $E$ a [[topological space]];


\item $\iota \in Z^n(E,\mathbb{R})$ a [[cocycle]] on $E$ for real-valued [[singular cohomology]] on $E$,



\end{itemize}
a \textbf{differential function} on a smooth [[manifold]] $U$ with values in $(E,\iota)$ is a triple $(c,h,\omega)$ with

\begin{itemize}%
\item $c : U \to E$ a continuous map;


\item $\omega \in \Omega^n(S)$ a smooth [[differential form]] on $S$;


\item $h \in C^{n-1}(U,\mathbb{R})$ a cochain in real cohomology on (the topological space underlying) $U$;



\end{itemize}
such that in the [[abelian group]] $Z^n(S,\mathbb{R})$ of singular cochains the equation

\begin{displaymath}
\omega = c^*\iota + \delta h
\end{displaymath}
holds, where

\begin{itemize}%
\item $\omega$ is here regarded as a singular cochain (that sends a chain to the integral of $\omega$ over it, as discussed at [[de Rham theorem]]),


\item $\delta$ denotes the coboundary operator,(the [[Moore complex]] differential of the [[singular simplicial complex]]).



\end{itemize}
\end{defn}
This is (\hyperlink{HopkinsSinger}{HopkinsSinger, def.4.1}).

In words this is: a continuous map to the topological space together with a \emph{smooth} refinement of the pullback of the chosen singular cochain.

\hypertarget{differential_function_complexes}{}\subsubsection*{{Differential function complexes}}\label{differential_function_complexes}

\begin{defn}
\label{DifferentialFunctionComplex}\hypertarget{DifferentialFunctionComplex}{}
For

\begin{itemize}%
\item $E$ be a [[topological space]] and


\item $\iota \in Z^n(E,\mathbb{R})$ a [[cocycle]] on $E$ for real-valued [[singular cohomology]] on $X$,


\item $U$ a [[smooth manifold]],



\end{itemize}
the \textbf{differential function complex}

\begin{displaymath}
(E,\iota)^U
\end{displaymath}
of all differential functions $S \to (X,\iota)$ is the [[simplicial set]] whose $k$-[[simplices]] are differential functions, def, \ref{DifferentialFunctions}

\begin{displaymath}
U \times \Delta^k_{Top} \to (E,\iota)
  \,.
\end{displaymath}
\end{defn}
For applications one needs certain sub-complex of this, filtered by the number of legs that $\omega$ has along the simplices.

\begin{defn}
\label{}\hypertarget{}{}
For $s \in \mathbb{N}$ write

\begin{itemize}%
\item $filt_s \Omega^\bullet(U \times \Delta^k)$

for the sub-simplicial set of differential forms that vanish when evaluated on more than $s$ [[vector field]]s tangent to the [[simplex]];


\item $filt_s (X,\iota)^S \subset (X,\iota)^S$

for the sub-simplicial set of those differential functions whose differential form component is in $filt_s \Omega^\bullet(U \times \Delta^k)$.



\end{itemize}
\end{defn}
This is (\hyperlink{HopkinsSinger}{HopkinsSinger, def. 4.5}).

\begin{prop}
\label{}\hypertarget{}{}
The complex $filt_s (E,\iota)^U$ is (up to [[weak homotopy equivalence|equivalence]], of course) the [[homotopy pullback]]

\begin{displaymath}
\itexarray{
    filt_s (E,\iota)^U &\to& 
      filt_s \Omega^n_{cl}(U \times \Delta^\bullet, \mathcal{V})
    \\
    \downarrow && \downarrow
    \\
    Sing E^U &\to& Z^\bullet(U \times \Delta^\bullet, \mathcal{V})
  }
\end{displaymath}
in [[sSet]] (regarded as equipped with its standard [[model structure on simplicial sets]]).

\end{prop}
Here $E^U$ is the [[internal hom]] in [[Top]] and $Sing(-)$ denotes the [[singular simplicial complex]].

The following proposition gives the [[simplicial homotopy groups]] of these differential function complexes in dependence of the parameter $s$.

\begin{prop}
\label{HomotopyGroupsOfDiffFunctionComplex}\hypertarget{HomotopyGroupsOfDiffFunctionComplex}{}
We have generally

\begin{displaymath}
\pi_k Z(S \times \Delta^\bullet_{Diff}, \mathcal{V}) = 
   H^{n-m}(S; \mathcal{V})
\end{displaymath}
(for instance by the [[Dold-Kan correspondence]]).

The [[simplicial homotopy group]]s of $filt_s \Omega^n_{cl}(S \times \Delta^\bullet_{Diff})$ are

\begin{displaymath}
\pi_k filt_s \Omega^n_{cl}(S \times \Delta^\bullet_{Diff})
  =
  \left\{
    \itexarray{
       H_{dR}^{n-k}(S, \mathcal{V}) & | k \lt s
       \\
       \Omega_{cl}(S; \mathcal{V})^{n-s} & | k = s
       \\
       0 & | k \gt s
    }
  \right\}
  \,.
\end{displaymath}
This implies [[isomorphism]]s

\begin{displaymath}
\pi_k filt_s(X; \iota)^S 
   \stackrel{\simeq}{\to}
  \left\{
    \itexarray{
       \pi_k X^S & | k \lt s
       \\
       H^{n-k-1}(S; \mathcal{V})/ \pi_{k+1} X^S
       |
       k \gt s
    }
  \right.
  \,.
\end{displaymath}
\end{prop}
This appears as \hyperlink{HopkinsSinger}{HopkinsSinger, p. 36 and corollary D15}.

\hypertarget{differential_cohomology_2}{}\subsubsection*{{Differential $E$-cohomology}}\label{differential_cohomology_2}

Let $E_\bullet$ be an [[Omega-spectrum]]. Let $\iota_\bullet$ be the canonical [[Chern character]] class (\ldots{}).

\begin{prop}
\label{}\hypertarget{}{}
For $S$ a [[smooth manifold]], and $s \in \mathbb{N}$, the sequence of differential function complexes, def. \ref{DifferentialFunctionComplex},

\begin{displaymath}
filt_{s + n}(E_n; \iota_n)^S
   \stackrel{\simeq}{\to}
  \Omega  filt_{s + (n + 1)}(E_{n+1}; \iota_{n+1})^S
\end{displaymath}
forms an [[Omega-spectrum]].

This is the \textbf{differential function spectrum} for $E$, $S$, $s$.

\end{prop}
This is (HopkinsSinger, section 4.6).

\begin{defn}
\label{}\hypertarget{}{}
The \textbf{differential $E$-cohomology group} of the smooth manifold $S$ in degree $n$ is

\begin{displaymath}
H_{diff}^n(S,E) :=
  \pi_0
  filt_0(E_n \iota_n)^S
\end{displaymath}
\end{defn}
This is (\hyperlink{HopkinsSinger}{HopkinsSinger, def. 4.34}).

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

\hypertarget{homotopy_groups}{}\subsubsection*{{Homotopy groups}}\label{homotopy_groups}

For reference, we repeat from above the central statements about the [[homotopy type]]s of the differential function complexes, def. \ref{DifferentialFunctionComplex}.

\begin{prop}
\label{LoopObjectsOfDiffFunctionComplexes}\hypertarget{LoopObjectsOfDiffFunctionComplexes}{}
For $E$ an [[Omega-spectrum]], $S$ a [[smooth manifold]], we have for all $s,n \in \mathbb{N}$, a [[weak homotopy equivalence]]

\begin{displaymath}
\Omega  filt_{s+1}(E_{n}; \iota_{n})^S
   \stackrel{\simeq}{\to}
  filt_{s}(E_{n-1}; \iota_{n-1})^S
  \,,
\end{displaymath}
identifying the [[loop space object]] (at the canonical base point) of the differential function complex of $E_{n}$ at filtration level $s+1$ with that differential function complex of $E_{n-1}$ at filtration level $s$.

\end{prop}
\hypertarget{relation_to_differential_cohomology_in_cohesive_toposes}{}\subsubsection*{{Relation to differential cohomology in cohesive $(\infty,1)$-toposes}}\label{relation_to_differential_cohomology_in_cohesive_toposes}

The following is a simple corollary or slight rephrasing of some of the above constructions, which may serve to show how differential function complexes present differential cohomology in the [[cohesive (∞,1)-topos]] of [[smooth ∞-groupoid]]s.

\begin{prop}
\label{}\hypertarget{}{}
For $E_\bullet$ a spectrum as above,\newline we have an [[(∞,1)-pullback]] square

\begin{displaymath}
\itexarray{  
    filt_0 (E_n; \iota_n)^{(-)} &\to& \prod_i \Omega^{n_i}_{cl}(-)
    \\
    \downarrow && \downarrow
    \\
    Disc E_n
    & 
     \stackrel{}{\to}
    &
    \prod_i \mathbf{B}^{n_i} \mathbb{R}_{disc}
  }
  \,.
\end{displaymath}
\end{prop}
\begin{proof}
By prop. \ref{HomotopyGroupsOfDiffFunctionComplex} we have that

\begin{itemize}%
\item $filt_\infty (E; \iota_n)^S \simeq Sing X^S$;


\item $filt_0 \Omega_{cl}(S \times \Delta^\bullet) \simeq \Omega_{cl}(S)$.



\end{itemize}
The statement then follows with the [[pasting law]] for [[homotopy pullback]]s

\begin{displaymath}
\itexarray{
     filt_0 (E_n; \iota_n)^S &\to&  \Omega^n_{cl}(S; \mathcal{V})
     \\
     \downarrow && \downarrow     
     \\  
     filt_\infty (E_n; \iota_n)^S &\to&  filt_\infty \Omega_{cl}(S \times \Delta^\bullet; \mathcal{V})
     \\
     \downarrow^{\mathrlap{\simeq}} && \downarrow^{\mathrlap{\simeq}}
     \\
     Sing X^S &\to& Z(S \times \Delta^\bullet; \mathcal{V})
  }
  \,.
\end{displaymath}
(\ldots{})

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

\begin{itemize}%
\item For $E = H \mathbb{Z}$ the [[Eilenberg-MacLane spectrum]], $H_{diff}^n(-,E)$ is [[ordinary differential cohomology]].


\item For $E = K U$ the [[K-theory spectrum]], $H_{diff}^n(-,E)$ is [[differential K-theory]].


\item For $E = M O, M U$ the [[Thom spectrum]], $H_{diff}^n(-,E)$ is [[differential cobordism cohomology]];


\item For $E = tmf$ the [[tmf]] spectrum, $H_{diff}^n(-,E)$ is [[differential cohomology]];



\end{itemize}
\hypertarget{line_bundles_with_connection}{}\subsubsection*{{Line bundles with connection}}\label{line_bundles_with_connection}

Let $X = \mathcal{B} U(1) \simeq K(\mathbb{Z},2)$ be the [[Eilenberg-MacLane space]] that is the [[classifying space]] for $U(1)$-[[principal bundle]]s. It carries the canonical [[cocycle]] $\iota := Id : \mathcal{B}U(1) \to \mathcal{B}U(1) \simeq K(\mathbb{Z},2)$ representing in $H^2(X,\mathbb{Z})$ the class of the universal complex [[line bundle]] $L \to X$ on $X$.

Accordingly, for $c : S\to \mathcal{B}U(1)$ a continuous map, we have the corresponding line bundle $c^* L$ on $S$.

One checks (\ldots{}details\ldots{}Example 2.7 in HopSin) that a refinement of $c$ to a differential function $(c,\omega,h)$ corresponds to equipping $c^* L$ with a [[connection on a bundle|smooth connection]].

Now consider $((c,\omega,h) \to (c',\omega', h')) \in filt_0  (\mathcal{B}U(1),Id)^S$ a morphism between two such $(\mathcal{B}U(1),Id)$-differential functions. By definition this is now a $U(1)$-principal bundle $\hat L$ with connection on $S \times \Delta^i_{Diff}$, whose curvature form $\hat \omega \in \Omega^2(S \times \Delta^1_{Diff})$ is of the form $g \cdot \tilde \omega$, where $\tilde \omega$ is a 2-form on $S$ and $g$ is a smooth function on $\Delta^1_{Diff}$, both pulled back to $S \times \Delta^1_{Diff}$ and multiplied there.

But since $\hat \omega$ is necessarily \emph{closed} it follows with $d (g \wedge \tilde \omega) = d t \frac{\partial g}{\partial t} \wedge \tilde \omega + g \wedge d_{S} \tilde \omega$ that $g$ is actually constant.

This means that that the parallel transoport of the connection $\hat \nabla$ on $S \times \Delta^1_{Diff}$ induces a insomorphism between the two line bundles on $S$ over the endpoints of $S \times \Delta^1_{Diff}$ that respects the connections.

\ldots{}

\hypertarget{differential_kcocycles}{}\subsubsection*{{Differential K-cocycles}}\label{differential_kcocycles}

(\ldots{})

\hypertarget{higher_filtration_degree}{}\subsubsection*{{Higher filtration degree}}\label{higher_filtration_degree}

\begin{example}
\label{HigherFilteringOfOrdinaryDiffCoh}\hypertarget{HigherFilteringOfOrdinaryDiffCoh}{}
For $E = H \mathbb{Z}$ the [[Eilenberg-MacLane spectrum]], prop. \ref{LoopObjectsOfDiffFunctionComplexes} states that $filt_{1}(H \mathbb{Z}^{n+1}; \iota_{n})^S$ is an [[n-groupoid]] such that the [[automorphism]]s of the 0-object form [[ordinary differential cohomology]] in degree $n$.

\begin{displaymath}
\Omega filt_{1}(H \mathbb{Z}^{n+1}; \iota_{n})^S
  \simeq 
  \mathbf{H}_{diff}^n(S)
  \,.
\end{displaymath}
\end{example}
\begin{remark}
\label{}\hypertarget{}{}
Example \ref{HigherFilteringOfOrdinaryDiffCoh} for $n = 4$ plays a central role in the description of [[T-duality]] by [[twisted differential K-theory]] in (\hyperlink{KahleValentino}{KahleValentino}).

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

Differential function complexes were introduced and studied in

\begin{itemize}%
\item [[Mike Hopkins]], I. Singer, \emph{[[Quadratic Functions in Geometry, Topology,and M-Theory]]}

\end{itemize}
For further references see [[differential cohomology]].

\begin{itemize}%
\item [[Alexander Kahle]], [[Alessandro Valentino]], \emph{[[T-Duality and Differential K-Theory]]}

\end{itemize}
[[!redirects differential function complexes]]



\end{document}