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\newtheorem{prop}{Proposition} \newtheorem{cor}{Corollary} \newtheorem*{utheorem}{Theorem} \newtheorem*{ulemma}{Lemma} \newtheorem*{uprop}{Proposition} \newtheorem*{ucor}{Corollary} \theoremstyle{definition} \newtheorem{defn}{Definition} \newtheorem{example}{Example} \newtheorem*{udefn}{Definition} \newtheorem*{uexample}{Example} \theoremstyle{remark} \newtheorem{remark}{Remark} \newtheorem{note}{Note} \newtheorem*{uremark}{Remark} \newtheorem*{unote}{Note} %------------------------------------------------------------------- \begin{document} %------------------------------------------------------------------- \section*{Deligne cohomology} \hypertarget{context}{}\subsubsection*{{Context}}\label{context} \hypertarget{differential_cohomology}{}\paragraph*{{Differential cohomology}}\label{differential_cohomology} [[!include differential cohomology - contents]] \hypertarget{topos_theory}{}\paragraph*{{$(\infty,1)$-Topos Theory}}\label{topos_theory} [[!include (infinity,1)-topos - 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{PreliminariesOnSheafCohomology}{Preliminaries on sheaf cohomology}\dotfill \pageref*{PreliminariesOnSheafCohomology} \linebreak \noindent\hyperlink{TheDeligneComplex}{The Deligne complex}\dotfill \pageref*{TheDeligneComplex} \linebreak \noindent\hyperlink{CechDeligneComplex}{Cech-Deligne complex}\dotfill \pageref*{CechDeligneComplex} \linebreak \noindent\hyperlink{cup_product_in_deligne_cohomology}{Cup product in Deligne cohomology}\dotfill \pageref*{cup_product_in_deligne_cohomology} \linebreak \noindent\hyperlink{Properties}{Properties}\dotfill \pageref*{Properties} \linebreak \noindent\hyperlink{CharacteristicMaps}{Curvature and characteristic classes}\dotfill \pageref*{CharacteristicMaps} \linebreak \noindent\hyperlink{TheChernCharacter}{The Chern character}\dotfill \pageref*{TheChernCharacter} \linebreak \noindent\hyperlink{ExactSequenceForCurvatureAndCharacteristicClass}{The exact sequences for curvature and characteristic classes}\dotfill \pageref*{ExactSequenceForCurvatureAndCharacteristicClass} \linebreak \noindent\hyperlink{TheExactDifferentialCohomologyHexagon}{The exact differential cohomology hexagon}\dotfill \pageref*{TheExactDifferentialCohomologyHexagon} \linebreak \noindent\hyperlink{GAGA}{GAGA}\dotfill \pageref*{GAGA} \linebreak \noindent\hyperlink{moduli_and_deformation_theory}{Moduli and deformation theory}\dotfill \pageref*{moduli_and_deformation_theory} \linebreak \noindent\hyperlink{interpretation_in_terms_of_higher_parallel_transport}{Interpretation in terms of higher parallel transport}\dotfill \pageref*{interpretation_in_terms_of_higher_parallel_transport} \linebreak \noindent\hyperlink{examples}{Examples}\dotfill \pageref*{examples} \linebreak \noindent\hyperlink{related_concepts}{Related concepts}\dotfill \pageref*{related_concepts} \linebreak \noindent\hyperlink{References}{References}\dotfill \pageref*{References} \linebreak \hypertarget{Idea}{}\subsection*{{Idea}}\label{Idea} \emph{Deligne cohomology} -- or \emph{[[Pierre Deligne|Deligne]]-[[Alexander Beilinson|Beilinson]] cohomology} -- is an [[abelian sheaf cohomology]] that models [[ordinary differential cohomology]], a refinement of the [[sheaf cohomology]] with [[coefficients]] in a [[locally constant sheaf|locally constant]] [[abelian sheaf]] (modeling [[ordinary cohomology]]) by [[differential form]] data. The \emph{Deligne complex} is like a truncated [[de Rham complex]] but, crucially, with the sheaf of 0-forms -- the [[structure sheaf]] $\mathcal{O}$ - replaced by the [[multiplicative group]] $\mathcal{O}^\times$ under the [[exponential map]] \begin{displaymath} \left[ \mathcal{O}^\times \stackrel{d log}{\longrightarrow} \Omega^1 \stackrel{d}{\longrightarrow} \Omega^2 \stackrel{d}{\longrightarrow} \cdots \stackrel{d}{\longrightarrow} \Omega^n \right] \,. \end{displaymath} Deligne cohomology $H^{n+1}_{conn}(X, \mathbb{Z})$ in degree $(n+1)$ is the [[abelian sheaf cohomology]] with [[coefficients]] in this [[chain complex]] of [[sheaves of abelian groups]] (``[[hypercohomology]]''). This was introduced in (\hyperlink{Deligne71}{Deligne 71}) in the context of [[analytic geometry]] (hence using [[holomorphic differential forms]]) as a [[Hodge filtration|Hodge-filtered]] version of [[singular cohomology]], designed to be a target for the [[Beilinson regulator]] from [[motivic cohomology]]. But the form of the definition applies more generally, in particular also in smooth [[differential geometry]], a fact amplified and popularized in (\hyperlink{Brylinski93}{Brylinski 93}). In smooth [[differential geometry]] the typical minor variant has the sheaf $\underline{U}(1) = C^\infty(-,U(1))$ of [[circle group]]-valued [[smooth functions]] in degree $n$: \begin{displaymath} \left[ C^\infty(-,U(1)) \stackrel{d log}{\longrightarrow} \Omega^1 \stackrel{d}{\longrightarrow} \Omega^2 \stackrel{d}{\longrightarrow} \cdots \stackrel{d}{\longrightarrow} \Omega^n \right] \,. \end{displaymath} Given any [[manifold]] $X$, then the resulting complex of abelian groups is, under the [[Dold-Kan correspondence]], the [[n-groupoid]] of [[circle n-bundles with connection]] whose underlying [[circle n-group|circle (n-1)-group]]-[[principal infinity-bundle]] is trivialized. Passing to the [[abelian sheaf cohomology]] implicitly corresponds to considering the [[infinity-stackification]] of this $n$-groupoid valued presheaf, and in this way Deligne cohomology computes equivalence classes of [[circle n-bundles with connection]]. Another way to say this is that under the [[Dold-Kan correspondence]] and [[infinity-stackification]], the above Deligne complex defines a [[smooth infinity-stack]] $\mathbf{B}^n U(1)_{conn}$ which is the universal [[moduli infinity-stack]] for [[circle n-bundles with connection]], and Deligne cohomology computes the [[homotopy classes]] of maps (of [[infinity-stacks]]) into this (\hyperlink{FSS10}{FSS 10}) \begin{displaymath} H^{n+1}_{conn}(X,\mathbb{Z}) \simeq \pi_0(X \to \mathbf{B}^n U(1)_{conn}) \,. \end{displaymath} In this way Deligne cohomology, or rather the collection of Deligne [[cocycles]] with [[coefficients]] in the Deligne complex that defines it, is considerably richer than other models for [[ordinary differential cohomology]] such as [[Cheeger-Simons differential characters]], which see only the [[cohomology group]], but not the full [[moduli infinity-stack|moduli n-stack]]. Explicitly, computing the [[abelian sheaf cohomology]] with coefficients in the [[Deligne complex]] via [[Cech cohomology]] gives that a [[cocycle]] $\overline{A}$ on some [[space]] $X$ is represented with respect to a suitable [[open cover]] $\{U_i \to X\}$ by a collection of [[differential forms]] and functions \begin{displaymath} \overline{A} = \left\{ A_{i_0, \cdots, i_k} \in \Omega^{n-k}(U_{i_0, \cdots i_k}) \right\}_{k = 0}^{n} \cup \{ g_{i_0, \cdots, i_n} \in \mathcal{O}^\times(U_{i_0, \cdots, i_{n+1}}) \} \end{displaymath} such that the failure of the $(n-k+1)$-forms to glue on $(k+1)$-fold intersections of charts is given by the de Rham differential of the $(n-k)$-forms \begin{displaymath} \sum_{j = 0}^k (-1)^j A_{i_0, \cdots, i_{j-1}, i_{j+1}, \cdots, i_{k+1}} = d_{dR} A_{i_0, \cdots, i_{k+1}} \,. \end{displaymath} This evidently generalizes the familiar Cech cocycle data for traditional [[line bundles with connection]]. As the notation indicates, Deligne cohomology is a [[differential cohomology]] refinement of [[ordinary cohomology]] with [[integer]] [[coefficients]], exhibited by a canonical forgetful map \begin{displaymath} \itexarray{ H^{n+1}_{conn}(X,\mathbb{Z}) \\ & \searrow \\ && H^{n+1}(X,\mathbb{Z}) } \end{displaymath} which is induced by the evident morphism of [[chain complexes]]. This is one map in an exact [[differential hexagon]] which exhibits Deligne cohomology as the differential refinement of ordinary integral cohomology by [[closed differential form|closed]] [[curvature]] [[differential form]] data. \begin{displaymath} \itexarray{ 0 & && && && & 0 \\ & \searrow && && && \nearrow \\ && \Omega^{n}(X)/\Omega^n_{int}(X) && \stackrel{\mathbf{d}}{\longrightarrow} && \Omega^{n+1}_{cl}(X) \\ & \nearrow && \searrow^{\mathrlap{a}} && {}^{\mathllap{F_{(-)}}}\nearrow && \searrow \\ H^{n}(X, \mathbb{R}) && && H^{n+1}_{conn}(X,\mathbb{Z}) && && H^{n+1}(X,\mathbb{R}) \\ & \searrow && \nearrow && \searrow^{\mathrlap{DD}} && \nearrow \\ && H^{n}(X,U(1)) && \underset{\beta}{\longrightarrow} && H^{n+1}(X,\mathbb{Z}) \\ & \nearrow && && && \searrow \\ 0 & && && && & 0 \\ \\ && {{connection\;forms} \atop {on\;trivial\;bundles}} && \stackrel{de\;Rham\;differential}{\longrightarrow} && {{curvature} \atop {forms}} \\ & \nearrow & & \searrow & & \nearrow_{\mathrlap{curvature}} && \searrow^{\mathrlap{de\;Rham\;theorem}} \\ {{flat} \atop {differential\;forms}} && && {{geometric\;bundles} \atop {with \;connection}} && && {{rationalized} \atop {bundles}} \\ & \searrow & & \nearrow & & \searrow^{\mathrlap{topol.\;class}} && \nearrow_{\mathrlap{Chern\;character}} \\ && {{flat} \atop {connections}} && \underset{Bockstein\,homomorphism}{\longrightarrow} && {{shape} \atop {of\;bundle}} } \end{displaymath} \hypertarget{definition}{}\subsection*{{Definition}}\label{definition} In any context where these symbols make the evident sense, the Deligne complex of degree $(n+1)$ is the [[chain complex]] $\mathcal{O}^\times \stackrel{d log}{\to}\Omega^1 \stackrel{d}{\to} \Omega^2 \to \cdots \to \Omega^n$, and Deligne cohomology in degree $(n+1)$ is the [[abelian sheaf cohomology]] with [[coefficients]] in this complex. More generally one considers any [[discrete group]] $A$ and inclusion $A \hookrightarrow \mathcal{O}$ into the [[structure sheaf]], then the corresponding Deligne complex is $A \hookrightarrow \mathcal{O} \stackrel{d}{\to} \Omega^1 \to \cdots \to \Omega^n$. For definiteness we consider here in detail the Deligne complex in the context of [[differential geometry]] modeled on [[smooth manifolds]]. All variants work essentially directly analogously, but it may be useful to have a specific case in hand. This discussion overlaps with and is put into a broader context at \emph{[[geometry of physics -- principal connections]]}. \hypertarget{PreliminariesOnSheafCohomology}{}\subsubsection*{{Preliminaries on sheaf cohomology}}\label{PreliminariesOnSheafCohomology} In order to be somewhat self-contained, this section reviews some elements of [[abelian sheaf cohomology]] specified to the context that we need. It also sets up some notation. The definition of the Deligne complex itself is \hyperlink{TheDeligneComplex}{below} in def. \ref{TheSmoothDeligneComplex}. \begin{defn} \label{Smooth0Types}\hypertarget{Smooth0Types}{} Write [[CartSp]] for the [[site]] whose \begin{itemize}% \item [[objects]] are [[Cartesian space]] $\mathbb{R}^n$ for $n \in \mathbb{N}$, \item [[morphisms]] are [[smooth functions]] $\mathbb{R}^{n_1} \to \mathbb{R}^{n_2}$ between these; \item whose [[coverage]] is given by ``differentially [[good open covers]]'', those [[open covers]] of $\mathbb{R}^n$s all whose finite non-empty intersections are [[diffeomorphism|diffeomorphic]] to an [[open ball]], hence again to $\mathbb{R}^n$. \end{itemize} Write $PSh(CartSp) = Func(CartSp^{op},Set)$ for the [[category of presheaves]] over this site. Write \begin{displaymath} Smooth0Type \coloneqq Sh(CartSp) \end{displaymath} for its [[category of sheaves]], also called the [[cohesive topos]] of [[smooth spaces]]. \end{defn} \begin{remark} \label{}\hypertarget{}{} Instead of the site [[CartSp]] of def. \ref{Smooth0Types} one could use the site [[SmoothMfd]] of all smooth manifolds. All of the statements and constructions in the following go through in that case just as well. In fact [[CartSp]] is a [[dense subsite]] of [[SmoothMfd]]. On the one hand this implies that the [[abelian sheaf cohomology]] is the same for both sites, but on the other hand means that it is convenient to restrict to the much ``smaller'' site of Cartesian spaces. In fact since the [[stalks]] of sheaves over smooth manifolds are evaluations on small [[open balls]] and since every open ball is [[diffeomorphism|diffeomorphic]] to a Cartesian space, many statements that are true (only) stalkwise over [[SmoothMfd]] are actually true globally over $CartSp$. It is the ``[[descent]]'' or ``[[infinity-stackification]]'' which is implicit in abelian sheaf cohomology that takes care of these global statements over [[CartSp]] to translate into the same local statements as one gets over [[SmoothMfd]]. \end{remark} \begin{example} \label{RepresentableSheavesAndTheirConstantVersion}\hypertarget{RepresentableSheavesAndTheirConstantVersion}{} The assignment $C^\infty(-,\mathbb{R}) \colon \mathbb{R}^n \mapsto C^\infty(\mathbb{R}^n,\mathbb{R})$ of [[smooth functions]] with values in the [[real numbers]] is a [[sheaf]]. Since this is [[representable functor|representable]] we are entitled to identify this with the [[smooth manifold]] $\mathbb{R}$ (the [[real line]]) itself, and just write $\mathbb{R} \in Smooth0Type$. Similarly for $X$ any other [[smooth manifold]], it represents a sheaf on [[CartSp]] and we just write $X \in Smooth0Type$ for this. Of particular interest below is the case where $X = S^1 = \mathbb{R}/\mathbb{Z} = U(1)$ is the [[circle]], to be regarded as the [[circle group]]. Notice that traditionally the sheaf represented by $\mathbb{R}$ or $U(1)$ is indicated by an underline as in $\underline{\mathbb{R}}$ and $\underline{U}(1)$, but we do not follow this tradition here. Instead, if we consider the other sheaf that might deserve to be denoted by $\mathbb{R}$, namely the [[constant sheaf]] on $\mathbb{R}$, which sends each $U \in CartSp$ to the set underlying $\mathbb{R}$, then we write $\flat \mathbb{R}$ for that. Similarly \begin{displaymath} \flat U(1) \in Smooth0Type \end{displaymath} is the sheaf sending each test manifold to the set of points in the circle, and each smooth function between Cartesian spaces to the identity function on that set. \end{example} \begin{example} \label{SheavesOfFormsAndDeRhamDifferential}\hypertarget{SheavesOfFormsAndDeRhamDifferential}{} For $k \in \mathbb{N}$ write \begin{displaymath} \mathbf{\Omega}^k \in Sh(CartSp) \end{displaymath} for the [[sheaf]] $\mathbf{\Omega}^k \colon U \mapsto \Omega^k(U)$ of smooth [[differential n-forms|differential k-forms]] on $X$. The [[de Rham differential]] extends to a morphism of sheaves \begin{displaymath} \mathbf{d} \colon \mathbf{\Omega}^k \to \mathbf{\Omega}^{k+1} \,. \end{displaymath} For positive $k$ its [[kernel]] is the [[subfunctor|sub-sheaf]] \begin{displaymath} 0 \to \mathbf{\Omega}^k_{cl} \hookrightarrow \mathbf{\Omega}^k \stackrel{\mathbf{d}}{\longrightarrow} \mathbf{\Omega}^{k+1} \end{displaymath} of closed differential forms; and for $k = 0$ its kernel is the [[subfunctor|sub-sheaf]] of constant functions \begin{displaymath} 0 \to \flat \mathbb{R} \hookrightarrow \mathbb{R} \stackrel{\mathbf{d}}{\longrightarrow} \mathbf{\Omega}^1 \,. \end{displaymath} \end{example} In the background, what plays a role for the following is the full [[cohesive (∞,1)-topos|cohesive]] [[homotopy theory]] of [[smooth ∞-groupoids]]. This receives a map from the following coarse [[homotopy theory]] of [[chain complexes]] of [[abelian sheaves]], which is all that is necessary for the present purpose. \begin{defn} \label{FibrantOnjectStructureOnChSmooth}\hypertarget{FibrantOnjectStructureOnChSmooth}{} Write \begin{displaymath} Ch_+(Smooth0Type) = Ch_+(Sh(CartSp)) = Sh(CartSp,Ch_+) \end{displaymath} for the [[category of chain complexes]] in the smooth sheaves of def. \ref{Smooth0Types}, hence for the [[1-category]] whose objects are [[chain complexes]] of [[abelian sheaves]] on $CartSp$. Regard this as equipped with the structure of a [[category of fibrant objects]] induced by the [[projective model structure on chain complexes]], hence with classes of [[morphisms]] labeled as follows: a [[chain map]] $f_\bullet \colon A_\bullet \to B_\bullet$ is called \begin{itemize}% \item a \emph{[[weak equivalence]]} if it is a [[quasi-isomorphism]], hence if it induces [[isomorphisms]] on ([[sheaves]] of) [[chain homology]] groups; \item a \emph{[[fibration]]} if it is an [[epimorphism]] (of [[abelian sheaves]]) in positive degree. \end{itemize} \end{defn} \begin{remark} \label{HomotopyFibersViaFactorizationLemma}\hypertarget{HomotopyFibersViaFactorizationLemma}{} For our purpose the main use of this structure is to compute [[homotopy fibers]] via the [[factorization lemma]]. Namely \begin{enumerate}% \item every chain map may be replaced, up to weak equivalence of its domain, by a fibration; \item the [[homotopy fiber]] of a chain map is the ordinary fiber of any of its fibration replacements. \end{enumerate} \end{remark} \begin{remark} \label{}\hypertarget{}{} That the properties in def. \ref{FibrantOnjectStructureOnChSmooth} are interpreted in sheaves simply means that they apply [[stalk]]-wise. For instance a morphism of chain complexes of presheaves $f_\bullet \colon A_\bullet \to B_\bullet$ is a weak equivalence precisely if the underlying presheaf of chain complexes becomes a [[quasi-isomorphism]] for each point $x$ in each Cartesian space $\mathbb{R}^n$ after restricting (via the presheaf structure maps) to a small enough [[open neighbourhood]] of that point. Similarly for epimorphisms. \end{remark} \begin{remark} \label{HomotopyPullbacksPreservedbyInclusionIntoAllSmoothHomotopyTypes}\hypertarget{HomotopyPullbacksPreservedbyInclusionIntoAllSmoothHomotopyTypes}{} There is a canonical map of [[homotopy theories]] from $Ch_+(Smooth0Type)$ to the full [[(∞,1)-topos]] [[Smooth∞Grpd]] which is given by applying the [[Dold-Kan correspondence]] followed by [[∞-stackification]]. The key point is that this map preserves [[homotopy fiber products]], which is the [[universal construction]] that already captures most of the relevant properties of the Deligne complex. In this way it is sufficient to concentrate on $Ch_+(Smooth0Type)$ for much of the theory. \end{remark} When writing out the components of chain complexes we will use square brackets always denote the group in degree-0 to the far right, and the group in degree $k$ being $k$ steps to the left from that. \begin{example} \label{DeloopingsOfAbelianSheaf}\hypertarget{DeloopingsOfAbelianSheaf}{} For $A \in Ab(Smooth0Type) = Sh(CartSp,Ab)$ any [[abelian sheaf]] and for $n \in \mathbb{N}$ we write \begin{displaymath} (\mathbf{B}^n A)_{\bullet} \coloneqq A[-n] = \left[ A \to 0 \to \cdots \to 0 \right] \end{displaymath} for the [[chain complex]] of sheaves concentrated on $A$ in degree $n$. \end{example} \begin{example} \label{LogDiffInSmoothContext}\hypertarget{LogDiffInSmoothContext}{} There is a weak equivalence, def. \ref{FibrantOnjectStructureOnChSmooth}, \begin{displaymath} \left[\mathbb{Z}\to \mathbb{R} \right] \stackrel{\simeq}{\longrightarrow} U(1) \end{displaymath} given by the [[chain map]] \begin{displaymath} \itexarray{ \mathbb{Z} &\hookrightarrow& \mathbb{R} \\ \downarrow && \downarrow^{\mathrlap{mod\,\mathbb{Z}}} \\ 0 &\to& U(1) } \end{displaymath} (which is just the [[exponential sequence]] regarded as a chain map). That this is a weak equivalence is the statement that every smooth $U(1)$-valued function is locally the quotient of a smooth $\mathbb{R}$-valued function by a $\mathbb{Z}$-valued function. In fact on [[Cartesian spaces]] this is of course true even globally. The de Rham differential [[extension|extends]] through this equivalence to produce a morphism denoted $\mathbf{d} log$: \begin{displaymath} \itexarray{ (\mathbb{Z} \to \mathbb{R}) &\stackrel{\mathbf{d}}{\longrightarrow}& \mathbf{\Omega}^1 \\ \downarrow^{\mathrlap{\simeq}} & \nearrow_{\mathrlap{\mathbf{d} log}} \\ U(1) \,. } \end{displaymath} On a given $U(1)$-valued function this is given by representing the function by a smooth $\mathbb{R}$-valued function under mod-$\mathbb{Z}$-reduction (which is always possible over a [[Cartesian space]]) and applying the de Rham differential to that. The [[kernel]] of that is the constant sheaf $\flat U(1)$ of example \ref{RepresentableSheavesAndTheirConstantVersion} \begin{displaymath} 0 \to \flat U(1) \hookrightarrow U(1) \stackrel{\mathbf{d} log}{\longrightarrow} \mathbf{\Omega}^1 \,. \end{displaymath} \end{example} Under addition of differential forms, the sheaves $\mathbf{\Omega}^k$ of example \ref{SheavesOfFormsAndDeRhamDifferential} becomes [[abelian sheaves]], and we will implicitly understand them this way now. \begin{defn} \label{DeRhamResolutionOfConstantFunctions}\hypertarget{DeRhamResolutionOfConstantFunctions}{} Write $\widehat{(\flat \mathbf{B}^{n+1}\mathbb{R})}_\bullet \in Ch(Smooth0Type)$ for the complex of sheaves given by the truncated [[de Rham complex]]: \begin{displaymath} \widehat{(\flat \mathbf{B}^{n+1}\mathbb{R})}_\bullet \coloneqq \left[ \mathbb{R} \stackrel{\mathbf{d}}{\to} \mathbf{\Omega}^1 \stackrel{\mathbf{d}}{\to} \mathbf{\Omega}^2 \stackrel{\mathbf{d}}{\to} \cdots \stackrel{\mathbf{d}}{\to} \mathbf{\Omega}^{n+1}_{cl} \right] \,. \end{displaymath} \end{defn} \begin{prop} \label{TheDeRhamResolutionOfConstantFunctions}\hypertarget{TheDeRhamResolutionOfConstantFunctions}{} The morphism \begin{displaymath} (\flat \mathbf{B}^{n+1}\mathbb{R})_\bullet \longrightarrow \widehat{(\flat \mathbf{B}^{n+1}\mathbb{R})}_\bullet \end{displaymath} given by the canonical [[chain map]] \begin{displaymath} \itexarray{ \flat \mathbb{R} &\stackrel{}{\to}& 0 &\stackrel{}{\to}& 0 &\stackrel{}{\to}& \cdots &\stackrel{}{\to}& 0 \\ \downarrow && \downarrow && \downarrow && \cdots && \downarrow \\ \mathbb{R} &\stackrel{\mathbf{d}}{\to}& \mathbf{\Omega}^1 &\stackrel{\mathbf{d}}{\to}& \mathbf{\Omega}^2 &\stackrel{\mathbf{d}}{\to}& \cdots &\stackrel{\mathbf{d}}{\to}& \mathbf{\Omega}^{n+1}_{cl} } \end{displaymath} is a weak equivalence in the sense of def. \ref{FibrantOnjectStructureOnChSmooth}. \end{prop} \begin{proof} By the [[Poincaré lemma]]. This \emph{is} the Poincar\'e{} Lemma. \end{proof} Every $A_\bullet \in Ch_+(Smooth0Type)$ serves as the [[coefficients]] for an [[abelian sheaf cohomology]] theory on smooth manifolds. Abelian sheaf cohomology has a general abstract characterization (see at \emph{[[cohomology]]}) in terms of [[derived hom-spaces]]. For definiteness, we recall the model for this construction given by [[Cech cohomology]] . \begin{defn} \label{CechComplex}\hypertarget{CechComplex}{} \textbf{(ech complex)} Let $X$ be a [[smooth manifold]] and let $A_\bullet \in Ch_+(Smooth0Type)$ be a sheaf of chain complexes. Let $\{U_i \to X\}$ be a [[good open cover]] of $X$, i.e. an open cover such that each finite non-empty intersection $U_{i_0, \cdots, i_k}$ is [[diffeomorphism|diffeomorphic]] to an [[open ball]]/[[Cartesian space]]. The \textbf{ech cochain complex} $C^\bullet((X,\{U_i\}),A_\bullet)$ of $X$ with respect to the cover $\{U_i \to X\}$ and with [[coefficients]] in $A_\bullet$ is in degree $k \in \mathbb{N}$ given by the [[abelian group]] \begin{displaymath} C^k((X,\{U_i\}),A_\bullet) \coloneqq \oplus_{{l,n} \atop {k = n-l}} \oplus_{i_0, i_1, \cdots, i_n} A_l(U_{i_0, \cdots, i_n}) \end{displaymath} which is the [[direct sum]] of the values of $A_\bullet$ on the given intersections as indicated; and whose [[differential]] \begin{displaymath} d \colon C^{k}((X,\{U_i\}),A_\bullet) \longrightarrow C^{k+1}((X,\{U_i\}),A_\bullet) \end{displaymath} is defined componentwise (see at [[matrix calculus]] for conventions on maps between [[direct sums]]) by \begin{displaymath} \begin{aligned} (d a)_{i_0, \cdots, i_{k+1}} & \coloneqq (\partial_A a + (-1)^k \delta a)_{i_0, \cdots, i_{k+1}} \\ & \coloneqq \partial_A a_{i_0, \cdots, i_{k+1}} + (-1)^k \sum_{0 \leq j \leq k+1} (-1)^{j} a_{i_0, \cdots, i_{j-1}, i_{j+1}, \cdots, i_{k+1}} |_{U_{i_0, \cdots, i_{k+1}}} \end{aligned} \end{displaymath} where on the right the sum is over all components of $a$ obtained via the canonical restrictions obtained by discarding one of the original $(k+1)$ subscripts. The \textbf{Cech cohomology} groups of $X$ with coefficients in $A_\bullet$ \textbf{relative to the given cover} are the [[chain homology]] groups of the Cech complex \begin{displaymath} H_{Cech}^k((X,\{U_i\}), A_\bullet) \coloneqq H^k(C^\bullet((X,\{U_i\}),A_\bullet)) \,. \end{displaymath} The \textbf{Cech cohomology} groups as such are the [[colimit]] (``[[direct limit]]'') of these groups over refinements of covers \begin{displaymath} H^k_{Cech}(X, A_\bullet) \coloneqq \underset{\longrightarrow}{\lim}_{\{U_i \to X\}} H_{Cech}^k((X,\{U_i\}), A_\bullet) \,. \end{displaymath} \end{defn} \begin{remark} \label{}\hypertarget{}{} Often Cech cohomology is considered for the case that $A_\bullet$ is concentrated in a single degree, in which case the first term in the sum defining the differential in def. \ref{CechComplex} disappears. When $A_\bullet$ is not concentrated in a single degree, then for emphasis one sometimes speaks of \emph{[[hypercohomology]]}. This is the case of relevance for Deligne cohomology. \end{remark} \begin{remark} \label{}\hypertarget{}{} The Cech chain complex in def. \ref{CechComplex} is the [[total complex]] of the [[double complex]] whose vertical differential is that of $A_\bullet$ and whose horizontal differential is the \emph{Cech differential} $\delta$ given by alternating sums over restrictions along patch inclusions \begin{displaymath} \itexarray{ \vdots && \vdots \\ \downarrow^{\mathrlap{\partial_A}} && \downarrow^{\mathrlap{\partial_A}} \\ \oplus_i A_1(U_{i_0}) &\stackrel{\delta}{\longrightarrow}& \oplus_{i_0, i_1} A_1(U_{i_0, i_1}) &\stackrel{\delta}{\longrightarrow}& \cdots \\ \downarrow^{\mathrlap{\partial_A}} && \downarrow^{\mathrlap{\partial_A}} \\ \oplus_i A_0(U_i) &\stackrel{\delta}{\longrightarrow}& \oplus_{i_0, i_1} A_0(U_{i_0, i_1}) &\stackrel{\delta}{\longrightarrow}& \cdots } \end{displaymath} \end{remark} \begin{remark} \label{GoodCoverCechCohomologyIsHomotopicallyGood}\hypertarget{GoodCoverCechCohomologyIsHomotopicallyGood}{} For analyzing the properties of Deligne cohomology \hyperlink{Properties}{below}, all one needs is the following fact about [[Cech cohomology]], which is discussed for instance at \emph{[[infinity-cohesive site]]}: For $X$ a [[smooth manifold]] (in particular [[paracompact topological space|paracompact]]), \begin{enumerate}% \item $X$ admits a [[good open cover]] $\{U_i \to X\}$ (by [[charts]] $U_i$ all whose finite non-empty intersections are [[diffeomorphism|diffeomorphic]] to an [[open ball]]/[[Cartesian space]] $\mathbb{R}^n$); \item for any such good open cover the \href{Čech+cohomology#CechComplex}{Cech complex} $C^\bullet((X,\{U_i\}),A_\bullet)$ already computes Cech cohomology (i.e. there is no further need to form the [[colimit]] of Cech complexes over refinements of covers); \item the functor $C^\bullet((X,\{U_i\}),-) \colon Ch_+(Smooth0Type) \to Ch_+$ preserves weak equivalences and fibrations. \end{enumerate} This means in particular that if $X_\bullet \to Y_\bullet \to Z_\bullet$ is a [[homotopy fiber sequence]] in $Ch_+(Smooth0Type)$, then also \begin{displaymath} C^\bullet((X,\{U_i\}), X_\bullet) \to C^\bullet((X,\{U_i\}), Y_\bullet) \to C^\bullet((X,\{U_i\}), Z_\bullet) \end{displaymath} is a homotopy fiber sequence of chain complexes, and therefore the [[cohomology groups]] sit in the [[long exact sequence in homology]] of this sequence of chain complexes. \end{remark} \begin{example} \label{OrdinaryCohomologyFromCechCohomology}\hypertarget{OrdinaryCohomologyFromCechCohomology}{} For $A_\bullet = (\mathbf{B}^{n+1}\mathbb{Z})_\bullet$ as in example \ref{DeloopingsOfAbelianSheaf}, then for $X$ a [[smooth manifold]] \begin{displaymath} H^0_{Cech}(X, (\mathbf{B}^{n+1}\mathbb{Z})_\bullet) \simeq H^{n+1}(X,\mathbb{Z}) \end{displaymath} is the [[ordinary cohomology]] of $X$ with [[integer]] [[coefficients]], the cohomology which is also computed as the [[singular cohomology]] of the underlying [[topological space]] of $X$. Similarly for $A_\bullet = (\flat \mathbf{B}^n U(1))_{\bullet}$ then \begin{displaymath} H^0_{Cech}(X, (\flat\mathbf{B}^{n}U(1))_\bullet) \simeq H^{n}(X,U(1)) \end{displaymath} is the [[ordinary cohomology]] of $X$ with [[circle group]] [[coefficients]], the cohomology which is also computed as the [[singular cohomology]] of the underlying [[topological space]] of $X$ with $U(1)$-coefficients. \end{example} \begin{example} \label{DeRhamTheorem}\hypertarget{DeRhamTheorem}{} Passing to [[abelian sheaf cohomology]] (e.g. via def. \ref{CechComplex}), then prop. \ref{TheDeRhamResolutionOfConstantFunctions} is the [[de Rham theorem]]. \end{example} We will have need to give names to truncations of the de Rham complex. One is this: \begin{defn} \label{TruncatedDeRham}\hypertarget{TruncatedDeRham}{} For $n \in \mathbb{N}$ write \begin{displaymath} \mathbf{\Omega}^{\bullet \leq n} \in Ch_+(Smooth0Type) \end{displaymath} for the [[chain complex]] of the form \begin{displaymath} \mathbf{\Omega}^{\bullet \leq n} \coloneqq \left[ \mathbb{R} \stackrel{\mathbf{d}}{\to} \mathbf{\Omega}^1 \stackrel{\mathbf{d}}{\to} \cdots \stackrel{\mathbf{d}}{\to} \mathbf{\Omega}^{n-1} \stackrel{\mathbf{d}}{\to} \mathbf{\Omega}^{n} \right] \end{displaymath} with all $n$-forms, not just the closed ones, in degree 0. \end{defn} \begin{example} \label{CohomologyWithCoefficientsInTruncatedDeRham}\hypertarget{CohomologyWithCoefficientsInTruncatedDeRham}{} The [[abelian sheaf cohomology]] of the truncated de Rham complex in def. \ref{TruncatedDeRham} is $\Omega^n(X)/im(\mathbf{d})$. \end{example} \hypertarget{TheDeligneComplex}{}\subsubsection*{{The Deligne complex}}\label{TheDeligneComplex} \begin{defn} \label{TheSmoothDeligneComplex}\hypertarget{TheSmoothDeligneComplex}{} For $n \in \mathbb{N}$ the \textbf{smooth Deligne complex} of degree $n$ \begin{displaymath} (\mathbf{B}^n U(1)_{conn})_\bullet \in Ch_+(Smooth0Type) \end{displaymath} is the [[chain complex]] of [[abelian sheaves]] given by \begin{displaymath} (\mathbf{B}^n U(1)_{conn})_\bullet \; \coloneqq \; \left[ U(1) \stackrel{\mathbf{d} log}{\to} \mathbf{\Omega}^1 \stackrel{\mathbf{d}}{\to} \cdots \stackrel{\mathbf{d}}{\to} \mathbf{\Omega}^n \right] \end{displaymath} with $U(1)$ in degree $n$ and with the differentials as in def. \ref{SheavesOfFormsAndDeRhamDifferential} and example \ref{LogDiffInSmoothContext}. We write \begin{displaymath} H^{n+1}_{conn}(X,\mathbb{Z}) \coloneqq H^0_{Cech}(X,(\mathbf{B}^n U(1)_{conn})_\bullet) \end{displaymath} for its [[abelian sheaf cohomology]]. \end{defn} \begin{remark} \label{RModZResolutionOfU1DeligneComplex}\hypertarget{RModZResolutionOfU1DeligneComplex}{} By example \ref{LogDiffInSmoothContext} the obvious [[chain map]] \begin{displaymath} \itexarray{ \mathbb{Z} &\hookrightarrow& \mathbb{R} &\stackrel{\mathbf{d}}{\to}& \mathbf{\Omega}^1 &\stackrel{\mathbf{d}}{\to}& \cdots &\stackrel{\mathbf{d}}{\to}& \mathbf{\Omega}^n \\ \downarrow && \downarrow^{\mathrlap{(-)/\mathbb{Z}}} && \downarrow^{id} && && \downarrow^{id} \\ 0 &\to& U(1) &\stackrel{\mathbf{d} log}{\to}& \mathbf{\Omega}^1 &\stackrel{\mathbf{d}}{\to}& \cdots &\stackrel{\mathbf{d}}{\to}& \mathbf{\Omega}^n } \end{displaymath} is a weak equivalence, def. \ref{FibrantOnjectStructureOnChSmooth}, and one could define the top chain complex here as ``the'' Deligne complex, just as well. In the context of [[homotopy theory]]/[[homological algebra]], all that matters is the complex up to [[zig-zags]] of weak equivalences. \end{remark} \begin{remark} \label{}\hypertarget{}{} In def. \ref{TheSmoothDeligneComplex} the de Rham complex is truncated to the right by discarding what would be the next differentials, without passing to their [[kernel]], i.e. in degree 0 the Deligne complex has all differential $n$-forms, not just the closed $n$-fomrs. This simple point is the key aspect of the Deligne complex. If one instead truncates while preserving the chain homology in the lowest degree, then one obtains the following complex with the sheaf $\mathbf{\Omega}^{n}_{cl}$ of \emph{closed} forms in lowest degree, which gives [[ordinary cohomology]]. \end{remark} \begin{defn} \label{FlatSmoothDeligneComplex}\hypertarget{FlatSmoothDeligneComplex}{} For $n \in \mathbb{N}$ the \textbf{flat smooth Deligne complex} of degree $n$ \begin{displaymath} (\mathbf{B}^n U(1)_{flat})_\bullet \in Ch_+(Smooth0Type) \end{displaymath} is the [[chain complex]] of [[abelian sheaves]] given by \begin{displaymath} (\mathbf{B}^n U(1)_{flat})_\bullet \; \coloneqq \; \left[ U(1) \stackrel{\mathbf{d} log}{\to} \mathbf{\Omega}^1 \stackrel{\mathbf{d}}{\to} \cdots \stackrel{\mathbf{d}}{\to} \mathbf{\Omega}^n_{cl} \right] \end{displaymath} with $U(1)$ in degree $n$ and with the differentials as in def. \ref{SheavesOfFormsAndDeRhamDifferential} and example \ref{LogDiffInSmoothContext}, and with the \emph{closed} $n$-forms on the right. \end{defn} \begin{defn} \label{FlatBnU1}\hypertarget{FlatBnU1}{} Write \begin{displaymath} (\flat\mathbf{B}^n U(1))_\bullet = (\mathbf{B}^n \flat U(1))_\bullet \in Ch_+(Smooth0Type) \end{displaymath} for the [[chain complex]] of [[abelian sheaves]] given by \begin{displaymath} (\flat \mathbf{B}^n U(1))_\bullet \; \coloneqq \; \left[ \flat U(1) \stackrel{}{\to} 0 \stackrel{}{\to} \cdots \stackrel{}{\to} 0 \right] \end{displaymath} with the constant sheaf $\flat U(1)$ of example \ref{RepresentableSheavesAndTheirConstantVersion} in degree $n$. \end{defn} \begin{prop} \label{TwoEquivalentModelsForFlatBnU1}\hypertarget{TwoEquivalentModelsForFlatBnU1}{} For $(\flat \mathbf{B}^n U(1))_\bullet$ as in def. \ref{FlatBnU1}, then the morphism \begin{displaymath} (\flat \mathbf{B}^n U(1))_\bullet \stackrel{\simeq}{\longrightarrow} (\mathbf{B}^n U(1)_{flat})_\bullet \end{displaymath} given by the [[chain map]] \begin{displaymath} \itexarray{ \flat U(1) &\to& 0 &\to& \cdots &\to& 0 \\ \downarrow && \downarrow && \cdots && \downarrow \\ U(1) &\stackrel{\mathbf{d}log}{\to}& \mathbf{\Omega}^1 &\stackrel{\mathbf{d}}{\to}& \cdots &\stackrel{\mathbf{d}}{\to}& \mathbf{\Omega}^n_{cl} } \end{displaymath} (with the vertical morphism on the left being the inclusion of example \ref{LogDiffInSmoothContext}) is a weak equivalence, def. \ref{FibrantOnjectStructureOnChSmooth}. \end{prop} \begin{proof} By the [[Poincaré lemma]], this is just an immediate variant of prop. \ref{TheDeRhamResolutionOfConstantFunctions}. \end{proof} \hypertarget{CechDeligneComplex}{}\subsubsection*{{Cech-Deligne complex}}\label{CechDeligneComplex} The Cech complex, def. \ref{CechComplex}, for Deligne cohomology of degree $(p+2)$ is the [[total complex]] of a [[double complex]] of the following form \begin{displaymath} \itexarray{ 0 &\longrightarrow& 0 &\longrightarrow& 0 &\longrightarrow & \cdots \\ \uparrow && \uparrow && \uparrow \\ \Omega^{p+1}(\coprod_i U_i) &\stackrel{\delta}{\longrightarrow}& \Omega^{p+1}(\coprod_{i,j} U_{ i j}) &\stackrel{\delta}{\longrightarrow}& \Omega^{p+1}(\coprod_{i,j, k} U_{i j k}) &\stackrel{\delta}{\longrightarrow}& \cdots \\ \uparrow^{\mathrlap{d}} && \uparrow^{\mathrlap{d}} && \uparrow^{\mathrlap{d}} && \\ \vdots && \vdots && \vdots \\ \uparrow^{\mathrlap{d}} && \uparrow^{\mathrlap{d}} && \uparrow^{\mathrlap{d}} && \\ \Omega^2(\coprod_i U_i) &\stackrel{\delta}{\longrightarrow}& \Omega^2(\coprod_{i,j} U_{ i j}) &\stackrel{\delta}{\longrightarrow}& \Omega^2(\coprod_{i,j, k} U_{i j k}) &\stackrel{\delta}{\longrightarrow}& \cdots \\ \uparrow^{\mathrlap{d}} && \uparrow^{\mathrlap{d}} && \uparrow^{\mathrlap{d}} && \\ \Omega^1(\coprod_i U_i) &\stackrel{\delta}{\longrightarrow}& \Omega^1(\coprod_{i,j} U_{ i j}) &\stackrel{\delta}{\longrightarrow}& \Omega^1(\coprod_{i,j, k} U_{i j k}) &\stackrel{\delta}{\longrightarrow}& \cdots \\ \uparrow^{\mathrlap{d log}} && \uparrow^{\mathrlap{d log}} && \uparrow^{\mathrlap{d log}} && \\ C^\infty(\coprod_i U_i, U(1)) &\stackrel{\delta}{\longrightarrow}& C^\infty(\coprod_{i,j} U_{i j}, U(1)) &\stackrel{\delta}{\longrightarrow}& C^\infty(\coprod_{i,j,k} U_{i j k}, U(1)) &\stackrel{\delta}{\longrightarrow}& \cdots } \end{displaymath} where vertically we have the [[de Rham differential]] and horizontally the Cech differential given by alternating sums of [[pullback of differential forms]]. The corresponding [[total complex]] has in degree $n$ the [[direct sum]] of the entries in this double complex which are on the $n$th nw-se off-diagonal and has the total differential \begin{displaymath} D = d + (-1)^{deg} \delta \end{displaymath} with $deg$ denoting form degree. \begin{example} \label{}\hypertarget{}{} A Cech-Deligne cocycle in degree $3$ (``[[bundle gerbe with connection]]'') is data $(\{B_{i}\}, \{A_{i j}\}, \{g_{i j k}\})$ such that \begin{displaymath} \itexarray{ \{B_i\} &\stackrel{\delta}{\longrightarrow}& {{\{B_j - B_i\}} = {d A_{i j}}} && && \\ && \uparrow^{\mathrlap{d}} && && \\ && \{A_{i j}\} &\stackrel{\delta}{\longrightarrow}& \{-A_{ j k} + A_{i k} - A_{i j}\} = \{d log g_{i j k}\} && \\ && && \uparrow^{\mathrlap{d log}} && \\ && && \{g_{i j k}\} &\stackrel{\delta}{\longrightarrow}& \{g_{j k l} g_{i k l}^{-1} g_{i j l} g_{i j k}^{-1} \} = 1 } \end{displaymath} \end{example} \hypertarget{cup_product_in_deligne_cohomology}{}\subsubsection*{{Cup product in Deligne cohomology}}\label{cup_product_in_deligne_cohomology} The [[cup product]] on [[ordinary cohomology]] refines to Deligne cohomology. For more on this see at \emph{[[Beilinson-Deligne cup-product]]}. \hypertarget{Properties}{}\subsection*{{Properties}}\label{Properties} \hypertarget{CharacteristicMaps}{}\subsubsection*{{Curvature and characteristic classes}}\label{CharacteristicMaps} We discuss the construction of two canonical morphisms out of Deligne cohomology, and two canonical morphisms into it. Below these are shown to form two interlocking [[exact sequences]] and in fact an exact [[differential cohomology hexagon]] which accurately characterizes Deligne cohomology as the [[differential cohomology]] extension of integral [[ordinary cohomology]] by [[differential forms]]. Throughout, for ease of notation, we assume $n \in \mathbb{N}$ to be positive, \begin{displaymath} n \geq 1 \,. \end{displaymath} The remaining case $n = 0$ describes ``circle 0-bundles with connection'', which are just $U(1)$-valued functions, and is hence essentially trivial in itself. In the following $X$ is any [[smooth manifold]]. \begin{defn} \label{UnderlyingBundleMap}\hypertarget{UnderlyingBundleMap}{} Let $(\mathbf{B}^{n+1}\mathbb{Z})_\bullet \in Ch_+(Smooth0Type)$ be as in example \ref{DeloopingsOfAbelianSheaf}. Write \begin{displaymath} (\mathbf{B}^n U(1)_{conn})_{\bullet} \stackrel{\simeq}{\longleftarrow} \stackrel{DD_\bullet}{\longrightarrow} (\mathbf{B}^{n+1} \mathbb{Z})_{\bullet} \end{displaymath} for the [[zig-zag]] of chain complexes where the left weak equivalence is that of remark \ref{RModZResolutionOfU1DeligneComplex}, i.e. for the [[chain maps]] given by \begin{displaymath} \itexarray{ \mathbb{Z} &\to& 0 &\to& 0 &\to& \cdots &\to& 0 \\ \uparrow^{\mathrlap{id}} && \uparrow^{\mathrlap{}} && \uparrow^{\mathrlap{}} && \cdots && \uparrow^{\mathrlap{}} \\ \mathbb{Z} &\hookrightarrow& \mathbb{R} &\stackrel{\mathbf{d}}{\to}& \mathbf{\Omega}^1 &\stackrel{\mathbf{d}}{\to}& \cdots &\stackrel{\mathbf{d}}{\to}& \mathbf{\Omega}^n \\ \downarrow^{\mathrlap{0}} && \downarrow^{\mathrlap{mod\,\mathbb{Z}}} && \downarrow^{\mathrlap{id}} && \cdots && \downarrow^{\mathrlap{id}} \\ 0 &\to& U(1) &\stackrel{\mathbf{d} log}{\to}& \mathbf{\Omega}^1 &\stackrel{\mathbf{d}}{\to}& \cdots &\stackrel{\mathbf{d}}{\to}& \mathbf{\Omega}^n } \end{displaymath} Passing to [[abelian sheaf cohomology]] this gives, by def. \ref{TheSmoothDeligneComplex} and example \ref{OrdinaryCohomologyFromCechCohomology}, a morphism \begin{displaymath} \itexarray{ H^{n+1}_{conn}(X,\mathbb{Z}) \\ & \searrow^{\mathrlap{DD}} \\ && H^{n+1}(X,\mathbb{Z}) } \end{displaymath} from Deligne cohomology to [[ordinary cohomology]] with [[integer]] [[coefficients]] in degree $n+1$. For $[\nabla] \in H^{n+1}_{conn}(X,\mathbb{Z})$ we call $DD(\nabla) \in H^{n+1}(X,\mathbb{Z})$\newline the [[Dixmier-Douady class]] of the \emph{underlying [[circle n-bundle]]}. \end{defn} \begin{defn} \label{CurvatureMap}\hypertarget{CurvatureMap}{} Write \begin{displaymath} (F_{(-)})_\bullet \colon (\mathbf{B}^n U(1)_{conn})_\bullet \stackrel{}{\longrightarrow} \mathbf{\Omega}^{n+1}_{cl} \end{displaymath} for the morphism given by the chain map which is just the de Rham differential in degree 0 \begin{displaymath} \itexarray{ 0 &\to& U(1) &\stackrel{\mathbf{d} log}{\to}& \mathbf{\Omega}^1 &\stackrel{\mathbf{d}}{\to}& \cdots &\stackrel{\mathbf{d}}{\to}& \mathbf{\Omega}^{n-1} &\stackrel{\mathbf{d}}{\to}& \mathbf{\Omega}^n \\ \downarrow && \downarrow && \downarrow && \cdots && \downarrow^{\mathrlap{}} && \downarrow^{\mathrlap{\mathbf{d}}} \\ 0 &\to& 0 &\to& 0 &\to& \cdots &\stackrel{}{\to}& 0 &\to& \mathbf{\Omega}^{n+1}_{cl} } \end{displaymath} Passing to [[abelian sheaf cohomology]] this gives a morphism of the form \begin{displaymath} \itexarray{ && \Omega^{n+1}_{cl}(X) \\ & {}^{\mathllap{F_{(-)}}}\nearrow \\ H^{n+1}_{conn}(X,\mathbb{Z}) } \end{displaymath} We call this the \emph{[[curvature]] map}, i.e. for $[\nabla] \in H^{n+1}_{conn}(X,\mathbb{Z})$ the class of a Deligne cocycle, we call \begin{displaymath} F_\nabla \in \Omega^{n+1}_{cl}(X) \end{displaymath} its \emph{[[curvature form]]}. \end{defn} \begin{defn} \label{InclusionOfGloballyDefinedConnectionForms}\hypertarget{InclusionOfGloballyDefinedConnectionForms}{} Consider the [[zig-zag]] \begin{displaymath} \mathbf{\Omega}^{\bullet\leq n} \stackrel{}{\longrightarrow} (\mathbf{B}^n U(1)_{conn})_\bullet \end{displaymath} out of the complex of def. \ref{TruncatedDeRham}, given by the [[chain maps]] \begin{displaymath} \itexarray{ 0 &\to& \mathbb{R} &\stackrel{\mathbf{d}}{\to}& \mathbf{\Omega}^1 &\stackrel{\mathbf{d}}{\to}& \cdots &\stackrel{\mathbf{d}}{\to}& \mathbf{\Omega}^{n-1} &\stackrel{\mathbf{d}}{\to}& \mathbf{\Omega}^n \\ \downarrow && \downarrow && \downarrow^{\mathrm{id}} && \cdots && \downarrow^{\mathrm{id}} && \downarrow^{\mathrm{id}} \\ \mathbb{Z} &\hookrightarrow& \mathbb{R} &\stackrel{\mathbf{d}}{\to}& \mathbf{\Omega}^1 &\stackrel{\mathbf{d}}{\to}& \cdots &\stackrel{\mathbf{d}}{\to}& \mathbf{\Omega}^{n-1} &\stackrel{\mathbf{d}}{\to}& \mathbf{\Omega}^n \\ \downarrow && \downarrow && \downarrow^{\mathrm{id}} && \cdots && \downarrow^{\mathrm{id}} && \downarrow^{\mathrm{id}} \\ 0 &\to& U(1) &\stackrel{\mathbf{d}log}{\to}& \mathbf{\Omega}^1 &\stackrel{\mathbf{d}}{\to}& \cdots &\stackrel{\mathbf{d}}{\to}& \mathbf{\Omega}^{n-1} &\stackrel{\mathbf{d}}{\to}& \mathbf{\Omega}^n } \end{displaymath} where the bottom [[quasi-isomorphism]] is from remark \ref{RModZResolutionOfU1DeligneComplex}. On passing to [[abelian sheaf cohomology]] this gives, by example \ref{CohomologyWithCoefficientsInTruncatedDeRham}, a morphism \begin{displaymath} \itexarray{ \Omega^n(X)/im(\mathbf{d}) \\ & \searrow \\ && H^{n+1}_{conn}(X,\mathbb{Z}) } \end{displaymath} \end{defn} \begin{defn} \label{InclusionOfFlatConnections}\hypertarget{InclusionOfFlatConnections}{} Consider the canonical morphism \begin{displaymath} (\flat \mathbf{B}^n U(1))_\bullet \stackrel{\simeq}{\longrightarrow} (\mathbf{B}^n U(1)_{flat})_\bullet \longrightarrow (\mathbf{B}^n U(1)_{conn})_\bullet \end{displaymath} via def. \ref{FlatSmoothDeligneComplex}, prop. \ref{TwoEquivalentModelsForFlatBnU1}. Passing to [[abelian sheaf cohomology]] this induces a morphism \begin{displaymath} \itexarray{ && H^{n+1}_{conn}(X,\mathbb{Z}) \\ & \nearrow \\ H^n(X,U(1)) } \end{displaymath} We call this map the \emph{inclusion of the [[flat infinity-connections]]} into all [[circle n-connections]]. \end{defn} Combining what we have so far: \begin{prop} \label{BocksteinHomomorphism}\hypertarget{BocksteinHomomorphism}{} The [[composition|composite]] of the morphisms of def. \ref{InclusionOfGloballyDefinedConnectionForms} and of the curvature morphism of def. \ref{CurvatureMap} \begin{displaymath} \itexarray{ \Omega^n(X)/im(\mathbf{d}) && \stackrel{\mathbf{d}}{\longrightarrow} && \Omega^{n+1}_{cl}(X) \\ & \searrow && \nearrow_{\mathrlap{F_{(-)}}} \\ && H^{n+1}_{conn}(X,\mathbb{Z}) } \end{displaymath} is given by the de Rham differential $\mathbf{d}$ on differential forms. The composite of the morphisms of def. \ref{InclusionOfFlatConnections} and def. \ref{UnderlyingBundleMap} is the [[Bockstein homomorphism]]: \begin{displaymath} \itexarray{ && H^{n+1}_{conn}(X,\mathbb{Z}) \\ & \nearrow && \searrow^{\mathrlap{DD}} \\ H^n(X,U(1)) && \underset{\beta}{\longrightarrow} && H^{n+1}(X,\mathbb{Z}) } \end{displaymath} \end{prop} \begin{proof} By composing the defining zig-zags of [[chain maps]] the statement is immediate. \end{proof} \hypertarget{TheChernCharacter}{}\subsubsection*{{The Chern character}}\label{TheChernCharacter} While the explicit definition of the Deligne complex in def. \ref{TheSmoothDeligneComplex} is easy enough, all its good abstract properties are best understood by realizing that it is the [[homotopy fiber product]] of a kind of higher abelian [[Chern character]] map with the closed differential forms $\mathbf{\Omega}^{n+1}_{cl}$. This is the content of prop. \ref{HomotopyFiberProductCharacterization} below. \begin{defn} \label{ChernCharacterMap}\hypertarget{ChernCharacterMap}{} Write \begin{displaymath} ch_\bullet \colon (\mathbf{B}^{n+1}\mathbb{Z})_\bullet \longrightarrow (\flat \mathbf{B}^{n+1}\mathbb{R})_\bullet \end{displaymath} for the morphism given as the composite \begin{displaymath} (\mathbf{B}^{n+1}\mathbb{Z})_\bullet = (\flat \mathbf{B}^{n+1}\mathbb{Z})_\bullet \longrightarrow (\flat \mathbf{B}^{n+1}\mathbb{R})_\bullet \end{displaymath} where the second morphism is induced by the canonical inclusion $\mathbb{Z} \hookrightarrow \mathbb{R}$. Passing to [[abelian sheaf cohomology]] this induces a morphism \begin{displaymath} \itexarray{ && H^{n+1}(X,\mathbb{R}) \\ & \nearrow_{\mathrlap{ch}} \\ H^{n+1}(X,\mathbb{Z}) } \end{displaymath} \end{defn} \begin{remark} \label{}\hypertarget{}{} The morphism in def. \ref{ChernCharacterMap} is just the traditional map from [[ordinary cohomology]] with [[integer]] coefficients to that with [[real numbers]] coefficients given for instance via [[singular cohomology]] simply by forming the [[tensor product of abelian groups]] with $\mathbb{R}$. In the broader context of [[differential cohomology]] however it is useful to think of this map as the \emph{[[Chern character]]}, whence the notation. While this is easy enough to construct in itself, the underlying [[chain map]] here is not a fibration in the sense of def. \ref{FibrantOnjectStructureOnChSmooth} (having as non-trivial component the inclusion $\mathbb{Z} \hookrightarrow \mathbb{R}$, which is evidently not an epimorphism). But the [[homotopy fiber]] of this map plays a crucial role in the theory, and so in view of remark \ref{HomotopyFibersViaFactorizationLemma} we consider now a fibration [[resolution]] of this map \end{remark} \begin{defn} \label{DifferentialResolutionOfIntegralCoefficients}\hypertarget{DifferentialResolutionOfIntegralCoefficients}{} Consider the chain complex \begin{displaymath} \widehat {(\mathbf{B}^{n+1}\mathbb{Z})}_\bullet \coloneqq \left[ \itexarray{ \mathbb{Z} &\stackrel{}{\hookrightarrow}& \mathbb{R} &\stackrel{\mathbf{d}}{\to}& \mathbf{\Omega}^1 &\stackrel{\mathbf{d}}{\to}& \cdots &\stackrel{\mathbf{d}}{\to}& \mathbf{\Omega}^{n-1} &\stackrel{\mathbf{d}}{\to}& \mathbf{\Omega}^n \\ \oplus &\nearrow_{-\mathrlap{id}}& \oplus &\nearrow_{+\mathrlap{id}}& \oplus &\nearrow_{-\mathrlap{id}}& \cdots &\nearrow_{\pm\mathrlap{id}}& \oplus &\nearrow_{\mp\mathrlap{id}}& \\ \mathbb{R} &\underset{\mathbf{d}}{\to}& \mathbf{\Omega}^1 &\underset{\mathbf{d}}{\to}& \mathbf{\Omega}^2 &\underset{\mathbf{d}}{\to}& \cdots &\underset{\mathbf{d}}{\to}& \mathbf{\Omega}^{n} && } \right] \end{displaymath} where we use [[matrix calculus]]-notation as for [[mapping cones]] (see at \emph{\href{mapping+cone#InChainComplexes}{mapping cone -- Examples -- In chain complexes}}). Write \begin{displaymath} \widehat {(\mathbf{B}^{n+1}\mathbb{Z})}_\bullet \stackrel{\widehat{ch}_\bullet}{\longrightarrow} \widehat {(\flat \mathbf{B}^{n+1}\mathbb{R})}_\bullet \end{displaymath} for the morphism to the chain complex of def. \ref{DeRhamResolutionOfConstantFunctions} which is given by the chain map that in positive degree projects onto the lower row in the above [[direct sum]] expression and in degree 0 is given by the de Rham differential: \begin{displaymath} \itexarray{ \mathbb{Z} &\stackrel{}{\hookrightarrow}& \mathbb{R} &\stackrel{\mathbf{d}}{\to}& \mathbf{\Omega}^1 &\stackrel{\mathbf{d}}{\to}& \cdots &\stackrel{\mathbf{d}}{\to}& \mathbf{\Omega}^{n-1} &\stackrel{\mathbf{d}}{\to}& \mathbf{\Omega}^n \\ \oplus &\nearrow_{-\mathrlap{id}}& \oplus &\nearrow_{+\mathrlap{id}}& \oplus &\nearrow_{-\mathrlap{id}}& \cdots &\nearrow_{\pm\mathrlap{id}}& \oplus &\nearrow_{\mp\mathrlap{id}}& \\ \mathbb{R} &\underset{\mathbf{d}}{\to}& \mathbf{\Omega}^1 &\underset{\mathbf{d}}{\to}& \mathbf{\Omega}^2 &\underset{\mathbf{d}}{\to}& \cdots &\underset{\mathbf{d}}{\to}& \mathbf{\Omega}^{n} && \\ \downarrow && \downarrow && \downarrow && \cdots && \downarrow && \downarrow^{\mathrlap{\mathbf{d}}} \\ \mathbb{R} &\stackrel{\mathbf{d}}{\to}& \mathbf{\Omega}^1 &\stackrel{\mathbf{d}}{\to}& \mathbf{\Omega}^2 &\stackrel{\mathbf{d}}{\to}& \cdots &\stackrel{\mathbf{d}}{\to}& \mathbf{\Omega}^{n} &\stackrel{\mathbf{d}}{\to}& \mathbf{\Omega}^{n+1} } \end{displaymath} Finally write \begin{displaymath} (\mathbf{B}^{n+1}\mathbb{Z})_\bullet \longrightarrow \widehat {(\mathbf{B}^{n+1}\mathbb{Z})}_\bullet \end{displaymath} for the morphism given by the chain map which in degree $n+1$ is given by \begin{displaymath} \mathbb{Z} \to \mathbb{Z} \oplus \mathbb{R} \end{displaymath} \begin{displaymath} n \mapsto (n,n) \,. \end{displaymath} \end{defn} \begin{lemma} \label{ChernCharacterResolution}\hypertarget{ChernCharacterResolution}{} The construction in def. \ref{DifferentialResolutionOfIntegralCoefficients} gives a fibration resolution of the Chern character morphism of def. \ref{ChernCharacterMap} in that it gives a [[commuting diagram]] of [[chain maps]] \begin{displaymath} \itexarray{ (\mathbf{B}^{n+1}\mathbb{Z})_\bullet &\stackrel{ch_\bullet}{\longrightarrow}& (\flat\mathbf{B}^{n+1} \mathbb{R})_\bullet \\ \downarrow^{\mathrlap{\simeq}} && \downarrow^{\mathrlap{\simeq}} \\ \widehat {(\mathbf{B}^{n+1}\mathbb{Z})}_\bullet &\stackrel{\widehat {ch}_\bullet}{\longrightarrow}& \widehat {(\flat \mathbf{B}^{n+1}\mathbb{R})}_\bullet } \end{displaymath} with, on the right, the weak equivalence of prop. \ref{TheDeRhamResolutionOfConstantFunctions}, where \begin{enumerate}% \item the left vertical morphism is a weak equivalence; \item the bottom horizontal morphism $\widehat{ch}_\bullet$ is a fibration. \end{enumerate} \end{lemma} \begin{proof} That the diagram commutes is a straightforward inspection, unwinding the definitions. That $\widehat{ch}_\bullet$ is a fibration according to def. \ref{FibrantOnjectStructureOnChSmooth} is by its very construction, being a [[projection]] in positive degree. That the left morphism is a weak equivalence comes down to the [[Poincaré lemma]], in a slight variant of the simple argument that proves prop. \ref{TheDeRhamResolutionOfConstantFunctions}. \end{proof} \begin{defn} \label{MapFromClosedFormsToHyperDeRhamCoefficients}\hypertarget{MapFromClosedFormsToHyperDeRhamCoefficients}{} Write \begin{displaymath} \mathbf{\Omega}^{n+1}_{cl} \stackrel{\iota_\bullet}{\longrightarrow} \widehat{\flat \mathbf{B}^{n+1}\mathbb{R}}_\bullet \stackrel{\simeq}{\longleftarrow} (\flat \mathbf{B}^{n+1}\mathbb{R})_\bullet \end{displaymath} for the [[zig-zag]] whose right morphism is the weak equivalence of prop. \ref{TheDeRhamResolutionOfConstantFunctions} and whose left morphism is given by the [[chain map]] \begin{displaymath} \itexarray{ 0 &\to& 0 &\to& \cdots &\to& 0 &\to& \mathbf{\Omega}^{n+1}_{cl} \\ \downarrow && \downarrow && \vdots && \downarrow && \downarrow^{\mathrlap{id}} \\ \mathbb{R} &\stackrel{\mathbf{d}}{\to}& \mathbf{\Omega}^1 &\stackrel{\mathbf{d}}{\to}& \cdots &\stackrel{\mathbf{d}}{\to}& \mathbf{\Omega}^{n} &\stackrel{\mathbf{d}}{\to}& \mathbf{\Omega}^{n+1}_{cl} } \,. \end{displaymath} \end{defn} \begin{prop} \label{HomotopyFiberProductCharacterization}\hypertarget{HomotopyFiberProductCharacterization}{} The chain maps \begin{itemize}% \item $(F_{(-)})_\bullet$, def. \ref{CurvatureMap}; \item $DD_\bullet$, def. \ref{UnderlyingBundleMap}; \item $\iota_\bullet$, def. \ref{MapFromClosedFormsToHyperDeRhamCoefficients} \item $\widehat{ch}_\bullet$, def. \ref{DifferentialResolutionOfIntegralCoefficients} \end{itemize} fit into a [[commuting diagram]] \begin{displaymath} \itexarray{ && \mathbf{\Omega}^{n+1}_{cl} \\ & {}^{\mathllap{(F_{(-)})_\bullet }}\nearrow && \searrow^{\mathrlap{\iota_\bullet}} \\ (\mathbf{B}^n U(1)_{conn})_\bullet && && \widehat{(\flat \mathbf{B}^{n+1}\mathbb{R})}_\bullet \\ & {}_{\mathllap{DD_\bullet}}\searrow && \nearrow_{\mathrlap{\widehat{ch}_\bullet}} \\ && \widehat{(\mathbf{B}^{n+1}\mathbb{Z})}_\bullet } \end{displaymath} which is a [[pullback diagram]] in $Ch_+(Smooth0Type)$. This exhibits the Deligne complex $(\mathbf{B}^n U(1)_{conn})_\bullet$ as the [[homotopy pullback]] of the inclusion of $\mathbf{\Omega}^{n+1}_{cl}$ along the Chern character map $ch_\bullet$. \end{prop} \begin{proof} The first statement follows straightforwardly by inspection, using that pullbacks of chain complexes are computed componentwise. From this the second statement follows then since by \ref{ChernCharacterResolution} $\widehat{ch}_\bullet$ is a fibration resolution of $ch_\bullet$. \end{proof} \hypertarget{ExactSequenceForCurvatureAndCharacteristicClass}{}\subsubsection*{{The exact sequences for curvature and characteristic classes}}\label{ExactSequenceForCurvatureAndCharacteristicClass} \begin{defn} \label{IntegralDifferentialForms}\hypertarget{IntegralDifferentialForms}{} Write \begin{displaymath} \Omega^{n+1}_{int}(X) \hookrightarrow \Omega^{n+1}_{cl}(X) \end{displaymath} for the inclusion of those closed [[differential forms]] whose [[periods]] ([[integration of differential forms|integration]] over $(n+1)$-cycles) takes values in the [[integers]]. \end{defn} \begin{prop} \label{ImageOfCurvatureInIntegralDifferentialForms}\hypertarget{ImageOfCurvatureInIntegralDifferentialForms}{} The [[image]] of the curvature map $F_{(-)} \colon H^{n+1}_{conn}(X,\mathbb{Z}) \longrightarrow \Omega^{n+1}_{cl}(X)$ of def. \ref{CurvatureMap} are the integral forms of def. \ref{IntegralDifferentialForms}. \end{prop} \begin{proof} The [[homotopy pullback]] characterization of of prop. \ref{HomotopyFiberProductCharacterization} implies that the image consists of precisely those closed differential forms which under the [[de Rham theorem]], remark \ref{DeRhamTheorem}, represent [[real cohomology]] classes that are in the image of integral cohomology classes. These are the differential forms with integral periods. \end{proof} \begin{prop} \label{CurvatureExactSequence}\hypertarget{CurvatureExactSequence}{} \textbf{(curvature exact sequence)} The Deligne cohomology group fits into a [[short exact sequence]] (of [[abelian groups]]) of the form \begin{displaymath} 0 \to H^n(X,U(1)) \longrightarrow H^{n+1}_{conn}(X,\mathbb{Z}) \stackrel{F_{(-)}}{\longrightarrow} \Omega^{n+1}_{int}(X) \to 0 \end{displaymath} where $F_{(-)}$ is the [[curvature]] map of def. \ref{CurvatureMap}. \end{prop} \begin{proof} By prop. \ref{ImageOfCurvatureInIntegralDifferentialForms} the morphism on the right is indeed an epimorphism. It remains to determine its [[kernel]]. To that end, consider the [[pasting diagram]] of [[homotopy pullbacks]] obtained form the homotopy pullback in prop. \ref{HomotopyFiberProductCharacterization}. Using the [[pasting law]] and the fact that the [[loop space object]] of a [[0-truncated]] object such as $\mathbf{\Omega}^{n+1}_{cl}$ is trivial, this is of the form \begin{displaymath} \itexarray{ 0 &\longrightarrow& \widehat{(\flat \mathbf{B}^{n}(\mathbb{R}/\mathbb{Z}))}_\bullet &\longrightarrow& 0 \\ \downarrow && \downarrow && \downarrow^{} \\ 0 &\longrightarrow& (\mathbf{B}^n U(1)_{conn})_\bullet &\stackrel{(F_{(-)}_\bullet)}{\longrightarrow}& \mathbf{\Omega}^{n+1}_{cl} \\ && \downarrow^{\mathrlap{DD_\bullet}} && \downarrow^{\mathrlap{\iota_\bullet}} \\ && \widehat{(\mathbf{B}^{n+1}\mathbb{Z})}_\bullet &\stackrel{\widehat{ch}_\bullet}{\longrightarrow}& \widehat{(\flat \mathbf{B}^{n+1}\mathbb{R})}_\bullet } \,. \end{displaymath} Passing to [[abelian sheaf cohomology]] and applying the induced [[long exact sequence in homology]], in view of remark \ref{GoodCoverCechCohomologyIsHomotopicallyGood}, implies the claim. \end{proof} \begin{prop} \label{CharacteristicClassExactSequence}\hypertarget{CharacteristicClassExactSequence}{} \textbf{(characteristic class exact sequence)} The Deligne cohomology group fits into a [[short exact sequence]] (of [[abelian groups]]) of the form \begin{displaymath} 0 \to \Omega^{n}(X)/\Omega^n_{int}(X) \stackrel{}{\longrightarrow} H^{n+1}_{conn}(X,\mathbb{Z}) \stackrel{DD}{\longrightarrow} H^{n+1}(X,\mathbb{Z}) \to 0 \end{displaymath} where $DD$ is the characteristic class map of def. \ref{UnderlyingBundleMap}. \end{prop} \begin{proof} The [[chain map]] that represents the [[Dixmier-Douady class]] by def. \ref{UnderlyingBundleMap} is manifestly a fibration in the sense of def. \ref{FibrantOnjectStructureOnChSmooth}. Therefore its ordinary [[fiber]] is already its [[homotopy fiber]]. That ordinary fiber is evidently the domain of the morphism constructed in def. \ref{InclusionOfGloballyDefinedConnectionForms}, in its second weakly equivalent incarnation as displayed there. Therefore the [[long exact sequence in homology]], induced by the [[chain map]] $DD_\bullet$ under passage to [[abelian sheaf cohomology]] (in view of remark \ref{GoodCoverCechCohomologyIsHomotopicallyGood}) goes as \begin{displaymath} \cdots \longrightarrow H^n(X,\mathbb{Z}) \longrightarrow \Omega^n(X)/im(\mathbf{d}) \stackrel{}{\longrightarrow} H^{n+1}_{conn}(X,\mathbb{Z}) \stackrel{DD}{\longrightarrow} H^{n+1}(X,\mathbb{Z}) \end{displaymath} As in the proof of prop. \ref{ImageOfCurvatureInIntegralDifferentialForms} it follows that the rightmost morphism is an [[epimorphism]]. Hence we get a [[short exact sequence]] by dividing out the image of $H^n(X,\mathbb{Z})$ in $\Omega^n/im(\mathbf{d})$. That image is $\Omega^n_{int}(X)$. Since this image contains $im(\mathbf{d})$ (as the closed differential forms all whose periods are $0 \in \mathbb{N}$ ) the resulting quotient is $\Omega^n(X)/Omega^n_{int}(X)$ and the claim follows. \end{proof} \begin{remark} \label{}\hypertarget{}{} In words the statement of prop. \ref{CurvatureExactSequence} and prop. \ref{CharacteristicClassExactSequence} is that Deligne [[cohomology groups]] $H^{n+1}_{conn}(X,\mathbb{Z})$ constitute a [[group extension]] \begin{enumerate}% \item of integral [[ordinary cohomology]] by differential $n$-forms modulo integral forms; \item of closed $(n+1)$-forms by $U(1)$-valued ordinary cohomology. \end{enumerate} The first statement is what gives the name to ``[[differential cohomology]]'' as it makes precise how $H^{n+1}_{conn}(X)$ is a combination of ordinary integral cohomology with \emph{[[differential form]]}-data. The second statement is secretly of the same flavor, if maybe not as manifestly so: the $U(1)$-valued ordinary cohomology is really what classifies \emph{[[flat infinity-connections|flat]]} [[circle n-connections]] on [[circle n-group]] [[principal infinity-bundles]] (either, depending on perspective, by def. \ref{FlatSmoothDeligneComplex} or else, more intrinsically, by the very statement of prop. \ref{CurvatureExactSequence}) and hence again describes a combination of underlying bundles with differential form data. \end{remark} \hypertarget{TheExactDifferentialCohomologyHexagon}{}\subsubsection*{{The exact differential cohomology hexagon}}\label{TheExactDifferentialCohomologyHexagon} Summing up, the homotopy pullback square of prop. \ref{HomotopyFiberProductCharacterization} together with the maps of prop. \ref{BocksteinHomomorphism} form a [[commuting diagram]] in $Ch_+(Smooth0Type)$ of the form. \begin{displaymath} \itexarray{ \mathbf{\Omega}^{\bullet \leq n} && \stackrel{\mathbf{d}}{\longrightarrow} && \mathbf{\Omega}^{n+1}_{cl} \\ &\searrow& & {}^{\mathllap{(F_{(-)})_\bullet }}\nearrow && \searrow^{\mathrlap{\iota_\bullet}} \\ && (\mathbf{B}^n U(1)_{conn})_\bullet && && \widehat{(\flat \mathbf{B}^{n+1}\mathbb{R})}_\bullet \\ &\nearrow& & {}_{\mathllap{DD_\bullet}}\searrow && \nearrow_{\mathrlap{\widehat{ch}_\bullet}} \\ \widehat{(\flat \mathbf{B}^n U(1))}_\bullet && \underset{\beta}{\longrightarrow} && \widehat{(\mathbf{B}^{n+1}\mathbb{Z})}_\bullet } \end{displaymath} \begin{prop} \label{TheDifferentialHexagonForDeligne}\hypertarget{TheDifferentialHexagonForDeligne}{} This extends to a diagram in $Ch_+(Smooth0Type)$ of the form \begin{displaymath} \itexarray{ && \mathbf{\Omega}^{\bullet \leq n} && \stackrel{\mathbf{d}}{\longrightarrow} && \mathbf{\Omega}^{n+1}_{cl} \\ & \nearrow& &\searrow& & {}^{\mathllap{(F_{(-)})_\bullet }}\nearrow && \searrow^{\mathrlap{\iota_\bullet}} \\ \widehat{(\flat \mathbf{B}^n \mathbb{R})}_{\bullet} && && (\mathbf{B}^n U(1)_{conn})_\bullet && && \widehat{(\flat \mathbf{B}^{n+1}\mathbb{R})}_\bullet \\ &\searrow & &\nearrow& & {}_{\mathllap{DD_\bullet}}\searrow && \nearrow_{\mathrlap{\widehat{ch}_\bullet}} \\ && \widehat{(\flat \mathbf{B}^n U(1))}_\bullet && \underset{\beta}{\longrightarrow} && \widehat{(\mathbf{B}^{n+1}\mathbb{Z})}_\bullet } \end{displaymath} such that \begin{enumerate}% \item both square are [[homotopy pullback]] squares; \item both diagonals are [[homotopy fiber sequences]]; \item the two outer sequences are [[long homotopy fiber sequences]]. \end{enumerate} \end{prop} \begin{proof} For the first statement consider the pasting of homotopy pullback diagrams as in the proof of prop. \ref{CurvatureExactSequence}, now extended to the left, via the [[pasting law]], as \begin{displaymath} \itexarray{ \widehat{(\flat \mathbf{B}^n \mathbb{R})}_\bullet &\longrightarrow& \widehat{(\flat \mathbf{B}^{n}(\mathbb{R}/\mathbb{Z}))}_\bullet &\longrightarrow& 0 \\ \downarrow && \downarrow && \downarrow^{} \\ \mathbf{\Omega}^{\bullet \leq n} &\longrightarrow& (\mathbf{B}^n U(1)_{conn})_\bullet &\stackrel{(F_{(-)}_\bullet)}{\longrightarrow}& \mathbf{\Omega}^{n+1}_{cl} \\ && \downarrow^{\mathrlap{DD_\bullet}} && \downarrow^{\mathrlap{\iota_\bullet}} \\ && \widehat{(\mathbf{B}^{n+1}\mathbb{Z})}_\bullet &\stackrel{\widehat{ch}_\bullet}{\longrightarrow}& \widehat{(\flat \mathbf{B}^{n+1}\mathbb{R})}_\bullet } \,. \end{displaymath} That the NE-diagonal is a homotopy fiber sequence is the statement in the proof of prop. \ref{CurvatureExactSequence}. That the SE-diagonal is a homotopy fiber sequence follows by inspection as remarked in the proof of prop. \ref{CharacteristicClassExactSequence}. From this the last statement now is implied by using the [[pasting law]] yet once more, as show in the proof \href{differential+cohomology+diagram#TheDifferentialDiagram}{here}. \end{proof} \begin{remark} \label{}\hypertarget{}{} The form of the exact hexagon characterizing the Deligne complex via prop. \ref{TheDifferentialHexagonForDeligne} is in fact a general abstract consequence of the fact that all universal constructions in $Ch_+(Smooth0Type)$ considered here indeed may be understood as taking place in the [[cohesive (∞,1)-topos|cohesive]] [[homotopy theory]] of [[smooth ∞-groupoids]], via remark \ref{HomotopyPullbacksPreservedbyInclusionIntoAllSmoothHomotopyTypes}. This is discussed in detail at \emph{[[differential cohomology hexagon]]}. \end{remark} \hypertarget{GAGA}{}\subsubsection*{{GAGA}}\label{GAGA} The Deligne complex is naturally defined in smooth [[differential geometry]] as well as in [[complex analytic geometry]] as well as in [[algebraic geometry]] over the complex numbers. In the spirit of [[GAGA]] it is of interest to know how Deligne cohomology in these different settings relates. One useful statement is: given an [[smooth scheme|smooth]] [[algebraic variety]] over the [[complex numbers]], then a sufficient condition for a complex-analytic Deligne cocycle over its [[analytification]] to lift to an algebraic Deligne cocycle is that its [[curvature form]] is an [[Kähler form|algebraic form]] (\hyperlink{Esnault89}{Esnault 89, corollary 1.3}). \hypertarget{moduli_and_deformation_theory}{}\subsubsection*{{Moduli and deformation theory}}\label{moduli_and_deformation_theory} The [[moduli spaces]] of holomorphic Deligne cohomology groups are closely related to [[intermediate Jacobians]], see there fore more. The [[deformation theory]] of Deligne cohomology groups is given by [[Artin-Mazur formal group]], see there for more [[!include moduli of higher lines -- table]] \hypertarget{interpretation_in_terms_of_higher_parallel_transport}{}\subsubsection*{{Interpretation in terms of higher parallel transport}}\label{interpretation_in_terms_of_higher_parallel_transport} There is a natural way to understand the Deligne complex of sheaves as a sheaf which assigns to each patch the Lie $n$-groupoid of smooth [[higher parallel transport]] [[n-functors]]. We start by discussing this in low degree. There is [[path groupoid]] $P_1(X)$ whose smooth space of objects is $X$ and whose smooth space of morphisms is a space of classes of smooth paths in $X$. Every smooth 1-form $A \in \Omega^1(X)$ induces a smooth [[functor]] $tra_A : P_1(X) \to \mathbf{B}U(1)$ from $P_1(X)$ to to the smooth [[groupoid]] $\mathbf{B} U(1)$ with one object and $U(1)$ as its smooth space of morphisms by sending each path $\gamma : [0,1] \to X$ to $\exp (2 \pi i\int_0^1 \gamma^* A)$. This map from 1-forms to smooth functors turns out to be bijective: every smooth functor of this form uniquely arises this way. Similarly, one finds that smooth [[natural transformation]] $\eta_f : tra_A \to tra_{A'}$ between two such functors is in components precisely a smooth function $f : X \to U(1)$ such that $A' = A + d log f$. Since the analogous statements are true for every open subset $U \subset X$ this defines a [[sheaf]] of [[Lie groupoid]]s \begin{displaymath} Funct^\infty(P_1(-), \mathbf{B}U(1)) : Op(X)^{op} \to LieGrpd \,. \end{displaymath} By the [[Dold-Kan correspondence]] this sheaf of groupoids corresponds to a sheaf of complexes of groups. This complex of sheaves is nothing but the degree 2 Deligne complex \begin{displaymath} Funct^\infty(\Pi_1(-), \mathbf{B}U(1)) \simeq \mathbb{Z}(2)^\infty_D \,. \end{displaymath} This way Deligne cohomology is realized as computing the [[(infinity,1)-sheafification|stackification]] of the pre-stack $Funct^\infty(P_1(-), \mathbf{B}(1))$ of smooth $U(1)$-valued parallel transport functors. The identification generalizes: for all $n$ there is a [[path n-groupoid]] $P_n(X)$ whose $k$-morphisms are $k$-dimensional smooth paths in $X$. Smooth $n$-functors $tra_C : _n(X) \to \mathbf{B}^n U(1)$ are canonically identified with smooth $n$-forms $C \in \Omega^n(X)$ and under the [[Dold-Kan correspondence]] the Deligne-complex in degree $n+1$ is identified with the sheaf of $n$-groupoids of such smooth $n$-functors \begin{displaymath} n Funct^\infty(P_n(-), \mathbf{B}^n) \simeq \mathbb{Z}(n+1)^\infty_D \,. \end{displaymath} See \begin{itemize}% \item John Baez, Urs Schreiber, \emph{Higher Gauge Theory} (\href{http://arxiv.org/abs/math/0511710}{arXiv}) \end{itemize} The full proof for $n=1$ this is in \begin{itemize}% \item Urs Schreiber, Konrad Waldorf, \emph{Parallel transport and functors} (\href{http://arxiv.org/abs/0705.0452}{arXiv}); \end{itemize} for $n=2$ in \begin{itemize}% \item Urs Schreiber, Konrad Waldorf, \emph{Smooth functors versus differential forms} (\href{http://arxiv.org/abs/0802.0663}{arXiv}) \end{itemize} For more on this see [[infinity-Chern-Weil theory introduction]]. For higher $n$ there is as yet no detailed proof in the literature, but the low dimensional proofs have obvious generalizations. \hypertarget{examples}{}\subsection*{{Examples}}\label{examples} As described in some detail at [[electromagnetic field]] in abelian higher [[gauge theory|gauge theories]] the background field naturally arises as a [[?ech cohomology|?ech]]--Deligne cocycle, i.e. a [[?ech cohomology|?ech cocycle]] representative with values in the Deligne complex. \begin{itemize}% \item Degree 2 Deligne cohomology classifies $U(1)$-[[principal bundle]]s [[connection on a bundle|with connection]]. The Deligne complex $\bar \mathbf{B}U(1)$ in this case coincides with the [[groupoid of Lie-algebra valued forms]] for the Lie algebra of $U(1)$. \begin{itemize}% \item In physics the [[electromagnetic field]] is modeled by a degree 2 Deligne cocycle. See there for a derivation of [[?ech cohomology|?ech]]--Deligne cohomology from physical input. \end{itemize} \item Degree 3 Deligne cohomology classifies [[bundle gerbe]]s with connection. \begin{itemize}% \item In [[electromagnetism]], degree 3 Deligne cocycles (with compact support, possibly) model [[magnetic charge]]. In formal high energy physics the [[Kalb-Ramond field]] is modeled by a Deligne 3-cocycle. \end{itemize} \item Degree 4 Deligne cohomology classifies [[bundle 2-gerbe]]s with connection. In particular Chern-Simons bundle 2-gerbes whose degree 4 curvature characteristic class is a multiple of the Pontryagin 4-form on some $SO(n)$-[[principal bundle]]. \begin{itemize}% \item In formal high energy physics degree 4 Deligne classes model the [[supergravity C-field]]. \end{itemize} \end{itemize} \hypertarget{related_concepts}{}\subsection*{{Related concepts}}\label{related_concepts} \begin{itemize}% \item [[intermediate Jacobian]] ([[self-dual higher gauge theory]]) \item [[Abel-Jacobi map]] \item [[Beilinson regulator]] \item [[circle n-bundle with connection]] \item \href{differential+cohomology+diagram#DeligneCoefficients}{differential cohomology diagram -- Examples -- Deligne coefficients} \end{itemize} \hypertarget{References}{}\subsection*{{References}}\label{References} Deligne cohomology was introduced in [[complex analytic geometry]] (by a [[chain complex]] of [[holomorphic differential forms]]) in \begin{itemize}% \item [[Pierre Deligne]], \emph{Th\'e{}orie de Hodge II} , IHES Pub. Math. (1971), no. 40, 5--57 (\href{http://www.math.jussieu.fr/~ai/d/Theory%20of%20Hodge%202.pdf}{pdf}) \end{itemize} with applications to [[Hodge theory]] and [[intermediate Jacobians]]. The same definition appears in \begin{itemize}% \item [[Barry Mazur]], [[William Messing]], \emph{Universal extensions and one-dimensional crystalline cohomology}, Springer lecture notes 370, 1974 \item [[Michael Artin]], [[Barry Mazur]], section III.1 of \emph{Formal Groups Arising from Algebraic Varieties}, Annales scientifiques de l'\'E{}cole Normale Sup\'e{}rieure, S\'e{}r. 4, 10 no. 1 (1977), p. 87-131 \href{http://www.numdam.org/item?id=ASENS_1977_4_10_1_87_0}{numdam}, \href{http://www.ams.org/mathscinet-getitem?mr=56:15663}{MR56:15663} \end{itemize} under the name ``multiplicative de Rham complex'' (and in the context of studying its [[deformation theory]] by [[Artin-Mazur formal groups]]). The theory was further developed in \begin{itemize}% \item [[Alexander Beilinson]] \emph{[[Higher regulators and values of L-functions]]}, Journal of Soviet Mathematics 30 (1985), 2036-2070, (\href{http://www.mathnet.ru/php/archive.phtml?wshow=paper&jrnid=intd&paperid=73&option_lang=eng}{mathnet (Russian)}, \href{http://dx.doi.org/10.1007%2FBF02105861}{DOI}) (reviewed in \hyperlink{EsnaultViehweg88}{Esnault-Viehweg 88}) \end{itemize} with the application to [[Beilinson regulators]]. Later the evident version of the Deligne complex in [[differential geometry]] over [[smooth manifolds]] gained more attention and is still referred to as ``Deligne cohomology''. Surveys and introductions in the context of [[differential geometry]] include \begin{itemize}% \item [[Jean-Luc Brylinski]], section 5 of \emph{Loop Spaces, Characteristic Classes and geometric Quantization}, Birkh\"a{}user 1993 \item [[Ulrich Bunke]], section 3 of \emph{Differential cohomology} (\href{http://arxiv.org/abs/1208.3961}{arXiv:1208.3961}) \end{itemize} Review with more emphasis on [[complex analytic geometry]] and the theory of (\hyperlink{Beilinson85}{Beilinson 85}) with more details spelled out is in \begin{itemize}% \item [[Hélène Esnault]], [[Eckart Viehweg]], \emph{Deligne-Beilinson cohomology} in Rapoport, Schappacher, Schneider (eds.) \emph{Beilinson's Conjectures on Special Values of L-Functions} . Perspectives in Math. 4, Academic Press (1988) 43 - 91 (\href{http://www.uni-due.de/~mat903/preprints/ec/deligne_beilinson.pdf}{pdf}) \item [[Hélène Esnault]], \emph{On the Loday-symbol in the Deligne-Beilinson cohomology}, K-theory 3, 1-28, 1989 (\href{http://www.mi.fu-berlin.de/users/esnault/preprints/helene/16-loday-symbol.pdf}{pdf}) \end{itemize} See also \begin{itemize}% \item [[Chris Peters]], [[Jozef Steenbrink]], section 7.2 of \emph{[[Mixed Hodge Structures]]}, Ergebisse der Mathematik (2008) (\href{http://www.arithgeo.ethz.ch/alpbach2012/Peters_Steenbrinck}{pdf}) \item [[Claire Voisin]], section 12 of \emph{[[Hodge theory and Complex algebraic geometry]] I,II}, Cambridge Stud. in Adv. Math. \textbf{76, 77}, 2002/3 \end{itemize} Discussion of Deligne cohomology in terms of [[simplicial presheaves]] and [[higher stacks]] includes \begin{itemize}% \item [[Domenico Fiorenza]], [[Urs Schreiber]], [[Jim Stasheff]], \emph{[[schreiber:Cech Cocycles for Differential characteristic Classes]]}, Advances in Theoretical and Mathematical Physics, Volume 16 Issue 1 (2012), pages 149-250 (\href{http://arxiv.org/abs/1011.4735}{arXiv:1011.4735}) \item [[Domenico Fiorenza]], [[Hisham Sati]], [[Urs Schreiber]], \emph{[[schreiber:Extended higher cup-product Chern-Simons theories]]}, Journal of Geometry and Physics, Volume 74, 2013, Pages 130--163 (\href{http://arxiv.org/abs/1207.5449}{arXiv:1207.5449}) \item [[Michael Hopkins]], [[Gereon Quick]], \emph{Hodge filtered complex bordism}, \href{http://arxiv.org/abs/1212.2173}{arXiv:1212.2173}. \item [[Domenico Fiorenza]], [[Hisham Sati]], [[Urs Schreiber]], \emph{A higher stacky perspective on Chern-Simons theory}, in Damien Calaque et al. (eds.) \emph{Mathematical Aspects of Quantum Field Theories}, Mathematical Physics Studies, Springer 2014 (\href{http://arxiv.org/abs/1301.2580}{arXiv:1301.2580}) \item [[Urs Schreiber]], \emph{[[schreiber:differential cohomology in a cohesive topos]]} (\href{http://arxiv.org/abs/1310.7930}{arXiv:1310.7930}) \end{itemize} See also the references given at \emph{\href{differential+cohomology+diagram#DeligneCoefficients}{differential cohomology hexagon -- Deligne coefficients}}. [[!redirects Deligne complex]] [[!redirects Deligne complexes]] [[!redirects Deligne-Beilinson cohomology]] [[!redirects Beilinson-Deligne cohomology]] [[!redirects Deligne cocycle]] [[!redirects Deligne cocycles]] \end{document}