Deligne cohomology

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+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Differential cohomology
+--{: .hide}
[[!include differential cohomology - contents]]
=--
#### $(\infty,1)$-Topos Theory
+--{: .hide}
[[!include (infinity,1)-topos - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea
 {#Idea}

_Deligne cohomology_ -- or _[[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 _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]]

$$
  \left[
     \mathcal{O}^\times
     \stackrel{d log}{\longrightarrow}
     \Omega^1
     \stackrel{d}{\longrightarrow}
     \Omega^2
     \stackrel{d}{\longrightarrow}
     \cdots
     \stackrel{d}{\longrightarrow}
     \Omega^n
  \right]
  \,.
$$

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 ([Deligne 71, Section 2.2](#Deligne71)) 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 ([Brylinski 93](#Brylinski93)).

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 $n$:

$$
  \left[
     C^\infty(-,U(1))
     \stackrel{d log}{\longrightarrow}
     \Omega^1
     \stackrel{d}{\longrightarrow}
     \Omega^2
     \stackrel{d}{\longrightarrow}
     \cdots
     \stackrel{d}{\longrightarrow}
     \Omega^n
  \right]
  \,.
$$

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 ([FSS 10](#FSS10))

$$
  H^{n+1}_{conn}(X,\mathbb{Z})
  \simeq
  \pi_0(X \to \mathbf{B}^n U(1)_{conn})
  \,.
$$

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

$$
  \overline{A}
  =
  \left\{
     A_{i_0, \cdots, i_k}
     \in \Omega^{n-k}(U_{i_0, \cdots i_k})
  \right\}_{k = 0}^{n}
  \cup
  \{
     g_{i_0, \cdots, i_n} \in \mathcal{O}^\times(U_{i_0, \cdots, i_{n+1}})
  \}
$$

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

$$
  \sum_{j = 0}^k
  (-1)^j
  A_{i_0, \cdots, i_{j-1}, i_{j+1}, \cdots, i_{k+1}}
  =
  d_{dR} A_{i_0, \cdots, i_{k+1}}
  \,.
$$

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

$$
  \array{
    H^{n+1}_{conn}(X,\mathbb{Z})
    \\
    & \searrow
    \\
    && H^{n+1}(X,\mathbb{Z})
  }
$$

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.

$$
  \array{
    0 & && && && & 0
    \\
    & \searrow && && &&  \nearrow
    \\
    && \Omega^{n}(X)/\Omega^n_{int}(X) && \stackrel{\mathbf{d}}{\longrightarrow} &&  \Omega^{n+1}_{cl}(X)
    \\
    & \nearrow && \searrow^{\mathrlap{a}} && {}^{\mathllap{F_{(-)}}}\nearrow && \searrow
    \\
    H^{n}(X, \mathbb{R})
    && &&
    H^{n+1}_{conn}(X,\mathbb{Z})
    && &&
    H^{n+1}(X,\mathbb{R})
    \\
    & \searrow && \nearrow && \searrow^{\mathrlap{DD}} && \nearrow
    \\
    && H^{n}(X,U(1)) && \underset{\beta}{\longrightarrow} && H^{n+1}(X,\mathbb{Z})
    \\
    & \nearrow && && &&  \searrow
    \\
    0 & && && && & 0
    \\
    \\
    &&  {{connection\;forms} \atop {on\;trivial\;bundles}} && \stackrel{de\;Rham\;differential}{\longrightarrow} &&
   {{curvature} \atop {forms}}
    \\
    & \nearrow & & \searrow & & \nearrow_{\mathrlap{curvature}} && \searrow^{\mathrlap{de\;Rham\;theorem}}
    \\
    {{flat} \atop {differential\;forms}}  && && {{geometric\;bundles} \atop {with \;connection}} && && {{rationalized} \atop {bundles}}
    \\
    & \searrow &  & \nearrow & & \searrow^{\mathrlap{topol.\;class}} && \nearrow_{\mathrlap{Chern\;character}}
    \\
    && {{flat} \atop {connections}} && \underset{Bockstein\,homomorphism}{\longrightarrow} && {{shape} \atop {of\;bundle}}
  }
$$

## 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 _[[geometry of physics -- principal connections]]_.

### Preliminaries on sheaf cohomology
 {#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 [below](#TheDeligneComplex) in def. \ref{TheSmoothDeligneComplex}.

+-- {: .num_defn #Smooth0Types}
###### Definition

Write [[CartSp]] for the [[site]] whose

* [[objects]] are [[Cartesian space]]  $\mathbb{R}^n$ for $n \in \mathbb{N}$,

* [[morphisms]] are [[smooth functions]] $\mathbb{R}^{n_1} \to \mathbb{R}^{n_2}$ between these;

*  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 $\mathbb{R}^n$.

Write $PSh(CartSp) = Func(CartSp^{op},Set)$ for the [[category of presheaves]] over this site. Write

$$
  Smooth0Type \coloneqq Sh(CartSp)
$$

for its [[category of sheaves]], also called the [[cohesive topos]] of [[smooth spaces]].

=--

+-- {: .num_remark}
###### Remark

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]].

=--

+-- {: .num_example #RepresentableSheavesAndTheirConstantVersion}
###### Example

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

$$
  \flat U(1) \in Smooth0Type
$$

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.

=--

+-- {: .num_example #SheavesOfFormsAndDeRhamDifferential}
###### Example

For $k \in \mathbb{N}$ write

$$
  \mathbf{\Omega}^k \in Sh(CartSp)
$$

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

$$
  \mathbf{d} \colon \mathbf{\Omega}^k \to \mathbf{\Omega}^{k+1}
  \,.
$$

For positive $k$ its [[kernel]] is the [[subfunctor|sub-sheaf]]

$$
  0 \to
  \mathbf{\Omega}^k_{cl} \hookrightarrow \mathbf{\Omega}^k
  \stackrel{\mathbf{d}}{\longrightarrow}
  \mathbf{\Omega}^{k+1}
$$

of closed differential forms; and for $k = 0$ its kernel is the
[[subfunctor|sub-sheaf]] of constant functions

$$
  0 \to
  \flat \mathbb{R} \hookrightarrow \mathbb{R}
  \stackrel{\mathbf{d}}{\longrightarrow}
  \mathbf{\Omega}^1
  \,.
$$

=--

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.

+-- {: .num_defn #FibrantOnjectStructureOnChSmooth}
###### Definition

Write

$$
  Ch_+(Smooth0Type)
  =
  Ch_+(Sh(CartSp))
  =
  Sh(CartSp,Ch_+)
$$

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

*  a _[[weak equivalence]]_ if it is a [[quasi-isomorphism]], hence if it induces [[isomorphisms]] on ([[sheaves]] of) [[chain homology]] groups;

* a _[[fibration]]_ if it is an [[epimorphism]] (of [[abelian sheaves]]) in positive degree.

=--

+-- {: .num_remark #HomotopyFibersViaFactorizationLemma}
###### Remark

For our purpose the main use of this structure is to compute [[homotopy fibers]] via the [[factorization lemma]]. Namely

1. every chain map may be replaced, up to weak equivalence of its domain, by a fibration;

1. the [[homotopy fiber]] of a chain map is the ordinary fiber of any of its fibration replacements.

=--

+-- {: .num_remark}
###### Remark

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.

=--

+-- {: .num_remark #HomotopyPullbacksPreservedbyInclusionIntoAllSmoothHomotopyTypes}
###### Remark

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.

=--

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.

+-- {: .num_example #DeloopingsOfAbelianSheaf}
###### Example

For $A \in Ab(Smooth0Type) = Sh(CartSp,Ab)$ any [[abelian sheaf]] and for $n \in \mathbb{N}$ we write

$$
  (\mathbf{B}^n A)_{\bullet}
  \coloneqq
  A[-n]
  =
  \left[
     A

     \to 0 \to \cdots \to 0
  \right]
$$

for the [[chain complex]] of sheaves concentrated on $A$ in degree $n$.

=--

+-- {: .num_example #LogDiffInSmoothContext}

###### Example

There is a weak equivalence, def. \ref{FibrantOnjectStructureOnChSmooth},

$$
  \left[\mathbb{Z}\to \mathbb{R} \right]
  \stackrel{\simeq}{\longrightarrow}
  U(1)
$$

given by the [[chain map]]

$$
  \array{
     \mathbb{Z} &\hookrightarrow& \mathbb{R}
     \\
     \downarrow && \downarrow^{\mathrlap{mod\,\mathbb{Z}}}
     \\
     0 &\to& U(1)
  }
$$

(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 $\mathbf{d} log$:

$$
  \array{
    (\mathbb{Z} \to \mathbb{R}) &\stackrel{\mathbf{d}}{\longrightarrow}&
    \mathbf{\Omega}^1
    \\
    \downarrow^{\mathrlap{\simeq}} & \nearrow_{\mathrlap{\mathbf{d} log}}
    \\
    U(1)
    \,.
  }
$$

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}

$$
  0 \to
  \flat U(1) \hookrightarrow U(1)
  \stackrel{\mathbf{d} log}{\longrightarrow}
  \mathbf{\Omega}^1
  \,.
$$

=--

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.

+-- {: .num_defn #DeRhamResolutionOfConstantFunctions}
###### Definition

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]]:

$$
  \widehat{(\flat \mathbf{B}^{n+1}\mathbb{R})}_\bullet
  \coloneqq
  \left[
    \mathbb{R}
    \stackrel{\mathbf{d}}{\to}
    \mathbf{\Omega}^1
    \stackrel{\mathbf{d}}{\to}
    \mathbf{\Omega}^2
    \stackrel{\mathbf{d}}{\to}
    \cdots
    \stackrel{\mathbf{d}}{\to}
    \mathbf{\Omega}^{n+1}_{cl}
  \right]
  \,.
$$

=--

+-- {: .num_prop #TheDeRhamResolutionOfConstantFunctions}
###### Proposition

The morphism

$$
  (\flat \mathbf{B}^{n+1}\mathbb{R})_\bullet
  \longrightarrow
  \widehat{(\flat \mathbf{B}^{n+1}\mathbb{R})}_\bullet
$$

given by the canonical [[chain map]]

$$
  \array{
    \flat \mathbb{R}
    &\stackrel{}{\to}&
    0
    &\stackrel{}{\to}&
    0
    &\stackrel{}{\to}&
    \cdots
    &\stackrel{}{\to}&
    0
    \\
    \downarrow && \downarrow && \downarrow && \cdots
    && \downarrow
    \\
    \mathbb{R}
    &\stackrel{\mathbf{d}}{\to}&
    \mathbf{\Omega}^1
    &\stackrel{\mathbf{d}}{\to}&
    \mathbf{\Omega}^2
    &\stackrel{\mathbf{d}}{\to}&
    \cdots
    &\stackrel{\mathbf{d}}{\to}&
    \mathbf{\Omega}^{n+1}_{cl}
  }
$$

is a weak equivalence in the sense of def. \ref{FibrantOnjectStructureOnChSmooth}.

=--

+-- {: .proof}
###### Proof

By the [[Poincaré lemma]]. This _is_ the Poincaré Lemma.

=--

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 _[[cohomology]]_) in terms of [[derived hom-spaces]]. For definiteness, we recall the model for this construction given by [[Cech cohomology]]
.
+-- {: .num_defn #CechComplex}
###### Definition
**(Čech 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 **Čech 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]]

$$
  C^k((X,\{U_i\}),A_\bullet)
  \coloneqq
  \oplus_{{l,n} \atop {k = n-l}}
  \oplus_{i_0, i_1, \cdots, i_n}
  A_l(U_{i_0, \cdots, i_n})
$$

which is the [[direct sum]] of the values of $A_\bullet$ on the given intersections as indicated; and whose [[differential]]

$$
  d
  \colon
  C^{k}((X,\{U_i\}),A_\bullet)
  \longrightarrow
  C^{k+1}((X,\{U_i\}),A_\bullet)
$$

is defined componentwise (see at [[matrix calculus]] for conventions on maps between [[direct sums]]) by

$$
  \begin{aligned}
    (d a)_{i_0, \cdots, i_{k+1}}
    & \coloneqq
    (\partial_A a + (-1)^k \delta a)_{i_0, \cdots, i_{k+1}}
    \\
    & \coloneqq
    \partial_A a_{i_0, \cdots, i_{k+1}}
    +
    (-1)^k
    \sum_{0 \leq j \leq k+1} (-1)^{j}
    a_{i_0, \cdots, i_{j-1}, i_{j+1}, \cdots, i_{k+1}}
    |_{U_{i_0, \cdots, i_{k+1}}}
  \end{aligned}
$$

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 **Cech cohomology** groups of $X$ with coefficients in $A_\bullet$ __relative to the given cover__ are the [[chain homology]] groups of the Cech complex

$$
  H_{Cech}^k((X,\{U_i\}), A_\bullet)
  \coloneqq
  H^k(C^\bullet((X,\{U_i\}),A_\bullet))
  \,.
$$

The **Cech cohomology** groups as such are the [[colimit]] ("[[direct limit]]") of these groups over refinements of covers

$$
  H^k_{Cech}(X, A_\bullet)
  \coloneqq
  \underset{\longrightarrow}{\lim}_{\{U_i \to X\}}
  H_{Cech}^k((X,\{U_i\}), A_\bullet)
  \,.
$$

=--

+-- {: .num_remark}
###### Remark

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 _[[hypercohomology]]_. This is the case of relevance for Deligne cohomology.

=--

+-- {: .num_remark}
###### Remark

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 _Cech differential_ $\delta$ given by alternating sums over restrictions along patch inclusions

$$
  \array{
     \vdots && \vdots
     \\
     \downarrow^{\mathrlap{\partial_A}}
     &&
     \downarrow^{\mathrlap{\partial_A}}
     \\
     \oplus_i A_1(U_{i_0})
     &\stackrel{\delta}{\longrightarrow}&
     \oplus_{i_0, i_1} A_1(U_{i_0, i_1})
     &\stackrel{\delta}{\longrightarrow}&
     \cdots
     \\
     \downarrow^{\mathrlap{\partial_A}}
     &&
     \downarrow^{\mathrlap{\partial_A}}
     \\
     \oplus_i A_0(U_i)
     &\stackrel{\delta}{\longrightarrow}&
     \oplus_{i_0, i_1} A_0(U_{i_0, i_1})
     &\stackrel{\delta}{\longrightarrow}&
     \cdots
  }
$$

=--

+-- {: .num_remark #GoodCoverCechCohomologyIsHomotopicallyGood}
###### Remark

For analyzing the properties of Deligne cohomology [below](#Properties), all one needs is the following fact about [[Cech cohomology]], which is discussed for instance at  _[[infinity-cohesive site]]_:

For $X$ a [[smooth manifold]] (in particular [[paracompact topological space|paracompact]]),

1. $X$ 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]] $\mathbb{R}^n$);

1. for any such good open cover the [Cech complex](Čech+cohomology#CechComplex) $C^\bullet((X,\{U_i\}),A_\bullet)$ already computes Cech cohomology (i.e. there is no further need to form the [[colimit]] of Cech complexes over refinements of covers);

1. the functor $C^\bullet((X,\{U_i\}),-) \colon Ch_+(Smooth0Type) \to Ch_+$ preserves weak equivalences and fibrations.

This means in particular that if $X_\bullet \to Y_\bullet \to Z_\bullet$ is a [[homotopy fiber sequence]] in $Ch_+(Smooth0Type)$, then also

$$
  C^\bullet((X,\{U_i\}), X_\bullet)
    \to
  C^\bullet((X,\{U_i\}), Y_\bullet)
    \to
  C^\bullet((X,\{U_i\}), Z_\bullet)
$$

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.

=--

+-- {: .num_example #OrdinaryCohomologyFromCechCohomology}
###### Example

For $A_\bullet = (\mathbf{B}^{n+1}\mathbb{Z})_\bullet$ as in example \ref{DeloopingsOfAbelianSheaf}, then for $X$ a [[smooth manifold]]

$$
  H^0_{Cech}(X, (\mathbf{B}^{n+1}\mathbb{Z})_\bullet)
  \simeq
  H^{n+1}(X,\mathbb{Z})
$$

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

$$
  H^0_{Cech}(X, (\flat\mathbf{B}^{n}U(1))_\bullet)
  \simeq
  H^{n}(X,U(1))
$$

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.

=--

+-- {: .num_example #DeRhamTheorem}
###### Example

Passing to [[abelian sheaf cohomology]] (e.g. via def. \ref{CechComplex}),
then prop. \ref{TheDeRhamResolutionOfConstantFunctions}
is the [[de Rham theorem]].

=--

We will have need to give names to truncations of the de Rham complex. One is this:

+-- {: .num_defn #TruncatedDeRham}
###### Definition

For $n \in \mathbb{N}$ write

$$
  \mathbf{\Omega}^{\bullet \leq n} \in Ch_+(Smooth0Type)
$$

for the [[chain complex]] of the form

$$
  \mathbf{\Omega}^{\bullet \leq n}
  \coloneqq
  \left[
     \mathbb{R}
     \stackrel{\mathbf{d}}{\to}
     \mathbf{\Omega}^1
     \stackrel{\mathbf{d}}{\to}
     \cdots
     \stackrel{\mathbf{d}}{\to}
     \mathbf{\Omega}^{n-1}
     \stackrel{\mathbf{d}}{\to}
     \mathbf{\Omega}^{n}
  \right]
$$

with all $n$-forms, not just the closed ones, in degree 0.

=--

+-- {: .num_example #CohomologyWithCoefficientsInTruncatedDeRham}
###### Example

The [[abelian sheaf cohomology]] of the truncated
de Rham complex in def. \ref{TruncatedDeRham}
is $\Omega^n(X)/im(\mathbf{d})$.

=--

### The Deligne complex
 {#TheDeligneComplex}

+-- {: .num_defn #TheSmoothDeligneComplex}
###### Definition

For $n \in \mathbb{N}$ the **smooth Deligne complex** of degree $n$

$$
  (\mathbf{B}^n U(1)_{conn})_\bullet
  \in
  Ch_+(Smooth0Type)
$$

is the [[chain complex]] of [[abelian sheaves]] given by

$$
  (\mathbf{B}^n U(1)_{conn})_\bullet
  \;
   \coloneqq
  \;
  \left[
    U(1) \stackrel{\mathbf{d} log}{\to}
    \mathbf{\Omega}^1
    \stackrel{\mathbf{d}}{\to}
     \cdots
    \stackrel{\mathbf{d}}{\to}
    \mathbf{\Omega}^n
  \right]
$$

with $U(1)$ in degree $n$ and with the differentials as
in def. \ref{SheavesOfFormsAndDeRhamDifferential} and example \ref{LogDiffInSmoothContext}.

We write

$$
  H^{n+1}_{conn}(X,\mathbb{Z})
  \coloneqq
  H^0_{Cech}(X,(\mathbf{B}^n U(1)_{conn})_\bullet)
$$

for its [[abelian sheaf cohomology]].

=--

+-- {: .num_remark #RModZResolutionOfU1DeligneComplex}
###### Remark

By example \ref{LogDiffInSmoothContext}
the obvious [[chain map]]

$$
  \array{
    \mathbb{Z}
    &\hookrightarrow&
    \mathbb{R}
      &\stackrel{\mathbf{d}}{\to}&
     \mathbf{\Omega}^1
     &\stackrel{\mathbf{d}}{\to}&
       \cdots
     &\stackrel{\mathbf{d}}{\to}&
     \mathbf{\Omega}^n
    \\
    \downarrow
    &&
    \downarrow^{\mathrlap{(-)/\mathbb{Z}}}
    &&
    \downarrow^{id}
    &&
    &&
    \downarrow^{id}
    \\
    0
    &\to&
    U(1)
      &\stackrel{\mathbf{d} log}{\to}&
     \mathbf{\Omega}^1
     &\stackrel{\mathbf{d}}{\to}&
       \cdots
     &\stackrel{\mathbf{d}}{\to}&
     \mathbf{\Omega}^n
  }
$$

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.

=--

+-- {: .num_remark}
###### Remark

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 _closed_ forms in lowest degree, which gives [[ordinary cohomology]].

=--

+-- {: .num_defn #FlatSmoothDeligneComplex}
###### Definition

For $n \in \mathbb{N}$ the **flat smooth Deligne complex** of degree $n$

$$
  (\mathbf{B}^n U(1)_{flat})_\bullet
  \in
  Ch_+(Smooth0Type)
$$

is the [[chain complex]] of [[abelian sheaves]] given by

$$
  (\mathbf{B}^n U(1)_{flat})_\bullet
  \;
   \coloneqq
  \;
  \left[
    U(1) \stackrel{\mathbf{d} log}{\to}
    \mathbf{\Omega}^1
    \stackrel{\mathbf{d}}{\to}
     \cdots
    \stackrel{\mathbf{d}}{\to}
    \mathbf{\Omega}^n_{cl}
  \right]
$$

with $U(1)$ in degree $n$ and with the differentials as
in def. \ref{SheavesOfFormsAndDeRhamDifferential} and example \ref{LogDiffInSmoothContext}, and with the _closed_ $n$-forms
on the right.

=--

+-- {: .num_defn #FlatBnU1}
###### Definition

Write

$$
  (\flat\mathbf{B}^n U(1))_\bullet
  =
  (\mathbf{B}^n \flat U(1))_\bullet
  \in
  Ch_+(Smooth0Type)
$$

for the [[chain complex]] of [[abelian sheaves]] given by

$$
  (\flat \mathbf{B}^n U(1))_\bullet
  \;
   \coloneqq
  \;
  \left[
    \flat U(1) \stackrel{}{\to}
    0
    \stackrel{}{\to}
     \cdots
    \stackrel{}{\to}
    0
  \right]
$$

with the constant sheaf $\flat U(1)$ of example \ref{RepresentableSheavesAndTheirConstantVersion}
in degree $n$.

=--

+-- {: .num_prop #TwoEquivalentModelsForFlatBnU1}
###### Proposition

For $(\flat \mathbf{B}^n U(1))_\bullet$ as in def. \ref{FlatBnU1},
then the morphism

$$
  (\flat \mathbf{B}^n U(1))_\bullet
  \stackrel{\simeq}{\longrightarrow}
  (\mathbf{B}^n U(1)_{flat})_\bullet
$$

given by the [[chain map]]

$$
  \array{
    \flat U(1) &\to& 0 &\to& \cdots &\to& 0
    \\
    \downarrow && \downarrow  && \cdots && \downarrow
    \\
    U(1)
     &\stackrel{\mathbf{d}log}{\to}&
    \mathbf{\Omega}^1
    &\stackrel{\mathbf{d}}{\to}&
    \cdots
    &\stackrel{\mathbf{d}}{\to}&
    \mathbf{\Omega}^n_{cl}
  }
$$

(with the vertical morphism on the left being the
inclusion of example \ref{LogDiffInSmoothContext})
is a weak equivalence, def. \ref{FibrantOnjectStructureOnChSmooth}.

=--

+-- {: .proof}
###### Proof

By the [[Poincaré lemma]], this is just an immediate variant of
prop. \ref{TheDeRhamResolutionOfConstantFunctions}.

=--

### Cech-Deligne complex
 {#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

$$
  \array{
    0 &\longrightarrow& 0 &\longrightarrow& 0 &\longrightarrow & \cdots
    \\
    \uparrow && \uparrow && \uparrow
    \\
    \Omega^{p+1}(\coprod_i U_i)
    &\stackrel{\delta}{\longrightarrow}&
    \Omega^{p+1}(\coprod_{i,j} U_{ i j})
    &\stackrel{\delta}{\longrightarrow}&
    \Omega^{p+1}(\coprod_{i,j, k} U_{i j k})
    &\stackrel{\delta}{\longrightarrow}&
    \cdots
    \\
    \uparrow^{\mathrlap{d}} && \uparrow^{\mathrlap{d}} && \uparrow^{\mathrlap{d}} &&
    \\
    \vdots && \vdots && \vdots
    \\
    \uparrow^{\mathrlap{d}} && \uparrow^{\mathrlap{d}} && \uparrow^{\mathrlap{d}} &&
    \\
    \Omega^2(\coprod_i U_i)
    &\stackrel{\delta}{\longrightarrow}&
    \Omega^2(\coprod_{i,j} U_{ i j})
    &\stackrel{\delta}{\longrightarrow}&
    \Omega^2(\coprod_{i,j, k} U_{i j k})
    &\stackrel{\delta}{\longrightarrow}&
    \cdots
    \\
    \uparrow^{\mathrlap{d}} && \uparrow^{\mathrlap{d}} && \uparrow^{\mathrlap{d}} &&
    \\
    \Omega^1(\coprod_i U_i)
    &\stackrel{\delta}{\longrightarrow}&
    \Omega^1(\coprod_{i,j} U_{ i j})
    &\stackrel{\delta}{\longrightarrow}&
    \Omega^1(\coprod_{i,j, k} U_{i j k})
    &\stackrel{\delta}{\longrightarrow}&
    \cdots
    \\
    \uparrow^{\mathrlap{d log}} && \uparrow^{\mathrlap{d log}} && \uparrow^{\mathrlap{d log}} &&
    \\
    C^\infty(\coprod_i U_i, U(1))
    &\stackrel{\delta}{\longrightarrow}&
    C^\infty(\coprod_{i,j} U_{i j}, U(1))
    &\stackrel{\delta}{\longrightarrow}&
    C^\infty(\coprod_{i,j,k} U_{i j k}, U(1))
    &\stackrel{\delta}{\longrightarrow}&
    \cdots
  }
$$
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

$$
  D = d + (-1)^{deg} \delta
$$

with $deg$ denoting form degree.

+-- {: .num_example}
###### Example

A Cech-Deligne cocycle in degree $3$ ("[[bundle gerbe with connection]]") is data $(\{B_{i}\}, \{A_{i j}\}, \{g_{i j k}\})$ such that

$$
  \array{
    \{B_i\}
    &\stackrel{\delta}{\longrightarrow}&
    {{\{B_j - B_i\}} = {d A_{i j}}}
    &&
    &&
    \\
    && \uparrow^{\mathrlap{d}} &&  &&
    \\
    &&
    \{A_{i j}\}
    &\stackrel{\delta}{\longrightarrow}&
    \{-A_{ j k} + A_{i k} - A_{i j}\} = \{d log g_{i j k}\}
    &&
    \\
     &&  && \uparrow^{\mathrlap{d log}} &&
    \\
    &&
    &&
    \{g_{i j k}\}
    &\stackrel{\delta}{\longrightarrow}&
    \{g_{j k l} g_{i k l}^{-1} g_{i j l} g_{i j k}^{-1} \} = 1
  }
$$

=--

### Cup product in Deligne cohomology

The [[cup product]] on [[ordinary cohomology]] refines
to Deligne cohomology.

For more on this see at _[[Beilinson-Deligne cup-product]]_.

## Properties
  {#Properties}

### Curvature and characteristic classes
 {#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,

$$
  n \geq 1
  \,.
$$

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]].

+-- {: .num_defn #UnderlyingBundleMap}
###### Definition

Let $(\mathbf{B}^{n+1}\mathbb{Z})_\bullet \in Ch_+(Smooth0Type)$
be as in example \ref{DeloopingsOfAbelianSheaf}. Write

$$
  (\mathbf{B}^n U(1)_{conn})_{\bullet}
  \stackrel{\simeq}{\longleftarrow}
  \stackrel{DD_\bullet}{\longrightarrow}
  (\mathbf{B}^{n+1} \mathbb{Z})_{\bullet}
$$

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

$$
  \array{
     \mathbb{Z}
     &\to&
     0
     &\to&
     0
     &\to&
     \cdots
     &\to&
     0
     \\
     \uparrow^{\mathrlap{id}}
     &&
     \uparrow^{\mathrlap{}}
     &&
     \uparrow^{\mathrlap{}}
     &&
     \cdots
     &&
     \uparrow^{\mathrlap{}}
     \\
     \mathbb{Z}
     &\hookrightarrow&
     \mathbb{R}
     &\stackrel{\mathbf{d}}{\to}&
     \mathbf{\Omega}^1
     &\stackrel{\mathbf{d}}{\to}&
     \cdots
     &\stackrel{\mathbf{d}}{\to}&
     \mathbf{\Omega}^n
     \\
     \downarrow^{\mathrlap{0}}
     &&
     \downarrow^{\mathrlap{mod\,\mathbb{Z}}}
     &&
     \downarrow^{\mathrlap{id}}
     &&
     \cdots
     &&
     \downarrow^{\mathrlap{id}}
     \\
     0 &\to&
     U(1)
     &\stackrel{\mathbf{d} log}{\to}&
     \mathbf{\Omega}^1
     &\stackrel{\mathbf{d}}{\to}&
     \cdots
     &\stackrel{\mathbf{d}}{\to}&
     \mathbf{\Omega}^n
  }
$$

Passing to [[abelian sheaf cohomology]] this gives, by def. \ref{TheSmoothDeligneComplex} and example \ref{OrdinaryCohomologyFromCechCohomology}, a morphism

$$
  \array{
     H^{n+1}_{conn}(X,\mathbb{Z})
     \\
     & \searrow^{\mathrlap{DD}}
     \\
     && H^{n+1}(X,\mathbb{Z})
  }
$$

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 $DD(\nabla) \in H^{n+1}(X,\mathbb{Z})$
the [[Dixmier-Douady class]] of the _underlying [[circle n-bundle]]_.

=--

+-- {: .num_defn #CurvatureMap}
###### Definition

Write

$$
  (F_{(-)})_\bullet
  \colon
  (\mathbf{B}^n U(1)_{conn})_\bullet
  \stackrel{}{\longrightarrow}
  \mathbf{\Omega}^{n+1}_{cl}
$$

for the morphism given by the chain map which is just the
de Rham differential in degree 0

$$
  \array{
     0 &\to&
     U(1)
     &\stackrel{\mathbf{d} log}{\to}&
     \mathbf{\Omega}^1
     &\stackrel{\mathbf{d}}{\to}&
     \cdots
     &\stackrel{\mathbf{d}}{\to}&
     \mathbf{\Omega}^{n-1}
     &\stackrel{\mathbf{d}}{\to}&
     \mathbf{\Omega}^n
     \\
     \downarrow
     &&
     \downarrow
     &&
     \downarrow
     &&
     \cdots
     &&
     \downarrow^{\mathrlap{}}
     &&
     \downarrow^{\mathrlap{\mathbf{d}}}
     \\
     0
     &\to&
     0
     &\to&
     0
     &\to&
     \cdots
     &\stackrel{}{\to}&
     0
     &\to&
     \mathbf{\Omega}^{n+1}_{cl}
  }
$$

Passing to [[abelian sheaf cohomology]] this gives a morphism of the form

$$
  \array{
     && \Omega^{n+1}_{cl}(X)
     \\
     & {}^{\mathllap{F_{(-)}}}\nearrow
     \\
     H^{n+1}_{conn}(X,\mathbb{Z})
  }
$$

We call this the _[[curvature]] map_, i.e. for
$[\nabla] \in H^{n+1}_{conn}(X,\mathbb{Z})$
the class of a Deligne cocycle, we call

$$
  F_\nabla \in \Omega^{n+1}_{cl}(X)
$$

its _[[curvature form]]_.

=--

+-- {: .num_defn #InclusionOfGloballyDefinedConnectionForms}
###### Definition

Consider the [[zig-zag]]

$$
  \mathbf{\Omega}^{\bullet\leq n}
  \stackrel{}{\longrightarrow}
  (\mathbf{B}^n U(1)_{conn})_\bullet
$$

out of the complex of def. \ref{TruncatedDeRham},
given by the [[chain maps]]

$$
  \array{
    0 &\to& \mathbb{R} &\stackrel{\mathbf{d}}{\to}& \mathbf{\Omega}^1 &\stackrel{\mathbf{d}}{\to}& \cdots &\stackrel{\mathbf{d}}{\to}& \mathbf{\Omega}^{n-1}
   &\stackrel{\mathbf{d}}{\to}& \mathbf{\Omega}^n
    \\
    \downarrow && \downarrow && \downarrow^{\mathrm{id}}
    && \cdots && \downarrow^{\mathrm{id}} && \downarrow^{\mathrm{id}}
    \\
    \mathbb{Z}
    &\hookrightarrow&
    \mathbb{R}
     &\stackrel{\mathbf{d}}{\to}&
    \mathbf{\Omega}^1
    &\stackrel{\mathbf{d}}{\to}&
    \cdots
    &\stackrel{\mathbf{d}}{\to}&
    \mathbf{\Omega}^{n-1}
    &\stackrel{\mathbf{d}}{\to}&
    \mathbf{\Omega}^n
    \\
    \downarrow && \downarrow && \downarrow^{\mathrm{id}}
    && \cdots && \downarrow^{\mathrm{id}} && \downarrow^{\mathrm{id}}
    \\
    0
    &\to&
    U(1)
     &\stackrel{\mathbf{d}log}{\to}&
    \mathbf{\Omega}^1
    &\stackrel{\mathbf{d}}{\to}&
    \cdots
    &\stackrel{\mathbf{d}}{\to}&
    \mathbf{\Omega}^{n-1}
    &\stackrel{\mathbf{d}}{\to}&
    \mathbf{\Omega}^n
  }
$$

where the bottom [[quasi-isomorphism]] is from remark  \ref{RModZResolutionOfU1DeligneComplex}.

On passing to [[abelian sheaf cohomology]] this gives,
by example \ref{CohomologyWithCoefficientsInTruncatedDeRham}, a morphism

$$
  \array{
    \Omega^n(X)/im(\mathbf{d})
    \\
    & \searrow
    \\
    && H^{n+1}_{conn}(X,\mathbb{Z})
  }
$$

=--

+-- {: .num_defn #InclusionOfFlatConnections}
###### Definition

Consider the canonical morphism

$$
  (\flat \mathbf{B}^n U(1))_\bullet
  \stackrel{\simeq}{\longrightarrow}
  (\mathbf{B}^n U(1)_{flat})_\bullet
  \longrightarrow
  (\mathbf{B}^n U(1)_{conn})_\bullet
$$

via def. \ref{FlatSmoothDeligneComplex},
prop. \ref{TwoEquivalentModelsForFlatBnU1}.

Passing to [[abelian sheaf cohomology]] this induces
a morphism

$$
  \array{
    && H^{n+1}_{conn}(X,\mathbb{Z})
    \\
    & \nearrow
    \\
    H^n(X,U(1))
  }
$$

We call this map the _inclusion of the [[flat infinity-connections]]_
into all [[circle n-connections]].

=--

Combining what we have so far:

+-- {: .num_prop #BocksteinHomomorphism}
###### Proposition

The [[composition|composite]] of the morphisms
of def. \ref{InclusionOfGloballyDefinedConnectionForms} and of the curvature morphism of def. \ref{CurvatureMap}

$$
  \array{
    \Omega^n(X)/im(\mathbf{d}) && \stackrel{\mathbf{d}}{\longrightarrow} &&
    \Omega^{n+1}_{cl}(X)
    \\
    & \searrow && \nearrow_{\mathrlap{F_{(-)}}}
    \\
    && H^{n+1}_{conn}(X,\mathbb{Z})
  }
$$

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]]:

$$
  \array{
    && H^{n+1}_{conn}(X,\mathbb{Z})
    \\
    & \nearrow && \searrow^{\mathrlap{DD}}
    \\
    H^n(X,U(1))
    &&
     \underset{\beta}{\longrightarrow}
    &&
    H^{n+1}(X,\mathbb{Z})
  }
$$

=--

+-- {: .proof}
###### Proof

By composing the defining zig-zags of [[chain maps]]
the statement is immediate.

=--

### The Chern character
 {#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.

+-- {: .num_defn #ChernCharacterMap}
###### Definition

Write

$$
  ch_\bullet
   \colon
  (\mathbf{B}^{n+1}\mathbb{Z})_\bullet
  \longrightarrow
  (\flat \mathbf{B}^{n+1}\mathbb{R})_\bullet
$$

for the morphism given as the composite

$$
  (\mathbf{B}^{n+1}\mathbb{Z})_\bullet
  =
  (\flat \mathbf{B}^{n+1}\mathbb{Z})_\bullet
  \longrightarrow
  (\flat \mathbf{B}^{n+1}\mathbb{R})_\bullet
$$

where the second morphism is induced by the canonical inclusion $\mathbb{Z} \hookrightarrow \mathbb{R}$.

Passing to [[abelian sheaf cohomology]] this induces a morphism

$$
  \array{
    && H^{n+1}(X,\mathbb{R})
    \\
    & \nearrow_{\mathrlap{ch}}
    \\
    H^{n+1}(X,\mathbb{Z})
  }
$$

=--

+-- {: .num_remark}
###### Remark

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 _[[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

=--

+-- {: .num_defn #DifferentialResolutionOfIntegralCoefficients}
###### Definition

Consider the chain complex

$$
  \widehat {(\mathbf{B}^{n+1}\mathbb{Z})}_\bullet
  \coloneqq
  \left[
  \array{
    \mathbb{Z} &\stackrel{}{\hookrightarrow}& \mathbb{R}
    &\stackrel{\mathbf{d}}{\to}& \mathbf{\Omega}^1  &\stackrel{\mathbf{d}}{\to}&
    \cdots
    &\stackrel{\mathbf{d}}{\to}& \mathbf{\Omega}^{n-1}
    &\stackrel{\mathbf{d}}{\to}& \mathbf{\Omega}^n
    \\
    \oplus
    &\nearrow_{-\mathrlap{id}}&
    \oplus
    &\nearrow_{+\mathrlap{id}}&
    \oplus
    &\nearrow_{-\mathrlap{id}}&
    \cdots
    &\nearrow_{\pm\mathrlap{id}}&
    \oplus
    &\nearrow_{\mp\mathrlap{id}}&
    \\
    \mathbb{R} &\underset{\mathbf{d}}{\to}& \mathbf{\Omega}^1
    &\underset{\mathbf{d}}{\to}&
    \mathbf{\Omega}^2 &\underset{\mathbf{d}}{\to}& \cdots
    &\underset{\mathbf{d}}{\to}&
    \mathbf{\Omega}^{n}
    &&
  }
  \right]
$$

where we use [[matrix calculus]]-notation as for [[mapping cones]]
(see at _[mapping cone -- Examples -- In chain complexes](mapping+cone#InChainComplexes)_).

Write

$$
  \widehat {(\mathbf{B}^{n+1}\mathbb{Z})}_\bullet
  \stackrel{\widehat{ch}_\bullet}{\longrightarrow}
  \widehat {(\flat \mathbf{B}^{n+1}\mathbb{R})}_\bullet
$$

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:

$$
  \array{
    \mathbb{Z} &\stackrel{}{\hookrightarrow}& \mathbb{R}
    &\stackrel{\mathbf{d}}{\to}& \mathbf{\Omega}^1
    &\stackrel{\mathbf{d}}{\to}&
    \cdots
    &\stackrel{\mathbf{d}}{\to}&
    \mathbf{\Omega}^{n-1}
    &\stackrel{\mathbf{d}}{\to}&
    \mathbf{\Omega}^n
    \\
    \oplus
    &\nearrow_{-\mathrlap{id}}&
    \oplus
    &\nearrow_{+\mathrlap{id}}&
    \oplus
    &\nearrow_{-\mathrlap{id}}&
    \cdots
    &\nearrow_{\pm\mathrlap{id}}&
    \oplus
    &\nearrow_{\mp\mathrlap{id}}&
    \\
    \mathbb{R} &\underset{\mathbf{d}}{\to}& \mathbf{\Omega}^1
    &\underset{\mathbf{d}}{\to}&
    \mathbf{\Omega}^2 &\underset{\mathbf{d}}{\to}& \cdots
    &\underset{\mathbf{d}}{\to}&
    \mathbf{\Omega}^{n}
    &&
    \\
    \downarrow && \downarrow && \downarrow && \cdots
    && \downarrow && \downarrow^{\mathrlap{\mathbf{d}}}
    \\
    \mathbb{R}
    &\stackrel{\mathbf{d}}{\to}&
    \mathbf{\Omega}^1
    &\stackrel{\mathbf{d}}{\to}&
    \mathbf{\Omega}^2
    &\stackrel{\mathbf{d}}{\to}&
    \cdots
    &\stackrel{\mathbf{d}}{\to}&
    \mathbf{\Omega}^{n}
    &\stackrel{\mathbf{d}}{\to}&
    \mathbf{\Omega}^{n+1}
  }
$$

Finally write

$$
  (\mathbf{B}^{n+1}\mathbb{Z})_\bullet
  \longrightarrow
  \widehat {(\mathbf{B}^{n+1}\mathbb{Z})}_\bullet
$$

for the morphism given by the chain map which in degree $n+1$ is given by

$$
  \mathbb{Z} \to \mathbb{Z} \oplus \mathbb{R}
$$

$$
  n \mapsto (n,n)
  \,.
$$

=--

+-- {: .num_lemma #ChernCharacterResolution}
###### Lemma

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]]

$$
  \array{
    (\mathbf{B}^{n+1}\mathbb{Z})_\bullet
    &\stackrel{ch_\bullet}{\longrightarrow}&
    (\flat\mathbf{B}^{n+1} \mathbb{R})_\bullet
    \\
    \downarrow^{\mathrlap{\simeq}} && \downarrow^{\mathrlap{\simeq}}
    \\
    \widehat {(\mathbf{B}^{n+1}\mathbb{Z})}_\bullet
    &\stackrel{\widehat {ch}_\bullet}{\longrightarrow}&
    \widehat {(\flat \mathbf{B}^{n+1}\mathbb{R})}_\bullet
  }
$$

with, on the right, the weak equivalence of prop. \ref{TheDeRhamResolutionOfConstantFunctions}, where

1. the left vertical morphism is a weak equivalence;

1. the bottom horizontal morphism $\widehat{ch}_\bullet$ is a fibration.

=--

+-- {: .proof}
###### 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}.

=--

+-- {: .num_defn #MapFromClosedFormsToHyperDeRhamCoefficients}
###### Definition

Write

$$
  \mathbf{\Omega}^{n+1}_{cl}
  \stackrel{\iota_\bullet}{\longrightarrow}
  \widehat{\flat \mathbf{B}^{n+1}\mathbb{R}}_\bullet
  \stackrel{\simeq}{\longleftarrow}
  (\flat \mathbf{B}^{n+1}\mathbb{R})_\bullet
$$

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]]

$$
  \array{
    0 &\to& 0 &\to& \cdots &\to& 0 &\to& \mathbf{\Omega}^{n+1}_{cl}
    \\
    \downarrow && \downarrow && \vdots && \downarrow && \downarrow^{\mathrlap{id}}
    \\
    \mathbb{R}
    &\stackrel{\mathbf{d}}{\to}&
    \mathbf{\Omega}^1
    &\stackrel{\mathbf{d}}{\to}&
    \cdots
    &\stackrel{\mathbf{d}}{\to}&
    \mathbf{\Omega}^{n}
    &\stackrel{\mathbf{d}}{\to}&
    \mathbf{\Omega}^{n+1}_{cl}
  }
  \,.
$$

=--

+-- {: .num_prop #HomotopyFiberProductCharacterization}
###### Proposition

The chain maps

* $(F_{(-)})_\bullet$, def. \ref{CurvatureMap};

* $DD_\bullet$, def. \ref{UnderlyingBundleMap};

* $\iota_\bullet$, def. \ref{MapFromClosedFormsToHyperDeRhamCoefficients}

* $\widehat{ch}_\bullet$, def. \ref{DifferentialResolutionOfIntegralCoefficients}

fit into a [[commuting diagram]]

$$
  \array{
     && \mathbf{\Omega}^{n+1}_{cl}
     \\
     & {}^{\mathllap{(F_{(-)})_\bullet }}\nearrow && \searrow^{\mathrlap{\iota_\bullet}}
     \\
     (\mathbf{B}^n U(1)_{conn})_\bullet
     && && \widehat{(\flat \mathbf{B}^{n+1}\mathbb{R})}_\bullet
     \\
     & {}_{\mathllap{DD_\bullet}}\searrow &&
     \nearrow_{\mathrlap{\widehat{ch}_\bullet}}
     \\
     && \widehat{(\mathbf{B}^{n+1}\mathbb{Z})}_\bullet
  }
$$

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 $ch_\bullet$.

=--

+-- {: .proof}
###### 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 $ch_\bullet$.

=--

### The exact sequences for curvature and characteristic classes
 {#ExactSequenceForCurvatureAndCharacteristicClass}

+-- {: .num_defn #IntegralDifferentialForms}
###### Definition

Write

$$
  \Omega^{n+1}_{int}(X)
  \hookrightarrow
  \Omega^{n+1}_{cl}(X)
$$

for the inclusion of those closed [[differential forms]] whose [[periods]] ([[integration of differential forms|integration]] over $(n+1)$-cycles) takes values in the [[integers]].

=--

+-- {: .num_prop #ImageOfCurvatureInIntegralDifferentialForms}
###### Proposition

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}.

=--

+-- {: .proof}
###### 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.

=--

+-- {: .num_prop #CurvatureExactSequence}
###### Proposition
**(curvature exact sequence)**

The Deligne cohomology group fits into a [[short exact sequence]]
(of [[abelian groups]]) of the form

$$
  0
  \to
  H^n(X,U(1))
  \longrightarrow
  H^{n+1}_{conn}(X,\mathbb{Z})
  \stackrel{F_{(-)}}{\longrightarrow}
  \Omega^{n+1}_{int}(X)
  \to 0
$$

where $F_{(-)}$ is the [[curvature]] map of def. \ref{CurvatureMap}.

=--

+-- {: .proof}
###### 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

$$
  \array{
     0 &\longrightarrow& \widehat{(\flat \mathbf{B}^{n}(\mathbb{R}/\mathbb{Z}))}_\bullet &\longrightarrow& 0
     \\
     \downarrow && \downarrow && \downarrow^{}
     \\
     0 &\longrightarrow& (\mathbf{B}^n U(1)_{conn})_\bullet
     &\stackrel{(F_{(-)}_\bullet)}{\longrightarrow}&
     \mathbf{\Omega}^{n+1}_{cl}
     \\
     && \downarrow^{\mathrlap{DD_\bullet}} && \downarrow^{\mathrlap{\iota_\bullet}}
     \\
     && \widehat{(\mathbf{B}^{n+1}\mathbb{Z})}_\bullet
     &\stackrel{\widehat{ch}_\bullet}{\longrightarrow}&
     \widehat{(\flat \mathbf{B}^{n+1}\mathbb{R})}_\bullet
  }
  \,.
$$

Passing to [[abelian sheaf cohomology]]
and applying the induced [[long exact sequence in homology]], in view of remark \ref{GoodCoverCechCohomologyIsHomotopicallyGood}, implies the claim.

=--

+-- {: .num_prop #CharacteristicClassExactSequence}
###### Proposition
**(characteristic class exact sequence)**

The Deligne cohomology group fits into a [[short exact sequence]]
(of [[abelian groups]]) of the form

$$
  0
  \to
  \Omega^{n}(X)/\Omega^n_{int}(X)
  \stackrel{}{\longrightarrow}
  H^{n+1}_{conn}(X,\mathbb{Z})
  \stackrel{DD}{\longrightarrow}
  H^{n+1}(X,\mathbb{Z})
  \to 0
$$

where $DD$ is the characteristic class map of
def. \ref{UnderlyingBundleMap}.

=--

+-- {: .proof}
###### 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

$$
  \cdots
  \longrightarrow
  H^n(X,\mathbb{Z})
  \longrightarrow
  \Omega^n(X)/im(\mathbf{d})
  \stackrel{}{\longrightarrow}
  H^{n+1}_{conn}(X,\mathbb{Z})
  \stackrel{DD}{\longrightarrow}
  H^{n+1}(X,\mathbb{Z})
$$

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.

=--

+-- {: .num_remark}
###### Remark

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]]

1. of integral [[ordinary cohomology]] by differential $n$-forms modulo integral forms;

1. of closed $(n+1)$-forms by $U(1)$-valued ordinary cohomology.

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 _[[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 _[[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.

=--

### The exact differential cohomology hexagon
 {#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.

$$
  \array{
     \mathbf{\Omega}^{\bullet \leq n} &&
     \stackrel{\mathbf{d}}{\longrightarrow} && \mathbf{\Omega}^{n+1}_{cl}
     \\
     &\searrow& & {}^{\mathllap{(F_{(-)})_\bullet }}\nearrow && \searrow^{\mathrlap{\iota_\bullet}}
     \\
     && (\mathbf{B}^n U(1)_{conn})_\bullet
     && && \widehat{(\flat \mathbf{B}^{n+1}\mathbb{R})}_\bullet
     \\
     &\nearrow& & {}_{\mathllap{DD_\bullet}}\searrow &&
     \nearrow_{\mathrlap{\widehat{ch}_\bullet}}
     \\
     \widehat{(\flat \mathbf{B}^n U(1))}_\bullet
     && \underset{\beta}{\longrightarrow} && \widehat{(\mathbf{B}^{n+1}\mathbb{Z})}_\bullet
  }
$$

+-- {: .num_prop #TheDifferentialHexagonForDeligne}
###### Proposition

This extends to a diagram in $Ch_+(Smooth0Type)$ of the form

$$
  \array{
     && \mathbf{\Omega}^{\bullet \leq n} &&
     \stackrel{\mathbf{d}}{\longrightarrow} && \mathbf{\Omega}^{n+1}_{cl}
     \\
     & \nearrow& &\searrow& & {}^{\mathllap{(F_{(-)})_\bullet }}\nearrow && \searrow^{\mathrlap{\iota_\bullet}}
     \\
     \widehat{(\flat \mathbf{B}^n \mathbb{R})}_{\bullet}
     && && (\mathbf{B}^n U(1)_{conn})_\bullet
     && && \widehat{(\flat \mathbf{B}^{n+1}\mathbb{R})}_\bullet
     \\
     &\searrow & &\nearrow& & {}_{\mathllap{DD_\bullet}}\searrow &&
     \nearrow_{\mathrlap{\widehat{ch}_\bullet}}
     \\
     && \widehat{(\flat \mathbf{B}^n U(1))}_\bullet
     && \underset{\beta}{\longrightarrow} && \widehat{(\mathbf{B}^{n+1}\mathbb{Z})}_\bullet
  }
$$

such that

1. both square are [[homotopy pullback]] squares;

1. both diagonals are [[homotopy fiber sequences]];

1. the two outer sequences are [[long homotopy fiber sequences]].

=--

+-- {: .proof}
###### 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

$$
  \array{
     \widehat{(\flat \mathbf{B}^n \mathbb{R})}_\bullet &\longrightarrow& \widehat{(\flat \mathbf{B}^{n}(\mathbb{R}/\mathbb{Z}))}_\bullet &\longrightarrow& 0
     \\
     \downarrow && \downarrow && \downarrow^{}
     \\
     \mathbf{\Omega}^{\bullet \leq n}
     &\longrightarrow& (\mathbf{B}^n U(1)_{conn})_\bullet
     &\stackrel{(F_{(-)}_\bullet)}{\longrightarrow}&
     \mathbf{\Omega}^{n+1}_{cl}
     \\
     && \downarrow^{\mathrlap{DD_\bullet}} && \downarrow^{\mathrlap{\iota_\bullet}}
     \\
     && \widehat{(\mathbf{B}^{n+1}\mathbb{Z})}_\bullet
     &\stackrel{\widehat{ch}_\bullet}{\longrightarrow}&
     \widehat{(\flat \mathbf{B}^{n+1}\mathbb{R})}_\bullet
  }
  \,.
$$

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 [here](differential+cohomology+diagram#TheDifferentialDiagram).

=--

+-- {: .num_remark}
###### Remark

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 _[[differential cohomology hexagon]]_.

=--

### GAGA
 {#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]] ([Esnault 89, corollary 1.3](#Esnault89)).

### 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]]

### 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
$$
  Funct^\infty(P_1(-), \mathbf{B}U(1)) : Op(X)^{op} \to LieGrpd
  \,.
$$
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

$$
  Funct^\infty(\Pi_1(-), \mathbf{B}U(1)) \simeq \mathbb{Z}(2)^\infty_D
  \,.
$$

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
$$
  n Funct^\infty(P_n(-), \mathbf{B}^n)
  \simeq
  \mathbb{Z}(n+1)^\infty_D
  \,.
$$

See

* John Baez, Urs Schreiber, _Higher Gauge Theory_ ([arXiv](http://arxiv.org/abs/math/0511710))

The full proof for $n=1$ this is in

* Urs Schreiber, Konrad Waldorf, _Parallel transport and functors_ ([arXiv](http://arxiv.org/abs/0705.0452));

for $n=2$ in

* Urs Schreiber, Konrad Waldorf, _Smooth functors versus differential forms_ ([arXiv](http://arxiv.org/abs/0802.0663))

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.

## 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.

* 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 $U(1)$.

  * 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.

* Degree 3 Deligne cohomology classifies [[bundle gerbe]]s with connection.

  * 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.

* 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]].

  * In formal high energy physics degree 4 Deligne classes model the [[supergravity C-field]].

## Related concepts

* [[Hodge-filtered differential cohomology]]

* [[intermediate Jacobian]] ([[self-dual higher gauge theory]])

* [[Abel-Jacobi map]]

* [[Beilinson regulator]]

* [[circle n-bundle with connection]]

* [differential cohomology diagram -- Examples -- Deligne coefficients](differential+cohomology+diagram#DeligneCoefficients)

## References
 {#References}

Deligne cohomology was introduced in [[complex analytic geometry]] (by a [[chain complex]] of [[holomorphic differential forms]]) in

* {#Deligne71} [[Pierre Deligne]], Section 2.2 of: _Théorie de Hodge II_, IHES Pub. Math. (1971), no. 40, 5–57 ([numdam:PMIHES_1971__40__5_0](http://www.numdam.org/item/?id=PMIHES_1971__40__5_0))

with applications to [[Hodge theory]] and [[intermediate Jacobians]]. The same definition appears in

* [[Barry Mazur]], [[William Messing]], Section 3.1.7 of: _Universal extensions and one-dimensional crystalline cohomology_, Springer lecture notes 370, 1974 ([doi:10.1007/BFb0061628](https://link.springer.com/book/10.1007/BFb0061628))

* {#ArtinMazur77} [[Michael Artin]], [[Barry Mazur]], section III.1 of _Formal Groups Arising from Algebraic Varieties_, Annales scientifiques de l'École Normale Supérieure, Sér. 4, 10 no. 1 (1977), p. 87-131  ([numdam:ASENS_1977_4_10_1_87_0](http://www.numdam.org/item?id=ASENS_1977_4_10_1_87_0), [MR56:15663](http://www.ams.org/mathscinet-getitem?mr=56:15663))

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

* {#Beilinson85} [[Alexander Beilinson]], _[[Higher regulators and values of L-functions]]_, Journal of Soviet Mathematics 30 (1985), 2036-2070, ([mathnet (Russian)](http://www.mathnet.ru/php/archive.phtml?wshow=paper&jrnid=intd&paperid=73&option_lang=eng), [DOI](http://dx.doi.org/10.1007%2FBF02105861))

   English translation: J Math Sci 30, 2036–2070 (1985) ([doi:10.1007/BF02105861](https://doi.org/10.1007/BF02105861))

  > (reviewed in [Esnault-Viehweg 88](#EsnaultViehweg88))

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".

Identifying the [[background field|background]] [[B-field]] in [[2d CFT]] ([[worldsheet]] [[string theory]]) as a Deligne 3-cocyle ([[bundle gerbe]] [[connection on a bundle gerbe|with connection]]):

* [[Krzysztof Gawędzki]], *Topological Actions in two-dimensional Quantum Field Theories*, in: _Nonperturbative quantum field theory_, Nato Science Series B **185**, Springer (1986) [[spire:257658](http://inspirehep.net/record/257658), [doi:10.1007/978-1-4613-0729-7_5](https://doi.org/10.1007/978-1-4613-0729-7_5), [[Gawedzki-TopologicalActions.pdf:file]]]

Surveys and introductions in the context of [[differential geometry]] include

* {#Brylinski93} [[Jean-Luc Brylinski]], section I.5 of: _Loop Spaces, Characteristic Classes and geometric Quantization_, Birkhäuser 1993 ([doi:10.1007/978-0-8176-4731-5](https://link.springer.com/book/10.1007/978-0-8176-4731-5))

* {#Bunke12} [[Ulrich Bunke]], section 3 of _Differential cohomology_ ([arXiv:1208.3961](http://arxiv.org/abs/1208.3961))

Review with more emphasis on [[complex analytic geometry]] and the theory of ([Beilinson 85](#Beilinson85)) with more details spelled out is in

* {#EsnaultViehweg88} [[Hélène Esnault]], [[Eckart Viehweg]], _Deligne-Beilinson cohomology_, in: [[Michael Rapoport]], [[Norbert Schappacher]], [[Peter Schneider]] (eds.), _[[Beilinson's Conjectures on Special Values of L-Functions]]_, Perspectives in Mathematics **4**, Academic Press, Inc. (1988) [ISBN:978-0-12-581120-0, [[EsnaultViehweg-DeligneBeilinsonCohomology.pdf:file]]]

* {#Esnault89} [[Hélène Esnault]], _On the Loday-symbol in the Deligne-Beilinson cohomology_, K-theory 3, 1-28, 1989 ([pdf](http://www.mi.fu-berlin.de/users/esnault/preprints/helene/16-loday-symbol.pdf))

See also

* {#PetersSteenbrink08} [[Chris Peters]], [[Jozef Steenbrink]], section 7.2 of _[[Mixed Hodge Structures]]_, Ergebisse der Mathematik (2008) ([pdf](http://www.arithgeo.ethz.ch/alpbach2012/Peters_Steenbrinck))

* {#Voisin02} [[Claire Voisin]], section 12 of _[[Hodge theory and Complex algebraic geometry]] I,II_,  Cambridge Stud. in Adv. Math. __76, 77__, 2002/3

Discussion of Deligne cohomology as classifying higher [[bundle gerbes]] ([[bundle 2-gerbes]], etc.) [[connection on a bundle gerbe|with connection]]:

* [[Pawel Gajer]], _Geometry of Deligne cohomology_, Invent. Math., 127(1):155-207 (1997) ([arXiv:alg-geom/9601025](http://arxiv.org/abs/alg-geom/9601025), [doi:10.1007/s002220050118](https://doi.org/10.1007/s002220050118))

Discussion of Deligne cohomology in terms of [[simplicial presheaves]] and [[higher stacks]] includes

* {#FSS10} [[Domenico Fiorenza]], [[Urs Schreiber]], [[Jim Stasheff]], _[[schreiber:Cech Cocycles for Differential characteristic Classes]]_, Advances in Theoretical and Mathematical Physics, Volume 16 Issue 1 (2012), pages 149-250 ([arXiv:1011.4735](http://arxiv.org/abs/1011.4735))

* [[Domenico Fiorenza]], [[Hisham Sati]], [[Urs Schreiber]], _[[schreiber:Extended higher cup-product Chern-Simons theories]]_, Journal of Geometry and Physics, Volume 74, 2013, Pages 130–163 ([arXiv:1207.5449](http://arxiv.org/abs/1207.5449))

* [[Michael Hopkins]], [[Gereon Quick]], _Hodge filtered complex bordism_, [arXiv:1212.2173](http://arxiv.org/abs/1212.2173)

  > (cf. [[Hodge-filtered differential cohomology]])

* [[Domenico Fiorenza]], [[Hisham Sati]], [[Urs Schreiber]], _A higher stacky perspective on Chern-Simons theory_, in Damien Calaque et al. (eds.) _Mathematical Aspects of Quantum Field Theories_, Mathematical Physics Studies, Springer 2014 ([arXiv:1301.2580](http://arxiv.org/abs/1301.2580))

* [[Urs Schreiber]], _[[schreiber:differential cohomology in a cohesive topos]]_ ([arXiv:1310.7930](http://arxiv.org/abs/1310.7930))

See also the references given at _[differential cohomology hexagon -- Deligne coefficients](differential+cohomology+diagram#DeligneCoefficients)_.

[[!redirects Deligne complex]]
[[!redirects Deligne complexes]]

[[!redirects Deligne-Beilinson cohomology]]
[[!redirects Beilinson-Deligne cohomology]]

[[!redirects Deligne cocycle]]
[[!redirects Deligne cocycles]]


Deligne completeness theorem

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Topos Theory
#### Model Theory
+-- {: .hide}
[[!include topos theory - contents]]
[[!include model theory - contents]]
=--
=--
=--

# Contents
* toc
{: toc}

## Idea

The _Deligne completeness theorem_, or **Deligne's theorem**, as it is also called, was initially proved by [[Pierre Deligne]] in an appendix of [[SGA4]][^sga] in the context of [[algebraic geometry]] as a theorem concerning the abundance of  _points_ for a [[coherent topos]].

When in the early 70s the connection between topos theory and logic became manifest [[William Lawvere]] (1975) pointed out that the theorem may be viewed as a variant of the classical [[Gödel-Henkin completeness theorem]] for [[first-order logic]]: since points of a topos $\mathcal{E}$ correspond to set-theoretic models of the theory [[classifying topos|classified]] by $\mathcal{E}$ it amounts not only to saying that a finitary [[geometric theory]] $\mathbb{T}$ has models in $Set$ but that the provability of a sequent in [[coherent logic]] relative to $\mathbb{T}$ is equivalent to its validity in all set-theoretical models of $\mathbb{T}$.

[^sga]: [SGA 4, vol. II](#SGA4II), exposé VI, p.336.

## Statement

Recall that a point of a topos $\mathcal{E}$ is simply a geometric morphism $p:Set\to\mathcal{E}$ and that $\mathcal{E}$ is said to have [[enough points]] when for any two distinct parallel $f,g:A\to B$ in $\mathcal{E}$ there is a point $p$ that separates $f$ and $g$: $p*(f)\neq p*(g)$.

+-- {: .num_theorem }
###### Theorem

A [[coherent topos]] has [[enough points]].

=--

As a corollary of the [[Deligne-Lurie completeness theorem]] this appears as ([[Spectral Schemes|Lurie SpecSchm, corollary 4.2]]).

## Remarks

Since a Grothendieck topos $\mathcal{E}$ has enough points iff it has a sufficient set of points in the sense that there is a [[surjective geometric morphism|surjection]] $Set/K\to \mathcal{E}$ with $K$ a set (cf. Johnstone [1977](#JT77), pp.224-229) and, furthermore, $Set/K\cong Sh(2^K)$,  one sees that Deligne's theorem yields a special form  $Sh(2^K)\to\mathcal{E}$ of [[Barr's theorem]].

A _general_ Grothendieck topos may fail to have enough points or even fail to have points at all, but it nevertheless has 'enough Boolean-valued points' (cf. [[Barr's theorem]]).

## Related entries

* [[Deligne-Lurie completeness theorem]]

* [[Barr's theorem]]

* [[coherent logic]]

* [[geometric theory]]

* [[coherent topos]]

* [[compactness theorem]]

## References

* {#SGA4II}M. Artin, A. Grothendieck, J. L. Verdier (eds.), _Théorie des Topos et Cohomologie Etale des Schémas - SGA 4. II_ , LNM **270** Springer Heidelberg 1972.

* B. Frot, _Gödel's Completeness Theorem and Deligne's Theorem_ , arXiv:1309.0389 (2013). ([pdf](http://arxiv.org/pdf/1309.0389v1))

* {#JT77}P. T. Johnstone, _Topos Theory_ , Academic Press New York 1977 (Dover reprint 2014). (ch. 7) {#Johnstone77b}

* F. W. Lawvere, _Continuously Variable Sets: Algebraic Geometry= Geometric Logic_ , pp.135-156 in _Proc. Logic Colloquium Bristol 1973_, North-Holland Amsterdam 1975.

* S. Mac Lane, I. Moerdijk, _Sheaves in Geometry and Logic_ , Springer Heidelberg 1994. (sec. IX.11, pp.521f)

* G. E. Reyes, _Sheaves and concepts: A model-theoretic interpretation of Grothendieck topoi_ , Cah. Top. Diff. Géo. Cat. **XVIII** no.2 (1977) pp.405-437. ([numdam](http://www.numdam.org/item?id=CTGDC_1977__18_2_105_0))


Deligne conjecture

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###Context###
#### Higher algebra
+--{: .hide}
[[!include higher algebra - contents]]
=--
#### Cohomology
+--{: .hide}
[[!include cohomology - contents]]
=--
#### Homological algebra
+--{: .hide}
[[!include homological algebra - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

### Statement

The original _Deligne conjecture_ on the structure of [[Hochschild cohomology]] of an [[associative algebra]] and more generally an [[A-∞ algebra]] states that Hochschild cohomology naturally has the structure of a [[BV-algebra]] and that moreover the full Hochschild cohomology complex has the structure of an [[algebra over an operad]] over the ([[framed little disk operad|framed]]) [[little disk operad]]. (The second statement implies the first since ([[BV-algebra|BV]]-)[[nLab:Gerstenhaber algebra]]s are [[algebras over an operad]] over the (framed) little disk operad.)

Proofs of the conjecture have been given first by  [McClure-Smith](#McClureSmith) and others, then also by [Kontsevich-Soibelman](#KontsevichSoibelman), and [Tamarkin](#Tamarkin).

A proof for general [[Ek-algebra]]s in [[symmetric monoidal (∞,1)-categories]] is in ([Lurie, section 2.5](#Lurie)).

### Geometric interpretation
 {#GeometricInterpretation}

A general geometric ([[higher geometry|higher geometric]]) interpretation has been indicated in [Ben-ZviFrancisNadler](#Ben-ZviFrancisNadler). They observed (see [[Hochschild cohomology]] for details) that the Hochschild _homology_ of $\mathcal{O}(X)$ is naturally interpreted as the algebra of functions on the [[derived loop space]] $\mathcal{L}X$. In the presence of good
[[geometric ∞-function theory]] this naturally induces an action of the [[little disk operad]] on $\mathcal{O}(\mathcal{L}X)$. Since a [[Gerstenhaber algebra]] is an algebra over the [[homology]] of the [[little disk operad]], this immediately explains the existence of this structure.

See also [this prop](Hochschild+cohomology#DeligneConjectureViaDerivedMappingSpaces).

### For higher algebras and higher monoidal categories

More generally, analogs of the statement of the Deligne conjecture exist and work for $(\infty,n)$-algebras: [[k-tuply monoidal n-categories]]. This is closely related to the statement (and proofs) of the [[delooping hypothesis]].

This case is discussed in ([Francis](#Francis)) and ([Lurie](#Lurie)).

In ([KockToen](#KockToen)) it is shown that in a [[monoidal model category]] that is also a [[simplicial model category]] the [[derived hom-space]] $\mathbb{R} Hom(I,I)$ from the tensor unit to itself is a [[Ek-algebra|E-2 algebra]].

### History

Historically it was first found that there is the structure of a [[Gerstenhaber algebra]] on $HH^\bullet(A,A)$.  By ([Cohen](#Cohen)) it was known that Gerstenhaber algebras arise as the [[homology]] of [[E2-algebra]]s in [[chain complex]]es. In a letter in 1993 Deligne wondered whether the [[Gerstenhaber algebra]] structure on the Hochschild cohomology $HH^\bullet(A,A)$ lifts to an [[E2-algebra]]-structure on the cochain complex $C^\bullet(A,A)$.

In [GerstenhaberVoronov (1994)](#GerstenhaberVoronov) a resolution of the Gerstenhaber algebra structure was given, but the relationship to $E_2$-algebras remained unclear.

In ([Tamarkin (1998)](#Tamarkin)) a genuine resolution in the [[model structure on operads]] of the Gerstenhaber operad was given and shown to act via the Gerstenhaber-Voronov construction on $C^\bullet(A,A)$. This proved Deligne's conjecture.

Various authors later further refined this result. A summary of this history can be found in ([Hess](#Hess)).

In [Hu-Kriz-Voronov (2003)](#HuKrizVoronov) it was further shown that for $A$ an [[En-algebra]], $C^\bullet(A,A)$ is an $E_{n+1}$-algebra.

## References

Direct proofs of the Deligne conjecture have been given in.

* James McClure, Jeffrey Smith, *A solution of Deligne's conjecture* ([arXiv:math/9910126](http://arxiv.org/abs/math/9910126))
{#McClureSmith}

* [[Maxim Kontsevich]], [[Yan Soibelman]], *Deformations of algebras over operads and Deligne's conjecture* ([arXiv:math/0001151](http://arxiv.org/abs/math/0001151))
{#KontsevichSoibelman}

* [[Dmitry Tamarkin]], *Another proof of M. Kontsevich formality theorem* ([arXiv:math/9803025](http://arxiv.org/abs/math/9803025))
{#Tamarkin}

* [[Clemens Berger]], [[Benoit Fresse]], *Combinatorial operad actions on cochains* ([arXiv:math/0109158](http://arxiv.org/abs/math/0109158))

A review is in

* [[Kathryn Hess]], _Deligne's Hochschild cohomology conjecture_ ([pdf](http://sma.epfl.ch/~hessbell/Pub_DeligneColloq.pdf))
{#Hess}

* Cohen
{#Cohen}

* Gertstenhaber-Voronov
{#GerstenhaberVoronov}

* Hu, Kriz, Voronov
{#HuKrizVoronov}

A transparent [[higher geometry|higher geometric]] interpretation in a suitably dualizable context is indicated in

* [[David Ben-Zvi]], [[John Francis]], [[David Nadler]],
  _[[geometric infinity-function theory|Integral transforms and Drinfeld centers in derived algebraic geometry]]_ ([arXiv:0805.0157](http://arxiv.org/abs/0805.0157))
{#Ben-ZviFrancisNadler}

* [[John Francis]], PhD thesis ([web](http://dspace.mit.edu/handle/1721.1/43792))
{#Francis}

Section 2.5 of

* [[Jacob Lurie]], _[[Ek-Algebras]]_
{#Lurie}

A construction in [[monoidal model categories]] is in

* [[Joachim Kock]], [[Bertrand Toen]], _Simplicial localization of monoidal structures, and a non-linear version of Deligne's conjecture_ Compositio Math. 141 (2005), 253-261 ([arXiv](http://arxiv.org/abs/math.AT/0304442))
{#KockToen}

[[!redirects Deligne conjectures]]


Deligne groupoid

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###Context###
#### $\infty$-Lie theory
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[[!include infinity-Lie theory - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Definition

For $\mathfrak{g}=\bigoplus_{i}\mathfrak{g}^i$ a [[differential graded Lie algebra]], let $MC(\mathfrak{g})$ be the set of [[Maurer-Cartan element]]s, i.e.,

$$
MC(\mathfrak{g})=\{x\in \mathfrak{g}^1 \;|\; dx+\frac{1}{2}[x,x]=0\}
$$

One thinks of elements in this set as _flat $\mathfrak{g}$-[[connection on a bundle|connection]]s: indeed

$$
x\in MC(\mathfrak{g}) \Leftrightarrow (d+[x,-])^2=0.
$$

The subspace $\mathfrak{g}^0$ of $\mathfrak{g}$ is a [[Lie algebra]]; the group

$$
  exp(\mathfrak{g}^0)
$$

acts on as a group of [[gauge transformation]]s on the set of $\mathfrak{g}$-connections (by conjugation), and this action preserves the subset of flat connections. Hence we have a _gauge action_ of  $exp(\mathfrak{g}^0)$ on $MC(\mathfrak{g})$:
$$
e^{a}(d+[x,-])e^{-a}=d+[e^a*x,-].
$$
Explicitly,
$$
e^a*x=x+\sum_{n=0}^\infty \frac{([a,-])^n}{(n+1)!}([a,x]-da)
$$

The _Deligne groupoid_ $Del(\mathfrak{g})$ of the dgla $\mathfrak{g}$ is the [[action groupoid]]

$$
  Del(\mathfrak{g})=MC(\mathfrak{g})// exp(\mathfrak{g}^0)
$$

## Examples

For $\mathfrak{g}$ a [[nLab:Lie algebra|Lie algebra]] this is the [[delooping|delooping groupoid]] $\mathbf{B} exp(\mathfrak{g})=*//exp(\mathfrak{g})$.

## Related concepts

* [[Lie integration]], [[deformation theory]]

## References

Some of the ideas of Deligne on deformation theory were transmitted via

* W. M. Goldman, J. J. Millson, _The deformation theory of representations of fundamental groups of compact Kähler manifolds_, Inst. Hautes Études Sci. Publ. Math. __67__ (1988), 43–96, [MR90b:32041](http://www.ams.org/mathscinet-getitem?mr=972343), [numdam](http://www.numdam.org/item?id=PMIHES_1988__67__43_0)

but the later study of the Deligne 2-groupoid is from a letter of Deligne to Breen from 1994 (see [[Ezra Getzler]]'s webpage; the letter page is not to be linked). See related

* [[E. Getzler]], _A Darboux theorem for Hamiltonian operators in the formal calculus of variations_, Duke Math. J. __111__, n. 3 (2002), 535-560, [MR2003e:32026](http://www.ams.org/mathscinet-getitem?mr=1885831), [doi](http://dx.doi.org/10.1215/S0012-7094-02-11136-3)

Other references

* V. Hinich,  _Descent of Deligne groupoids_, Int. Math. Research Notices,  1997, n. 5, 223-239, [alg-geom/9606010](http://arxiv.org/abs/alg-geom/9606010); _DG coalgebras as formal stacks_, J. Pure Appl. Algebra __162__ (2001), 209-250, [pdf](http://math.haifa.ac.il/hinich/WEB/mypapers/dgc.pdf)

* Amnon Yekutieli, _MC elements in complete DG Lie algebras_, [arXiv/1103.1035](http://arxiv.org/abs/1103.1035)

A careful analysis extends the assignment of the Deligne groupoid to a Maurer-Cartan pseudofunctor, see part 2 of

* Alexander I. Efimov, Valery A. Lunts, [[Dmitri Orlov|Dmitri O. Orlov]], _Deformation theory of objects in homotopy and derived categories_
   * _I: general theory_ [arXiv:math/0702838](http://arxiv.org/abs/math/0702838);
   * _II: pro-representability of the deformation functor_ [arXiv:math/0702839](http://arxiv.org/abs/math/0702839);
   * _III: abelian categories_ [arXiv:math/0702840](http://arxiv.org/abs/math/0702840)

Parts of the above text is taken from

* [[Domenico Fiorenza]], _[[domenicofiorenza:The periods map of a complex projective manifold. An $infty$-category perspective]]_

[[!redirects Deligne groupoids]]


Deligne line bundle

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### Context
#### Differential cohomology
+-- {: .hide}
[[!include differential cohomology - contents]]
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#### Complex geometry
+--{: .hide}
[[!include complex geometry - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

The [[Beilinson-Deligne cup product]] of two [[holomorphic functions]] $f,g \in H^0(X,\mathbb{G}_m)$ is a [[holomorphic line bundle]] with connection, hence a degree-2 [[cocycle]] in [[Deligne cohomology]]

$$
  f \cup g \in H^2(X, \mathbb{Z} \to \Omega^0 \to \Omega^1)
  \,.
$$

This kind of line bundle is often referred to as the _Deligne line bundle_, following ([Deligne 91](#Deligne91)).

They were originally motivated and used in a geometric construction of the [[Beilinson regulator]] $c_{2,2}$, see at _[Beilinson regulator -- Geometic construction](higher+regulator#GeometricConstruction)_.

The universal version of this is the Deligne bundle on $\mathbb{G}_m \times \mathbb{G}_m$ with $f$ and $g$ the [[projection]] onto the first and onto the second factor, respectively (e.g. [Brylinski 00, p. 15](#Brylinski00)).

## References

* {#Deligne91} [[Pierre Deligne]], _Le symbole modéré_. Publ. Math. IHES 73, (1991), 147-181.

Review in the general context of [[Deligne cohomology]] is in

* {#EsnaultViehweg88} [[Hélène Esnault]], [[Eckart Viehweg]], p.5 of _Deligne-Beilinson cohomology_ in Rapoport, Schappacher, Schneider (eds.) _Beilinson's Conjectures on Special Values of L-Functions_ . Perspectives in Math. 4, Academic Press (1988) 43 - 91 ([pdf](http://www.uni-due.de/~mat903/preprints/ec/deligne_beilinson.pdf))

Review in the context of [[holomorphic line 2-bundles]] and [[Chern-Simons line 3-bundles]] is in

* {#Brylinski00} [[Jean-Luc Brylinski]], p. 15 of _Gerbes on complex reductive Lie groups_ ([arXiv:math/0002158](http://arxiv.org/abs/math/0002158))

[[!redirects Deligne line bundles]]


Deligne tensor product of abelian categories

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###Context###
#### Additive and abelian categories
+--{: .hide}
[[!include additive and abelian categories - contents]]
=--
#### Monoidal categories
+--{: .hide}
[[!include monoidal categories - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

The natural [[tensor product]] operation on [[finite abelian category|finite abelian categories]] is known as the _Deligne tensor product_ or _Deligne box product_, introduced in ([Deligne 90](#Deligne)).

For $A$ and $B$ two [[abelian categories]], their Deligne tensor product $A \boxtimes B$ is the abelian category such that for any other abelian category $C$ right [[exact functors]] of the form $A \boxtimes B \to C$ are equivalent to functors $A \times B \to C$ that are right exact in each argument separately.

This tensor product exists for [[finite abelian categories]] but not generally on all [[abelian categories]]. However a slight variant does: instead of abelian categories one can consider categories with [[finite colimits]], see [Franco 12](#Franco12) and ([Chirvasitu&Johnson-Freyd 11, remark 2.2.8](#ChirvasituJohnson-Freyd11)).

## Properties

### In terms of categories of modules and tensor product of algebras

Recall that for every [[finite abelian category]] over $k$, there is a finite-dimensional [[associative algebra|algebra]] $A$ over $k$ and a $k$-linear [[equivalence of categories]]

$$
  \mathcal{C} \simeq A Mod_{fd}
  \,
$$

where the right side consists of $A$-modules which are finite-dimensional as vector spaces over $k$.   $A$ is uniquely determined up to [[Morita equivalence]].

+-- {: .num_prop}
###### Proposition

For $A, B \in Alg_k$ two finite-dimensional [[associative algebras]] over a field $k$, the Deligne tensor product of their categories of finite-dimensional modules is the category of finite-dimensional modules of the [[tensor product of algebras]] $A \otimes_k B$:

$$
  A Mod_{fd}
  \boxtimes
  B Mod_{fd}
  \simeq
  (A \otimes_k B) Mod_{fd}
  \,.
$$

=--

This appears for instance as ([EGNO, prop. 1.46.2](#EGNO)). Without the finiteness constraints and using the tensor product of categories with finite colimits, this appears as ([Chirvasitu&Johnson-Freyd 11, remark 2.2.8](#ChirvasituJohnson-Freyd11)).

## Related concepts

* [[tensor product of presentable (infinity,1)-categories]]

## References

The construction was introduced in

* [[Pierre Deligne]], _Catégories tannakiennes_, The Grothendieck
Festschrift, Vol. II. Progr. Math. 87, 111–195. Birkhäuser
Boston. 1990 (1990)
 {#Deligne}

A survey is in

* [[Ignacio López Franco]], _Tensor products of finitely cococomplete and abelian categories_ (2012) ([pdf](http://www.mat.uc.pt/~workCT/slides/Ignacio.pdf))
 {#Franco12}

Here the author points out that while Deligne's tensor product always exists for finite abelian categories, it does not always exist for general [[abelian categories]].  He argues that in this case it is better to use Kelly's tensor product of finitely cocomplete categories, because it always exists, and it agrees with Deligne's tensor product when the latter exists.

Kelly's tensor product can be found in section 6.5 of

* [[Max Kelly]], _Basic concepts of enriched category theory_, London Math. Soc. Lec. Note Series __64__, Cambridge Univ. Press 1982, 245 pp..  Reprint: TAC reprints 10, [tac](http://www.tac.mta.ca/tac/reprints/articles/10/tr10abs.html),[pdf](http://www.tac.mta.ca/tac/reprints/articles/10/tr10.pdf)

Similar remarks (in the context of [[2-rings]]/[[2-modules]]) are from corollary 2.2.5 on in

* [[Alexandru Chirvasitu]], [[Theo Johnson-Freyd]], _The fundamental pro-groupoid of an affine 2-scheme_, Applied Categorical Structures, Vol 21, Issue 5 (2013), pp. 469–522. ([DOI](http://dx.doi.org/10.1007/s10485-011-9275-y), [arXiv:1105.3104](http://arxiv.org/abs/1105.3104))
 {#ChirvasituJohnson-Freyd11}

In this set of lecture notes

* [[Pavel Etingof]], Shlomo Gelaki, Dmitri Nikshych, [[Victor Ostrik]], _Topics in Lie theory and Tensor categories_, Lecture notes (Spring 2009) ([web](http://ocw.mit.edu/courses/mathematics/18-769-topics-in-lie-theory-tensor-categories-spring-2009/lecture-notes/))
 {#EGNO}

the Deligne tensor product is discussed in [lecture 9](http://ocw.mit.edu/courses/mathematics/18-769-topics-in-lie-theory-tensor-categories-spring-2009/lecture-notes/MIT18_769S09_lec09.pdf).

[[!redirects tensor product of abelian categories]]

[[!redirects Deligne tensor product]]
[[!redirects Deligne tensor products]]


Deligne's theorem

[[Pierre Deligne]] proved a number of theorems. Some that go by his name are

* [[Deligne completeness theorem]]

* [[Deligne's theorem on tensor categories]]

Of course he also proved statements like the [[Weil conjectures]], on [[Tannaka duality]] and such. But this page here is to  disambiguate those statements which people ambiguously refer to as "Deligne's theorem".

[[!redirects Deligne's theorems]]

[[!redirects Deligne theorem]]
[[!redirects Deligne theorems]]

category: disambiguation