(2,1)-quasitopos?
structures in a cohesive (∞,1)-topos
Deligne cohomology – or Deligne-Beilinson cohomology – is an abelian sheaf cohomology that models ordinary differential cohomology, a refinement of the sheaf cohomology with coefficients in a 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
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) in the context of analytic geometry (hence using holomorphic differential forms) as a 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).
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 $n$:
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-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)
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 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
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
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
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? curvature differential form data.
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.
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 in def. 6.
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 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
for its category of sheaves, also called the cohesive topos of smooth spaces.
Instead of the site CartSp of def. 1 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 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.
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 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
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.
For $k \in \mathbb{N}$ write
for the sheaf $\mathbf{\Omega}^k \colon U \mapsto \Omega^k(U)$ of smooth differential k-forms on $X$. The de Rham differential extends to a morphism of sheaves
For positive $k$ its kernel is the sub-sheaf
of closed differential forms; and for $k = 0$ its kernel is the sub-sheaf of constant functions
In the background, what plays a role for the following is the full 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.
Write
for the category of chain complexes in the smooth sheaves of def. 1, 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.
For our purpose the main use of this structure is to compute homotopy fibers via the factorization lemma. Namely
every chain map may be replaced, up to weak equivalence of its domain, by a fibration;
the homotopy fiber of a chain map is the ordinary fiber of any of its fibration replacements.
That the properties in def. 2 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.
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.
For $A \in Ab(Smooth0Type) = Sh(CartSp,Ab)$ any abelian sheaf and for $n \in \mathbb{N}$ we write
for the chain complex of sheaves concentrated on $A$ in degree $n$.
There is a weak equivalence, def. 2,
given by the chain map
(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 extends through this equivalence to produce a morphism denoted $\mathbf{d} log$:
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 1
Under addition of differential forms, the sheaves $\mathbf{\Omega}^k$ of example 2 becomes abelian sheaves, and we will implicitly understand them this way now.
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:
The morphism
given by the canonical chain map
is a weak equivalence in the sense of def. 2.
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 .
(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 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
which is the direct sum of the values of $A_\bullet$ on the given intersections as indicated; and whose differential
is defined componentwise (see at matrix calculus for conventions on maps between direct sums) by
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
The Cech cohomology groups as such are the colimit (“direct limit”) of these groups over refinements of covers
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. 4 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.
The Cech chain complex in def. 4 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
For analyzing the properties of Deligne cohomology below, 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),
$X$ admits a good open cover $\{U_i \to X\}$ (by charts $U_i$ all whose finite non-empty intersections are diffeomorphic to an open ball/Cartesian space $\mathbb{R}^n$);
for any such good open cover the 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);
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
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.
For $A_\bullet = (\mathbf{B}^{n+1}\mathbb{Z})_\bullet$ as in example 3, then for $X$ a smooth manifold
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
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.
Passing to abelian sheaf cohomology (e.g. via def. 4), then prop. 1 is the de Rham theorem.
We will have need to give names to truncations of the de Rham complex. One is this:
For $n \in \mathbb{N}$ write
for the chain complex of the form
with all $n$-forms, not just the closed ones, in degree 0.
The abelian sheaf cohomology of the truncated de Rham complex in def. 5 is $\Omega^n(X)/im(\mathbf{d})$.
For $n \in \mathbb{N}$ the smooth Deligne complex of degree $n$
is the chain complex of abelian sheaves given by
with $U(1)$ in degree $n$ and with the differentials as in def. 2 and example 4.
We write
for its abelian sheaf cohomology.
By example 4 the obvious chain map
is a weak equivalence, def. 2, 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.
In def. 6 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.
For $n \in \mathbb{N}$ the flat smooth Deligne complex of degree $n$
is the chain complex of abelian sheaves given by
with $U(1)$ in degree $n$ and with the differentials as in def. 2 and example 4, and with the closed $n$-forms on the right.
Write
for the chain complex of abelian sheaves given by
with the constant sheaf $\flat U(1)$ of example 1 in degree $n$.
For $(\flat \mathbf{B}^n U(1))_\bullet$ as in def. 8, then the morphism
given by the chain map
(with the vertical morphism on the left being the inclusion of example 4) is a weak equivalence, def. 2.
By the Poincaré lemma, this is just an immediate variant of prop. 1.
The Cech complex, def. 4, for Deligne cohomology of degree $(p+2)$ is the total complex of a double complex of the following form
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
with $deg$ denoting form degree.
A Cech-Deligne cocycle in degree $3$ (“bundle gerbe with connection”) is data $(\{B_{i}\}, \{A_{i j}\}, \{g_{i j k}\})$ such that
The cup product on ordinary cohomology refines to Deligne cohomology.
For more on this see at Beilinson-Deligne cup-product.
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,
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.
Let $(\mathbf{B}^{n+1}\mathbb{Z})_\bullet \in Ch_+(Smooth0Type)$ be as in example 3. Write
for the zig-zag of chain complexes where the left weak equivalence is that of remark 8, i.e. for the chain maps given by
Passing to abelian sheaf cohomology this gives, by def. 6 and example 5, a morphism
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.
Write
for the morphism given by the chain map which is just the de Rham differential in degree 0
Passing to abelian sheaf cohomology this gives a morphism of the form
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
its curvature form.
Consider the zig-zag
out of the complex of def. 5, given by the chain maps
where the bottom quasi-isomorphism is from remark 8.
On passing to abelian sheaf cohomology this gives, by example 7, a morphism
Consider the canonical morphism
Passing to abelian sheaf cohomology this induces a morphism
We call this map the inclusion of the flat infinity-connections into all circle n-connections.
Combining what we have so far:
The composite of the morphisms of def. 11 and of the curvature morphism of def. 10
is given by the de Rham differential $\mathbf{d}$ on differential forms.
The composite of the morphisms of def. 12 and def. 9 is the Bockstein homomorphism:
By composing the defining zig-zags of chain maps the statement is immediate.
While the explicit definition of the Deligne complex in def. 6 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. 4 below.
Write
for the morphism given as the composite
where the second morphism is induced by the canonical inclusion $\mathbb{Z} \hookrightarrow \mathbb{R}$.
Passing to abelian sheaf cohomology this induces a morphism
The morphism in def. 13 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. 2 (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 2 we consider now a fibration resolution of this map
Consider the chain complex
where we use matrix calculus-notation as for mapping cones (see at mapping cone – Examples – In chain complexes).
Write
for the morphism to the chain complex of def. 3 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:
Finally write
for the morphism given by the chain map which in degree $n+1$ is given by
The construction in def. 14 gives a fibration resolution of the Chern character morphism of def. 13 in that it gives a commuting diagram of chain maps
with, on the right, the weak equivalence of prop. 1, where
the left vertical morphism is a weak equivalence;
the bottom horizontal morphism $\widehat{ch}_\bullet$ is a fibration.
That the diagram commutes is a straightforward inspection, unwinding the definitions. That $\widehat{ch}_\bullet$ is a fibration according to def. 2 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. 1.
Write
for the zig-zag whose right morphism is the weak equivalence of prop. 1 and whose left morphism is given by the chain map
The chain maps
$(F_{(-)})_\bullet$, def. 10;
$DD_\bullet$, def. 9;
$\iota_\bullet$, def. 15
$\widehat{ch}_\bullet$, def. 14
fit into a commuting diagram
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 $ch_\bullet$.
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 1 $\widehat{ch}_\bullet$ is a fibration resolution of $ch_\bullet$.
Write
for the inclusion of those closed differential forms whose periods (integration over $(n+1)$-cycles) takes values in the integers.
The image of the curvature map $F_{(-)} \colon H^{n+1}_{conn}(X,\mathbb{Z}) \longrightarrow \Omega^{n+1}_{cl}(X)$ of def. 10 are the integral forms of def. 16.
The homotopy pullback characterization of of prop. 4 implies that the image consists of precisely those closed differential forms which under the de Rham theorem, remark 6, represent real cohomology classes that are in the image of integral cohomology classes. These are the differential forms with integral periods.
(curvature exact sequence)
The Deligne cohomology group fits into a short exact sequence (of abelian groups) of the form
By prop. 5 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. 4. 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
Passing to abelian sheaf cohomology and applying the induced long exact sequence in homology, in view of remark 7, implies the claim.
(characteristic class exact sequence)
The Deligne cohomology group fits into a short exact sequence (of abelian groups) of the form
where $DD$ is the characteristic class map of def. 9.
The chain map that represents the Dixmier-Douady class by def. 9 is manifestly a fibration in the sense of def. 2. Therefore its ordinary fiber is already its homotopy fiber. That ordinary fiber is evidently the domain of the morphism constructed in def. 11, 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 7) goes as
As in the proof of prop. 5 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.
In words the statement of prop. 6 and prop. 7 is that Deligne cohomology groups $H^{n+1}_{conn}(X,\mathbb{Z})$ constitute a group extension
of integral ordinary cohomology by differential $n$-forms modulo integral forms;
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 circle n-connections on circle n-group principal infinity-bundles (either, depending on perspective, by def. 7 or else, more intrinsically, by the very statement of prop. 6) and hence again describes a combination of underlying bundles with differential form data.
Summing up, the homotopy pullback square of prop. 4 together with the maps of prop. 3 form a commuting diagram in $Ch_+(Smooth0Type)$ of the form.
This extends to a diagram in $Ch_+(Smooth0Type)$ of the form
such that
both square are homotopy pullback squares;
both diagonals are homotopy fiber sequences;
the two outer sequences are long homotopy fiber sequences.
For the first statement consider the pasting of homotopy pullback diagrams as in the proof of prop. 6, now extended to the left, via the pasting law, as
That the NE-diagonal is a homotopy fiber sequence is the statement in the proof of prop. 6. That the SE-diagonal is a homotopy fiber sequence follows by inspection as remarked in the proof of prop. 7.
From this the last statement now is implied by using the pasting law yet once more, as show in the proof here.
The form of the exact hexagon characterizing the Deligne complex via prop. 8 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 homotopy theory of smooth ∞-groupoids, via remark 4. This is discussed in detail at differential cohomology hexagon.
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 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 algebraic form (Esnault 89, corollary 1.3).
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
moduli spaces of line n-bundles with connection on $n$-dimensional $X$
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 groupoids
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
This way Deligne cohomology is realized as computing the 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
See
The full proof for $n=1$ this is in
for $n=2$ in
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.
As described in some detail at electromagnetic field in abelian higher gauge theories the background field naturally arises as a Čech–Deligne cocycle, i.e. a Čech cocycle representative with values in the Deligne complex.
Degree 2 Deligne cohomology classifies $U(1)$-principal bundles 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 $U(1)$.
Degree 3 Deligne cohomology classifies bundle gerbes with connection.
Degree 4 Deligne cohomology classifies bundle 2-gerbes 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.
Deligne cohomology was introduced in complex analytic geometry (by a chain complex of holomorphic differential forms) in
with applications to Hodge theory and intermediate Jacobians. The same definition appears in
Barry Mazur, William Messing, Universal extensions and one-dimensional crystalline cohomology, Springer lecture notes 370, 1974
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, MR56:15663
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
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
Jean-Luc Brylinski, section 5 of Loop Spaces, Characteristic Classes and geometric Quantization, Birkhäuser 1993
Ulrich Bunke, section 3 of Differential cohomology (arXiv:1208.3961)
Review with more emphasis on complex analytic geometry and the theory of (Beilinson 85) with more details spelled out is in
Hélène Esnault, Eckart Viehweg, 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)
Hélène Esnault, On the Loday-symbol in the Deligne-Beilinson cohomology, K-theory 3, 1-28, 1989 (pdf)
See also
Chris Peters, Jozef Steenbrink, section 7.2 of Mixed Hodge Structures, Ergebisse der Mathematik (2008) (pdf)
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 in terms of simplicial presheaves and higher stacks includes
Domenico Fiorenza, Urs Schreiber, Jim Stasheff, Cech Cocycles for Differential characteristic Classes, Advances in Theoretical and Mathematical Physics, Volume 16 Issue 1 (2012), pages 149-250 (arXiv:1011.4735)
Domenico Fiorenza, Hisham Sati, Urs Schreiber, Extended higher cup-product Chern-Simons theories, Journal of Geometry and Physics, Volume 74, 2013, Pages 130–163 (arXiv:1207.5449)
Michael Hopkins, Gereon Quick, Hodge filtered complex bordism, arXiv:1212.2173.
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)
Urs Schreiber, differential cohomology in a cohesive topos (arXiv:1310.7930)
See also the references given at differential cohomology hexagon – Deligne coefficients.