geometry of physics -- principal connections

> This entry contains one chapter of _[[geometry of physics]]_.

> previous chapters:  _[[geometry of physics -- flat connections|flat connections]]_, _[[geometry of physics -- de Rham coefficients|de Rham coefficients]]_

> next chapters: _[[geometry of physics -- prequantum geometry|prequantum geometry]]_, _[[geometry of physics -- WZW terms|WZW terms]]_

***

#Contents#
* table of contents
{:toc}

## **Principal connections**
 {#PrincipalConnections}

### Model Layer

#### Motivation

* [[Dirac charge quantization]] implies that the [[electromagnetic field]] is not just given by a closed [[differential 2-form]] but by a refinement of that to a [[circle group]] [[principal connection]]

* analogous arguments show that more generally a configuration of the _[[Yang-Mills field]]_ for some [[gauge group]] $G$ is a $G$-[[principal connection]];

* analogous arguments say that the [[B-field]] and the [[C-field]] are given by [[circle n-bundles with connection]] for $n = 2$ and $n = 3$, respectively.

In this way all of [[gauge theory]] is much the study of [[principal connections]] and all of [[higher gauge theory]] is much the study of [[principal infinity-connections]].

More is true: also a configuration of the [[field (physics)|field]] of [[gravity]] is represented by a connection, in this case a _[[Cartan connection]]_, which is a [[Poincare group]]-[[principal connection]] satisfying an extra condition (that "[[vielbein|solders]]" it to the underlying manifold).

#### Deligne cohomology
 {#DeligneCohomology}

It turns out that one of the many equivalent incarnations of [[circle n-bundles with connection]] are [[cocycles]] in [[Deligne cohomology]]. That this is so we discuss below in the [Semantics layer](#SemanticsLayer). Here instead we discuss [[Deligne cohomology]] taken at face value.

##### 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}.

Recall the following definition, which is discussed in much detail in the chapter _[[geometry of physics -- smooth sets]]_.

+-- {: .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 $\mathbb{R} \in Sooth0Type$.

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).

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]]

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

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

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

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]] ("[[directed 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

$$
  \array{
     \vdots && \vdots
     \\
     \downarrow^{\mathrlap{\partial_A}}
     &&
     \downarrow^{\mathrlap{\partial_A}}
     \\
     \oplus_i A_1(U_i)
     &\stackrel{\delta}{\longrightarrow}&
     \oplus_{i_1, i_2} A_1(U_{i_1, i_2})
     &\stackrel{\delta}{\longrightarrow}&
     \cdots
     \\
     \downarrow^{\mathrlap{\partial_A}}
     &&
     \downarrow^{\mathrlap{\partial_A}}
     \\
     \oplus_i A_0(U_i)
     &\stackrel{\delta}{\longrightarrow}&
     \oplus_{i_1, i_2} A_0(U_{i_1, i_2})
     &\stackrel{\delta}{\longrightarrow}&
     \cdots
  }
$$

=--

+-- {: .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}.

=--

##### The Cech-Deligne complex

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

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

=--

##### 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]]_.

=--

### Semantic Layer
 {#SemanticsLayer}

We give now a general abstract formulation of principal connections in any [[cohesive (∞,1)-topos]] $\mathbf{H}$. The structures discussed above turn out to be models for these axioms.

Throughout, let $\mathbb{G}$ be a [[braided ∞-group]] in the given [[cohesive (∞,1)-topos]], write $\mathbf{B}\mathbb{G}$ for its [[delooping]] and assume that this is precisely [[n-truncated]]. Write $\mathbf{B}^2 \mathbb{G}$ for the second delooping, given by the braidedness.

Write $\flat$ for the [[flat modality]] and recall the [[homotopy fiber sequence]] of its [[counit of a comonad|counit]]

$$
  \mathbf{B}\mathbb{G}
  \stackrel{\theta_{\mathbf{B}\mathbb{G}}}{\longrightarrow}
  \flat_{dR} \mathbf{B}^2 \mathbb{G}
   \stackrel{}{\longrightarrow}
  \flat \mathbf{B}^2\mathbb{G} \longrightarrow \mathbf{B}^2 \mathbb{G}
$$

where $\theta_{\mathbf{B}\mathbb{G}}$ is the _[[Maurer-Cartan form]]_ of the group $\mathbf{B}\mathbb{G}$. This will appear as the [[curvature characteristic form]] for $\mathbb{G}$-[[principal ∞-bundles]] now.

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

A $\mathbb{G}$-[[Hodge filtration]] is a choice of [[filtration]] of $\flat \mathbf{B}^2 \mathbb{G}$ (using the [[flat modality]]) such that the first stage
$F^1 \flat \mathbf{B}^2 \mathbb{G}\longrightarrow F^0 \flat \mathbf{B}^2\mathbb{G}$ is equivalent to the canonical morphism $\flat_{dR} \mathbf{B}\mathbb{G} \longrightarrow \flat \mathbf{B}^2 \mathbb{G}$.

For the $n+1$st stage of the filtration we write

$$
  \Omega^2_{cl}(-,\mathbb{G}) \coloneqq F^{n+1}\flat\mathbf{B}^2 \mathbb{G}
  \,.
$$

=--

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

Consider $\mathbf{H} = $ [[Smooth∞Grpd]] and for $n \gt 1$ consider $\mathbb{G} \coloneqq \mathbf{B}^{n-1} U(1)$ the [[circle n-group]]. Then  (as in prop. \ref{TheDeRhamResolutionOfConstantFunctions})

$$
  \flat \mathbf{B}^2\mathbb{G}
  =
  \flat \mathbf{B}{n+1} U(1)
  \simeq
  L DK
  \left[
    U(1) \stackrel{\mathbf{d}log}{\to} \mathbf{\Omega}^1 \stackrel{\mathbf{d}}{\to}
    \cdots
    \stackrel{\mathbf{d}}{\to}
    \mathbf{\Omega}^{n+1}_{cl}
  \right]
  \,,
$$

where on the right we have the image of the truncated [[de Rham complex]]/flat smooth Deligne complex, def. \ref{FlatSmoothDeligneComplex}, under the [[Dold-Kan correspondence]] $DK$ followed by [[∞-stackification]] $L$.

The traditional [[Hodge filtration]] provides a filtration on this, in the general abstract sense of def. \ref{HodgeFiltration}, whence the terminology.

=--

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

Given a $\mathbb{G}$-[[Hodge filtration]], def. \ref{HodgeFiltration}, write $\mathbf{B}\mathbb{G}_{conn^\bullet}$ for the induced cofiltration given by [[homotopy pullback]] of the [[Maurer-Cartan form]] $\theta_{\mathbf{B}\mathbb{G}} \longrightarrow \flat_{dR}\mathbf{B}^2 \mathbb{G}$ along the Hodge filtration, i.e. the left sequence in the following [[pasting diagram]] of [[homotopy pullbacks]]

$$
  \array{
    \flat \mathbf{B}\mathbb{G} &\longrightarrow& \ast & = & \{0\}
    \\
    \downarrow && \downarrow
    \\
    \mathbf{B}\mathbb{G}_{conn} &\longrightarrow& F^{n+1}\flat \mathbf{B}^2\mathbb{G} & \simeq & \Omega^2_{cl}(-,\mathbb{G})
    \\
    \downarrow && \downarrow
    \\
    \vdots && \vdots
    \\
    \downarrow && \downarrow
    \\
    \mathbf{B}\mathbb{G}_{conn^2} &\longrightarrow& F^3 \flat \mathbf{B}^2 \mathbb{G}
    \\
    \downarrow && \downarrow
    \\
    \mathbf{B}\mathbb{G}_{conn^1} &\longrightarrow& F^2 \flat \mathbf{B}^2 \mathbb{G}
    \\
    \downarrow && \downarrow
    \\
    \mathbf{B}\mathbb{G} &\stackrel{\theta_{\mathbf{B}\mathbb{G}}}{\longrightarrow}& F^1 \flat \mathbf{B}^2 \mathbb{G}
    &\simeq& \flat_{dR} \mathbf{B}^2 \mathbb{G}
  }
$$

We say that $\mathbf{B}\mathbb{G}_{conn}$ here is the coefficient object for $\mathbb{G}$-[[differential cohomology]] (with the given choice of Hodge filtration understood).

=--

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

With the choices as in example \ref{TraditionalHodgeFiltrationAsExampleOfGeneralConcept}, then
the general abstract def. \ref{BraidedDifferentialCohomology} of $\mathbb{G}$-differential coefficient object is represented by the smooth Deligne complex, def. \ref{TheSmoothDeligneComplex}:

$$

  \mathbf{B}^n U(1)_{conn}
  \simeq
  L DK
  (\mathbf{B}^n U(1)_{conn})_\bullet
$$

where $L DK$ denotes the [[Dold-Kan correspondence]] followed by [[∞-stackification]].

=--

### Syntactic Layer

> under construction

#### The dependent curvature type

The universal curvature characteristic, def. \ref{UniversalCurvatureCharacteristic}, has the [[syntax]]

$$
  \vdash curv_{G} \colon \mathbf{B}G \to \flat_{dR} \mathbf{B}^2 G
  \,.
$$

Regarded as a [[dependent type]] in the de Rham coefficient [[context]] this is

$$
    \omega \colon \flat_{dR}\mathbf{B}^2 G
    \;
    \vdash
    \;
    \underset{\mathbf{c} \colon \mathbf{B}G}{\sum}
    \left(
      curv_G\left(\mathbf{c}\right)
      \simeq \omega
    \right)
  \colon
  Type
$$

Therefore the [[syntax]] for a domain object $F \colon X \to \flat_{dR} \mathbf{B}^2 G$ in this [[context]] is

$$
  \omega \colon \flat_{dR} \mathbf{B}^2 G
  \;\vdash\;
  \underset{x \colon X}{\sum}
  \left(
    F_x \simeq \omega
  \right)
  \colon Type
$$

and the [[syntax]] for a [[cocycle]]

$$
  \array{
    X &&\stackrel{\bar P}{\to}&& \mathbf{B}G
    \\
    & {}_{\mathllap{F}}\searrow &\swArrow_{\nabla}& \swarrow_{\mathrlap{curv_G}}
    \\
    && \flat_{dR} \mathbf{B}^2 G
  }
$$

in [[differential cohomology]], def. \ref{GeneralConcreteDifferentialCohomology}, on $(X,F)$ is hence

$$
  \vdash \;
  (\bar P,\nabla)
  \colon
  \underset{\omega \colon \flat_{dR} \mathbf{B}^2 G}{\prod}
  \left(
    \left(
    \underset{x \colon X}{\sum}
    \left(
      F_x \simeq \omega
    \right)
    \right)
    \to
    \left(
    \underset{\mathbf{c} \colon \mathbf{B}G}{\sum}
    \left(
      curv_G(\mathbf{c}) \simeq \omega
    \right)
    \right)
  \right)
$$

#### Fixed curvature twists

$$
  \begin{aligned}
    (\mathbf{B}^n \mathbb{G} \colon Type)_{conn} \colon & Type
    \\
     \coloneqq &
     \sum_{\mathbf{c} \colon \mathbf{B}\mathbb{G}}
     \sum_{\omega \colon \Omega^{n+1}_{cl}}
     \left(
       curv(\mathbf{c}) = \omega
     \right)
  \end{aligned}
$$


geometry of physics -- quantum mechanics

> This is a sub-entry of _[[geometry of physics]]_.

***

#Contents#
* table of contents
{:toc}

So far we have discussed [[extended prequantum field theory]]: [[Lagrangians]] and their induced [[action functionals]] and [[prequantum n-bundles]]. Now we turn to actual [[quantum field theory]]. A [[prequantum field theory]] is supposed to induce a [[quantum field theory]] under the last step of [[higher geometric quantization]]: a choice of [[polarization]] (or equivalent) and the passage to the corresponding [[space of states]] of polarized [[sections]] of the [[prequantum n-bundles]]. This step that connects [[prequantum field theory]] with [[quantum field theory]] we discuss below in _[Geometric Quantization](#GeometricQuantization)_.

Here we discuss the structure of the _outcome_ of this process.

## **Quantum mechanics**
 {#QuantumMechanics}

### Model Layer

* [FQFT -- Exposition and Introduction -- Quantum mechanics in Schrödinger picture](FQFT#QuantumMechanicsInSchroedingerPicture)

* [[quantum mechanics]]

#### Worldvolumes and cobordisms

* [[cobordism]]

#### Spectral triple

* [[spectral triple]]

### Semantic Layer

#### Internal categories in an $\infty$-topos

##### Simplicial objects in an $\infty$-topos

* [[simplicial object in an (∞,1)-category]]

##### Category objects in an $\infty$-topos

* [[Segal condition]]

* [[internal category in an (∞,1)-category]]

### Syntactic Layer

#### Directed homotopy type theory

* [[directed homotopy type theory]]


geometry of physics -- representations and associated bundles

> This entry contains one chapter of _[[geometry of physics]]_. See there for background and context.

> previous chapters: _[[geometry of physics -- groups|groups]]_, _[[geometry of physics|principal bundles]]_

> next chapter: _[[geometry of physics -- modules|modules]]_

***

#Contents#
* table of contents
{:toc}

## Representations and associated bundles

The mathematical term _group_ is short for _group of [[symmetries]]_, namely of symmetries of some object. That a group $G$ is the group of symmetries of some $V$ is technically expressed by there being an [[action]] of $G$ on $X$. Generally, or at least if $V$ and this representation are suitably linear, this is also called a [[representation]] of $G$ (namely a representation of the abstract group as an actual group of symmetries).

The chapter _[[geometry of physics -- groups]]_ discusses in detail how in geometric [[homotopy theory]] groups $G$ are equivalent to [[groupoids]] $\mathbf{B}G$ which have a single object and $G$ as the [[automorphisms]] of that object, if only $\mathbf{B}G$ is regarded as a [[pointed object]] via the canonical base point inclusion $\ast \to \mathbf{B}G$.

This perspective turns out to be exceedingly useful for the discussion of the representations of $G$: these turn out to be equivalent simply to $\mathbf{B}G$-[[dependent types]], hence to any [[bundles]] $E \to \mathbf{B}G$ over $G$. Given such, then the object that $V$ that this encodes an action on is the [[homotopy fiber]] of this map

$$
  \array{
     V &\longrightarrow& E
     \\
     && \downarrow
     \\
     && \mathbf{B}G
  }
  \,.
$$

In traditional literature this is familiar in special cases, where the perspective is usually the opposite: given an [[action]] of $G$ on $V$, then there is the [[associated bundle]] $E = \mathbf{E}G \times_G V$ which is associated to the $G$-[[universal principal bundle]] via the action.

Indeed, in the generality of geometric homotopy theory, this association is an equivalence, so that actions and universal associated bundles are essentially the same concept.

### Model Layer

#### 1-Representations of 1-Groups
 {#1RepresentationsOf1Groups}

We discuss here ordinary [[groups]] (i.e. [[infinity-groups]] which are just 1-groups), and their ordinary [[actions]] and ordinary [[associated bundles]]. Even that ordinary case profits from its formulation via [[action groupoids]], but its key advantage is that this formulation seamlessly generalizes.

#### Actions
 {#ActionsOf1Groups}

We discuss here traditional concept of [[discrete groups]] [[action|acting]] on a [[sets]] ("[[permutation representations]]") but phrased in terms of [[action groupoids]] [[slice (infinity,1)-category|sliced]] over [[delooping]] groupoids. The discussion immediately, and essentially verbatim, generalizes to pre-smooth groupoids and to [[smooth groupoids]] proper.

Write [[Grpd]] for the [[(2,1)-category]] of [[groupoids]], the [[full sub-(infinity,1)-category]] of [[∞Grpd]] on the [[1-truncated objects]].

We write

$$
  X_\bullet = (X_1 \stackrel{\longrightarrow}{\longrightarrow} X_0)
$$

for a [[groupoid object]] given by an explicit choice of set of objects and of morphisms and then write $X \in Grpd$ for the object that this presents in the $(2,1)$-category. Given any such $X$, we recover a presentation by choosing any [[essentially surjective functor]] $S \to X$  (an [[atlas]]) out of a set $S$ (regarded as a groupoid) and setting

$$
  X_\bullet = (S \underset{X}{\times} S \stackrel{\longrightarrow}{\longrightarrow} S)
$$

hence taking $S$ as the set of objects and the [[homotopy fiber product]] of $S$ with itself over $X$ as the set of morphism.

For $G$ a [[discrete group]], then $\mathbf{B}G$ denotes the [[groupoid]] presented by $(\mathbf{B}G)_\bullet = (G \stackrel{\longrightarrow}{\longrightarrow}\ast)$ with [[composition]] operation given by the product in the group. Of the two possible ways of making this identification, we agree to use

$$
  \array{
    && \ast
    \\
    & {}^{\mathllap{g_1}}\nearrow && \searrow^{\mathrlap{g_2}}
    \\
    \ast && \underset{g_1 \cdot g_2}{\longrightarrow} && \ast
  }
  \,.
$$

+-- {: .num_defn #Action1Groupoid}
###### Definition

Given a [[discrete group]] $G$ and an [[action]] $\rho$ of $G$ on a [[set]] $S$

$$
  \rho \colon S \times G \longrightarrow S
$$

then the corresponding _[[action groupoid]]_ is

$$
  (S//G)_\bullet
  \coloneqq
  \left(
    S\times G
    \stackrel{\overset{p_1}{\longrightarrow}}{\underset{\rho}{\longrightarrow}}
    S
  \right)
$$

with [[composition]] given by the product in $G$. Hence the [[objects]] of $S$ are the elements of $S$, and the morphisms $s \stackrel{}{\longrightarrow } t$ are labeled by elements $g\in G$ and are such that $t = \rho(s)(g)$.

=--

Schematically:

$$
  (S//G)_\bullet =
  \left\{
    \array{
      && \rho(s)(g)
      \\
      & {}^{\mathllap{g_1}}\nearrow && \searrow^{\mathrlap{g_2}}
      \\
      s && \underset{g_1 g_2}{\longrightarrow} && \rho(s)(g_1 g_2)
    }
  \right\}
  \,.
$$

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

For the unique and trivial $G$-action on the singleton set $\ast$, we have

$$
  \ast//G \simeq \mathbf{B}G
  \,.
$$

=--

This makes it clear that:

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

In the situation of def. \ref{Action1Groupoid},
there is a canonical morphism of groupoids

$$
  (p_\rho)_\bullet \;\colon\; (S//G)_\bullet \longrightarrow (\mathbf{B}G)_\bullet
$$

which, in the above presentation, forgets the labels of the objects and is the identity on the labels of the morphisms.

This morphism is an [[isofibration]].

=--

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

For $G$ a [[discrete group]], given two $G$-[[actions]] $\rho_1$ and $\rho_2$ on sets $S_1$ and $S_2$, respectively, then there is a [[natural equivalence]] between the set of action [[homomorphisms]] ("[[intertwiners]]") $\rho_1 \to \rho_2$, regarded as a groupoid with only identity morphisms, and the [[hom groupoid]] of the [[slice (infinity,1)-category|slice]] $Grpd_{/\mathbf{B}G}$ between their [[action groupoids]] regarded in the slice via the maps from prop. \ref{MapFromActionGroupoidOnSetBackToBG}

$$
  G Act(\rho_1,\rho_2)
  \simeq
  Grpd_{/\mathbf{B}G}(p_{\rho_1}, p_{\rho_2})
  \,.
$$

=--

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

One quick way to see this is to use, via the discussion at _[[slice (infinity,1)-category]]_, that the [[hom-groupoid]] in the slice is given by the [[homotopy pullback]] of unsliced hom-groupoids

$$
  \array{
    Grpd_{/\mathbf{B}G}(p_{\rho_1}, p_{\rho_2})
    &\longrightarrow&
    Grpd(S_1//G, S_2//G)
    \\
    \downarrow &(pb)& \downarrow^{\mathrlap{Grpd(S_1//G,p_{\rho_2})}}
    \\
    \ast &\stackrel{}{\longrightarrow}& Grpd(S_1//G, \mathbf{B}G)
  }
  \,.
$$

Now since $(p_{\rho_2})_\bullet$ is an [[isofibration]], so is $Grpd((S_1//G)_\bullet, (p_{\rho_2})_\bullet)$, and hence this is computed as an ordinary pullback (in the above presentation). That in turn gives the [[hom-set]] in the 1-categorical slice. This consists of functors

$$
  \phi_\bullet \colon (S_1//G)_\bullet \longrightarrow (S_1//G)_\bullet
$$

which strictly preserves the $G$-labels on the morphisms. These are manifestly the intertwiners.

$$
  \phi_\bullet
  \;\colon\;
  \left(
  \array{
    s
    \\
    \downarrow^{\mathrlap{g}}
    \\
    \rho(s)(g)
  }
  \right)
  \mapsto
  \left(
  \array{
    \phi(s)
    \\
    \downarrow^{\mathrlap{g}}
    \\
    \phi(\rho(s)(g)) & = \rho(\phi(s))(g)
  }
  \right)
  \,.
$$

=--

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

The [[homotopy fiber]] of the morphism in prop. \ref{MapFromActionGroupoidOnSetBackToBG} is [[equivalence of groupoids|equivalent]] to the set $S$, regarded as a groupoid with only identity morphisms, hence we have a [[homotopy fiber sequence]] of the form

$$
  \array{
    S &\longrightarrow& S//G
    \\
    && \downarrow^{\mathrlap{p_\rho}}
    \\
    && \mathbf{B}G
  }
  \,.
$$

=--

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

In the presentation $(S//G)_\bullet$ of def. \ref{Action1Groupoid}, $p_\rho$ is an [[isofibration]], prop. \ref{MapFromActionGroupoidOnSetBackToBG}. Hence the [[homotopy fibers]] of $p_\rho$ are equivalent to the ordinary fibers of $(p_\rho)_\bullet$ computed in the 1-category of 1-groupoids. Since $(p_\rho)_\bullet$ is the identity on the labels of the morphisms in this presentation, this ordinary fiber is precisely the sub-groupoid of $(S//G)_\bullet$ consisting of only the identity morphismss, hence is the set $S$ regarded as a groupoid.

=--

Conversely, the following construction extract a group action from a homotopy fiber sequence of groupoids of this form.

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

Given a [[homotopy fiber sequence]] of [[groupoids]] of the form

$$
  \array{
    S &\stackrel{i}{\longrightarrow}& E
    \\
    && \downarrow^{\mathrlap{p}}
    \\
    && \mathbf{B}G
  }
$$

such that $S$ is [[equivalence of groupoids|equivalent]] to a [[set]] $S$, define a $G$-[[action]] on this set as follows.

Consider the [[homotopy fiber product]]

$$
  S \underset{E}{\times} S
  \stackrel{\overset{}{\longrightarrow}}{\underset{}{\longrightarrow}}
  S
$$

of $i$ with itself. By the [[pasting law]] applied to the total homotopy pullback diagram

$$
  \array{
    S \underset{E}{\times} S &\longrightarrow& S
    \\
    \downarrow && \downarrow^{\mathrlap{i}}
    \\
    S &\stackrel{i}{\longrightarrow}& E
    \\
    \downarrow && \downarrow^{\mathrlap{p}}
    \\
    \ast &\longrightarrow& \mathbf{B}G
  }
  \;\;\;\;
  \simeq
  \;\;\;\;
  \array{
    S\times G &\stackrel{p_1}{\longrightarrow}& S
    \\
    \downarrow && \downarrow
    \\
    G &\stackrel{}{\longrightarrow}& \ast
    \\
    \downarrow && \downarrow
    \\
    \ast &\longrightarrow& \mathbf{B}G
  }
$$

there is a canonical [[equivalence of groupoids]]

$$
  S \underset{E}{\times} S \simeq S \times G
$$

such that one of the two canonical maps from the fiber product to $S$ is projection on the first factor. The _other_ map under this equivalence we denote by $\rho$:

$$
  S \times G
  \stackrel{\overset{p_1}{\longrightarrow}}{\underset{\rho}{\longrightarrow}}
  S
  \,.
$$

=--

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

The functor $i \colon S \to E$ is clerly [[essentially surjective functor|essentially surjective]] (every connected component of $E$ has a homotopy fiber under its map to $\mathbf{B}G$). This implies that $E$ is presented by

$$
  E_\bullet \coloneqq (S \underset{E}{\times}S \stackrel{\overset{p_1}{\longrightarrow}}{\underset{p_2}{\longrightarrow}} S)
$$

and hence, via the construction in def. \ref{ActionMapFromFiberSequenceSetToGroupoidToBG}, by

$$
  E_\bullet \simeq (S \times G \stackrel{\overset{p_1}{\longrightarrow}}{\underset{\rho}{\longrightarrow}} S)
  \,.
$$

=--

But this already exhibits $E$ as an [[action groupoid]], in particular it mans that $\rho$ is really an [[action]]:

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

The morphism $\rho$ constructed in def. \ref{ActionMapFromFiberSequenceSetToGroupoidToBG} is a $G$-[[action]] in that it satisfies the action propery, which says that the [[diagram]] (of [[sets]])

$$
  \array{
    S\times G \times G &\stackrel{(id,(-)\cdot(-))}{\longrightarrow}& S \times G
    \\
    \downarrow^{\mathrlap{(\rho,id)}} && \downarrow^{\mathrlap{\rho}}
    \\
    S \times G &\stackrel{\rho}{\longrightarrow}& S
  }
$$

[[commuting diagram|commutes]].

=--

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

For $G$ a [[discrete group]], there is an [[equivalence of categories]]

$$
  G Act(Set)
  \stackrel{\simeq}{\longrightarrow}
  (Grpd_{/\mathbf{BG}})_{\leq 0}
$$

between the category of [[permutation representations]] of $G$ and the full subcategory of the [[slice (infinity,1)-category|slice (2,1)-category]] of [[Grpd]] over $\mathbf{B}G$ on the [[0-truncated objects]].

This equivalence takes an action to its [[action groupoid]].

=--

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

By remark \ref{ActionGroupoidFromFiberSequence} the construction of action groupoids is [[essentially surjective functor|essentially surjective]].
By prop. \ref{IntertwinersOfPermutationActionAsSliceHoms} it is [[fully faithful functor|fully faithful]].

=--

##### Examples of actions

One remarkable consequence of prop. \ref{EquivalenceOfPermutationRepresentationsWithActionGroupodsInSlice} is that it says that categories of actions are [[slice (infinity,1)-category|slices]] of [[(2,1)-toposes]], hence are [[slice (infinity,1)-topos|slice (2,1)-toposes]] hence in particular are themselves [[(2,1)-topos]]. In particular there is an [[internal hom]] of actions. This is the [[conjugation action]] construction.

+-- {: .num_defn #ConjugationActionForDiscrete1Groups}
###### Definition

Given a [[discrete group]] $G$ and two $G$-actions $\rho_1$ and $\rho_2$ on [[sets]] $S_1$ and $S_2$, respectively, then the [[function set]] $[S_1, S_2]$ is naturally equipped with the [[conjugation action]]

$$
  Ad \;\colon \;  [S_1, S_2] \times G \longrightarrow [S_1,S_2]
$$

which takes $((S_1 \stackrel{f}{\to} S_2), g)$ to

$$
  \rho_2(-)(g)\circ f \circ  \rho_1(-)(g^{-1})
  \;\colon\;
  S_1 \stackrel{\rho_1(-)(g^{-1})}{\longrightarrow} S_1 \stackrel{f}{\longrightarrow} S_2\stackrel{\rho_2(-)(g)}{\longrightarrow} S_2
  \,.
$$

=--

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

The [[conjugation action]] construction of def. \ref{ConjugationActionForDiscrete1Groups} is the [[internal hom]] in the [[category]] of actions.

=--

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

We need to show that for any three [[permutation representations]],  [[functions]]

$$
  \phi \;\colon\; S_3 \longrightarrow [S_1,S_2]
$$

which [[intertwiner|intertwine]] the $G$-action on $S_3$ with the conjugation action on $[S_1,S_2]$ are in [[natural bijection]] with functions

$$
  \tilde \phi \;\colon\; S_3 \times S_1 \longrightarrow S_2
$$

which intertwine the diagonal action on the [[Cartesian product]] $S_3 \times S_1$ with the action on $S_2$.

The condition on $\phi$ means that for all $g\in G$ and $s_3 \in S_3$ it sends

$$
  \phi
   \;\colon\;
  \rho_3(s_3)(g)
  \mapsto
  \left(
    s_1 \mapsto
    \rho_2\left(
     \phi\left(s_3\right)\left( \rho_1\left(s_1\right)\left(g^{-1}\right) \right)\right)\left(g\right)
  \right)
 \,.
$$

This is equivalently a function $\tilde \phi$ of two variables which sends

$$
  \tilde \phi
  \;\colon\;
  (\rho_3(s_3)(g), s_1)
  \mapsto
  \rho_2
  (
     \phi(s_3)(
       \rho_1(s_1)(g^{-1})
     )
  )(g)
 \,.
$$

Since this has to hold for all values of the variables, it has to hold when substituing $s_1$ with $\rho_1(s_1)(g)$. After this substitution the above becomes

$$
  \tilde \phi
  \;\colon\;
  (\rho_3(s_3)(g),
  \rho_1(s_1)(g))
  \mapsto
   \rho_2(\phi(s_3)(s_1 ))(g)
  \,.
$$

This is the intertwining condition on $\tilde \phi$. Conversely, given $\tilde \phi$ satisfying this for all values of the variables, then running the argument backwards shows that its hom-[[adjunct]] $\phi$ satisfies its required intertwining condition.

=--

The following is immediate but conceptually important:

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

The [[invariants]] of the conjugation action on $[S_1,S_2]$ is the set of action [[homomorphisms]]/[[intertwiners]].

=--

Hence the inclusion of invariants into the conjugation action gives the inclusion of the external [[hom set]] of the category of $G$-actions into the set underlying the [[internal hom]]

$$
  G Act(\rho_1,\rho_2)\hookrightarrow [\rho_1,\rho_2]
  \,.
$$

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

Given any $X$ with its canonical action of its [[automorphism group]] $Aut(X)$, regard any $Y$ as equipped with the trivial $Aut(Y)$-action.

Then the [[conjugation action]], def. \ref{ConjugationActionForDiscrete1Groups}, on $[X,Y]$ is the action by precomposition with automorphisms of $X$.

=--

#### Associated bundles

At _[[geometry of physics -- principal bundles]]_ in the section _[Smooth principal bundles via smooth groupoids](geometry%20of%20physics%20--%20principal%20bundles#PrincipalBundlesViaSmoothGroupoids)_ is discussed how smooth [[principal bundles]] for a [[Lie group]] $G$ over a [[smooth manifold]] $X$ are equivalently the [[homotopy fibers]] of morphisms of [[smooth groupoids]] ([[smooth stacks]]) of the form

$$
  X \stackrel{}{\longrightarrow} \mathbf{B}G
  \,.
$$

Now given an [[action]] $\rho$ of $G$ on some [[smooth manifold]] $V$, and regardiing this action via its [[action groupoid]] projection $p_\rho \colon V//G \to \mathbf{B}G$ as discussed [above](#ActionsOf1Groups), then we may consider these two morphisms into $\mathbf{B}G$ jointly

$$
  \array{
    && V//G
    \\
    && \downarrow^{\mathrlap{p_\rho}}
    \\
    X &\stackrel{g}{\longrightarrow}& \mathbf{B}G
  }
$$

and so it is natural to construct their [[homotopy fiber product]].

We now discuss that this is equivalently the [[associated bundle]] which is associated to the principal bundle $P \to X$ via the action $\rho$.

+-- {: .num_prop #Associated1BundleAsPullbackOfActionGroupoid}
###### Proposition

For $G$ a [[smooth group]] (e.g. a [[Lie group]]), $X$ a [[smooth manifold]], $P \to X$ a smooth $G$-[[principal bundle]] over $X$ and $\rho$ a smooth [[action]] of $G$ on some [[smooth manifold]] $V$, then the [[associated bundle|associated]] $V$-[[fiber bundle]] $P \times_G V\to X$ is equivalently (regarded as a [[smooth groupoid]]) the [[homotopy pullback]] of the [[action groupoid]]-projection $p_\rho \colon V//G \to \mathbf{B}G$ along a morphism $g \colon X\to\mathgbf{B}G$ which [[modulating morphism|modulates]] $P$

$$
  \array{
    P\times_G V &\longrightarrow& V//G
    \\
    \downarrow && \downarrow^{\mathrlap{p_\rho}}
    \\
    X &\stackrel{g}{\longrightarrow}& \mathbf{B}G
  }
  \,.
$$

=--

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

By the discussion at _[[geometry of physics -- principal bundles]]_ in the section _[Smooth principal bundles via smooth groupoids](geometry%20of%20physics%20--%20principal%20bundles#PrincipalBundlesViaSmoothGroupoids)_, the morphism $g$ of smooth groupoids is presented by a morphism of pre-smooth groupoids after choosing an [[open cover]] $\{U_i \to X\}$ over wich $P$ trivialize and choosing a trivialization, by the [[zig-zag]]

$$
  \array{
    C(\{U_i\})_\bullet &\stackrel{g_\bullet}{\longrightarrow}& (\mathbf{B}G)_\bullet
    \\
    \downarrow^{\mathrlap{\simeq_{lwe}}}
    \\
    X
  }
$$

where the top morphism is the [[Cech cohomology|Cech cocycle]] of the given local trivialization regarded as a morphism out of the [[Cech groupoid]] of the given cover.

Moreover, by prop. \ref{MapFromActionGroupoidOnSetBackToBG} the morphism $(p_\rho)_\bullet$ is a global fibration of pre-smooth groupoids, hence, by the discussion at _[[geometry of physics -- smooth homotopy types]]_, the homotopy pullback in question is equivalently computed as the ordinary pullback of pre-smooth groupoids of $(p_\rho)_\bullet$ along this $g_\bullet$

$$
  \array{
    C(\{U_i\})_\bullet \underset{(\mathbf{B}G)_\bullet}{\times} (V//G)_\bullet &\longrightarrow& (V//G)_\bullet
    \\
    \downarrow && \downarrow^{\mathrlap{(p_\rho)_\bullet}}
    \\
    C(\{U_i\})_\bullet &\stackrel{g_\bullet}{\longrightarrow}&
    (\mathbf{B}G)_\bullet
    \\
    \downarrow^{\mathrlap{\simeq_{lwe}}}
    \\
    X
  }
  \,.
$$

This pullback is computed componentwise. Hence it is the pre-smooth groupoid whose morphisms are pairs consisting of a morphism $(x,i)\to (x,j)$ in the Cech groupoid as well as a morphism $s \stackrel{g}{\to} \rho(s)(g)$ in the action groupoid, such that the group label $g$ of the latter equals the cocycle $g_{i j}(x)$ of the cocycle on the former. Schematically:

$$
  C(\{U_i\})_\bullet \underset{(\mathbf{B}G)_\bullet}{\times} (V//G)_\bullet
  =
  \left\{
     ((x,i),s) \stackrel{g_{i j}(x)}{\longrightarrow} ((x,j),\rho(s)(g))
  \right\}
  \,.
$$

This means that the pullback groupoid has at most one morphism between every ordered pair of objects. Accordingly this groupoid is [[equivalence of groupoids]] equivalent to the [[quotient]] of its space of objects by the [[equivalence relation]] induced by its morphisms:

$$
  \cdots \simeq
  \left(
    \underset{i}{\coprod} U_i \times V
  \right)/_\sim
  \,.
$$

This is a traditional description of the [[associated bundle]] in question.

=--

#### Invariants and sections

One advantage of the perspective on representations via action groupoids is that it gives a good formulation of the [[invariants]] and the [[coinvariants]] of actions. The invariants are the _[[sections]]_ of the action groupoid projection, while the coivariants in fact are the action groupoid itself.

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

For $G$ a [[discrete group]], $\rho$ a $G$-[[action]] on some set $S$, then the set of [[invariants]] of that action is equivalent to the groupoid of [[sections]] of the [[action groupoid]] projection of prop. \ref{MapFromActionGroupoidOnSetBackToBG}, corresponding to the action via prop. \ref{EquivalenceOfPermutationRepresentationsWithActionGroupodsInSlice}.

=--

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

The sections in question are diagrams in [[Grpd]] of the form

$$
  \array{
    \mathbf{B}G && \stackrel{\sigma}{\longrightarrow} && S//G
    \\
    & {}_{\mathllap{id}}\searrow
    &\swArrow_{\mathrlap{\simeq}}&
    \swarrow_{\mathrlap{p_\phi}}
    \\
    && \mathbf{B}G
  }
  \,,
$$

hence the groupoid which they form is equivalently the [[hom-groupoid]]

$$
  Grpd_{/\mathbf{B}G}(id_{\mathbf{B}G}, p_\rho)
  \in Grpd
$$

in the [[slice (infinity,1)-category|slice]] of [[Grpd]] over $\mathbf{B}G$.
As in the proof of prop. \ref{IntertwinersOfPermutationActionAsSliceHoms}, with the fibrant presentation $(p_\rho)_\bullet$ of prop. \ref{MapFromActionGroupoidOnSetBackToBG}, this is equivalently given by strictly commuting diagrams of the form

$$
  \array{
    (\mathbf{B}G)_\bullet && \stackrel{\sigma_\bullet}{\longrightarrow} && (S//G)_\bullet
    \\
    & {}_{\mathllap{id_\bullet}}\searrow &=& \swarrow_{\mathrlap{(p_\phi)_\bullet}}
    \\
    && (\mathbf{B}G)_\bullet
  }
  \,.
$$

These $\sigma$ now are manifestly functors that are the identiy on the group labels of the morphisms

$$
  \sigma_\bullet
  \;\colon\;
  \left(
  \array{
    \ast
    \\
    \downarrow^{\mathrlap{g}}
    \\
    \ast
  }
  \right)
  \;\;
  \mapsto
  \;\;
  \left(
  \array{
    \sigma(\ast)
    \\
    \downarrow^{\mathrlap{g}}
    \\
    \sigma(\ast) & = \rho(\sigma(\ast)(g))
  }
  \right)
  \,.
$$

This shows that they pick precisely those elements $\sigma(\ast) \in S$ which are fixed by the $G$-action $\rho$.

Moreover, since these functors are identity on the group labels, there are no non-trivial [[natural isomorphisms]] between them, and hence the groupoid of sections is indeed a set, the set of invariant elements.

=--

More generally, we may consider sections of these groupoid projections after pulling them back along some cocycle:

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

Given an [[associated bundle]] $P \times_G V\to X$ [[modulating morphism|modulated]], as in prop. \ref{Associated1BundleAsPullbackOfActionGroupoid}, by a morphism of [[smooth groupoids]] of the form $g \colon X \longrightarrow \mathbf{B}G$, then its set of [[sections]] is equivalently the groupoid of diagrams

$$
  \array{
    X && \stackrel{\sigma}{\longrightarrow} && S//G
    \\
    & {}_{\mathllap{g}}\searrow
    &\swArrow_{\mathrlap{\simeq}}& \swarrow_{\mathrlap{p_\phi}}
    \\
    && \mathbf{B}G
  }
  \,,
$$

hence the groupoid of sections is equivalently the slice [[hom-groupoid]]

$$
  \Gamma_X(P\times_G V)
  \simeq
  Grpd_{/\mathbf{B}G}(g, p_\rho)
  \,.
$$

=--

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

By the defining [[universal property]] of the [[homotopy pullback]] in prop. \ref{Associated1BundleAsPullbackOfActionGroupoid}.

=--

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

Taken together this means that [[invariants]] of group actions are equivalently the sections of the corresponding [[universal principal bundle|universal]] [[associated bundle]].

=--

#### Stabilizer groups

Given an ordinary [[discrete group]] $G$ and an [[action]] $\rho \colob G \times V \longrightarrow V$ of the group on some [[set]] $V$, then for $x \in V$ any element, the [[stabilizer group]]

$$
  Stab_x(V) \hookrightarrow G
$$

is the [[subgroup]] of $G$ consisting of those group elements under whose action the element $v$ does not change (is "stable"):

$$
  Stab_G(x)
  =
  \left\{
    g \in G
    \;|\;
    \rho(g)(v) = v
  \right\}
  \,.
$$

We observe that this has the following intrinsic reformulation.  Write again

$$
  \array{
     V &\stackrel{i}{\longrightarrow}& V//G
     \\
     && \downarrow^{\mathrlap{p_\rho}}
     \\
     && \mathbf{B}G
  }
$$

for the [[action groupoid]] [[homotopy fiber sequence]] that corresponds to $\rho$
via prop. \ref{EquivalenceOfPermutationRepresentationsWithActionGroupodsInSlice}.

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

The stabilizer group of $x$ under $G$ acting on $V$ is equivalently the [[looping]] of the [[action groupoid]] at the point $i(x)$:

$$
  Stab_G(x)
  \simeq
  \Omega_{i(x)} (V//G)
$$

=--

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

Since $V//G$ is a [[1-groupoid]], the [[loop space object]] $\Omega_{i(x)} (V//G)$ is simply the [[automorphism group]] of $x$ regarded as an object in the action groupoid, hence the group of [[morphisms]] in $(V//G)_\bullet$ from $x$ to $x$. Comparison with def. \ref{Action1Groupoid} shows that these morphisms are precisely those labeled by elements $g \in G$ for which $\rho(g)(x) = x$.

=--

#### $\infty$-Representations of 1-groups

The [above](#1RepresentationsOf1Groups) perspective on ordinary representations of ordinary groups on sets via their [[action groupoid]] projection has the advantage that it immediately generalizes to a definition where 1-groups act on more general [[homotopy types]] up to [[coherence|coherent]] [[homotopy]], hence to _[[infinity-representations]]_ or _[[infinity-actions]]_.

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

Given a [[discrete group]] $G$ and a [[Kan complex]] $V_\bullet$, then an _[[infinity-representation]]_ or _[[infinity-action]]_ of $G$ on $V$ is another [[Kan complex]], to be denoted $(V//G)_\bullet$, equipped with a simplicial map $(p_\rho) \colon (V//G)_\bullet \longrightarrow N(\mathbf{B}G)_\bullet$ to the [[nerve]] of $(\mathbf{B}G_\bullet)$, such that the [[homotopy fiber]] of that map is [[weak homotopy equivalence|weakly homotopy equivalent]] to $V_\bullet$.

=--

##### Examples of $\infty$-Representations

Given an [[abelian group]] $A$ and $n \in \mathbb{N}$,
write $(\mathbf{B}^n A)_\bullet$ for the [[Kan complex]] which is the image under the [[Dold-Kan correspondence]] of the [[chain complex]] that is concentrated on $A$ in degree $n$.

Then for $G$ a [[discrete group]], the [[mapping complex]]

$$
  [\mathbf{B}G,\mathbf{B}^n A] \in KanCplx
$$

is the [[infinity-groupoid]] whose [[objects]] are the degree-$n$ [[group cohomology|group cocycles]] on $G$ with [[coefficients]] in $A$ (regarded as a $G$-[[module]] with trivial [[action]]), whose [[morphisms]] are the [[coboundaries]] between these cocycles, and whose higher morphisms are higher order coboundaries-of-coboundaries.

Being a [[mapping space]], this naturally carries a precomposition action by the [[automorphism infinity-group]] of $\mathbf{B}G$, which is also known as the [[automorphism 2-group]] of $G$. Restricting this to _pointed_ automorphisms is is the 1-group $Aut_{Grp}(G)$ of invertible group homomorphisms of $G$.

### Semantic Layer

We now consider any [[(∞,1)-topos]] $\mathbf{H}$ and formulate the group actions and their associated bundles in [[general abstract|general abstractly]] here.

By the discussion at _[[geometry of physics -- groups]]_, [[group objects in an (∞,1)-category]] are equivalently pointed connected objects which we write $(\ast\to\mathbf{B}G)\in \mathbf{H}^{\ast/}$, the group itself being the [[loop space object]]

$$
  G \simeq \Omega \mathbf{B}G
$$

formed at the given base point.

#### $\infty$-Actions

In view of this, the characterization of ordinary group action according to prop. \ref{EquivalenceOfPermutationRepresentationsWithActionGroupodsInSlice}
has an immediate generalization to [[∞-groups]] in any [[(∞,1)-topos]]

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

Given an [[∞-group]] $G \in \mathbf{H}$ and any object $V \in \mathbf{H}$, then an _[[∞-action]]_ $\rho$ of $G$ on $V$ is a [[homotopy fiber sequence]] in $\mathbf{H}$ of the form

  $$
    \array{
      V &\to& V\sslash G
      \\
      && \downarrow^{\mathrlap{p_\rho}}
      \\
      && \mathbf{B}G
    }
    \,.
  $$

The object $V/G$ defined thereby we call the [[homotopy quotient]] of $V$ by $G$ via this action.

=--

#### Associated $\infty$-bundles

* [[associated infinity-bundle]]

$$
  \array{
    E &\to& V\sslash G
    \\
    \downarrow &pb& \downarrow
    \\
    \tilde X &\to& \mathbf{B}G
    \\
    \downarrow^{\mathrlap{\simeq}}
    \\
    X
  }
$$

* [[section]]

$$
  \array{
    X &&\stackrel{\sigma}{\to}&& V \sslash G
    \\
    & \searrow &\swArrow_{\simeq}& \swarrow
    \\
    && \mathbf{B}G
  }
$$

#### Stabilizer $\infty$-Groups

We had seen above in prop. \ref{OrdinaryStabilizer} that the traditional concept of stabilizer groups of group actions is equivalent to groups of loops in the [[action groupoid]] of the given action. This equivalent reformulation has an immediate generalization to [[∞-actions]], def. \ref{InfinityAction}.

For $\mathbf{H}$ an [[(∞,1)-topos]],  $G\in \mathbf{H}$ an object equipped with [[∞-group]] structure, hence with a [[delooping]] $\mathbf{B}$G, and for $\rho$ an [[∞-action]] of $G$ on some $V$, exhibited by a [[homotopy fiber sequence]] of the form

$$
  \array{
    V &\stackrel{i}{\longrightarrow}& V/G
    \\
    && \downarrow^{\mathrlap{p_\rho}}
    \\
    && \mathbf{B}G
  }
  \,.
$$

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

Given a [[global element]] of $V$

$$
  x \colon \ast \to X
$$

then the **stabilizer $\infty$-group** $Stab_\rho(x)$ of the $G$-action at $x$ is the [[loop space object]]

$$
  Stab_\rho(x) \coloneqq \Omega_{i(x)} (X/G)
 \,.
$$

=--

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

Equivalently, def. \ref{StabilizerInInfinityTopos}, gives the [[loop space object]] of the [[1-image]] $\mathbf{B}Stab_\rho(x)$ of the morphism

$$
  \ast \stackrel{x}{\to} X \to X/G
  \,.
$$

As such the [[delooping]] of the stabilizer $\infty$-group sits in a [[1-epimorphism]]/[[1-monomorphism]] factorization $\ast \to \mathbf{B}Stab_\rho(x) \hookrightarrow X/G$ which combines with the homotopy fiber sequence of prop. \ref{InfinityAction} to a diagram of the form

$$
  \array{
    \ast &\stackrel{x}{\longrightarrow}& X &\stackrel{}{\longrightarrow}& X/G
    \\
    \downarrow^{\mathrlap{epi}} && & \nearrow_{\mathrlap{mono}} & \downarrow
    \\
    \mathbf{B} Stab_\rho(x)
    &=&
    \mathbf{B} Stab_\rho(x)
    &\longrightarrow&
    \mathbf{B}G
  }
 \,.
$$

In particular there is hence a canonical homomorphism of $\infty$-groups

$$
  Stab_\rho(x) \longrightarrow G
  \,.
$$

However, in contrast to the classical situation, this morphism is not in general a monomorphism anymore, hence the stabilizer $Stab_\rho(x)$ is not a _sub_-group of $G$ in general.

=--

### Syntactic Layer

#### The context of a pointed connected type: representation theory

(...)

#### Dependent product over a pointed connected type: invariants

(...)

#### Dependent sum over a pointed connected type: quotients

(...)