circle n-bundle with connection

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+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Bundles
+-- {: .hide}
[[!include bundles - contents]]
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#### Differential cohomology
+--{: .hide}
[[!include differential cohomology - contents]]
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#### Cohesive $\infty$-Toposes
+--{: .hide}
[[!include cohesive infinity-toposes - contents]]
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#Contents#
* table of contents
{:toc}

## Idea

We discuss the refinement to [[higher differential geometry]] of the concept of a [[circle group]]-[[principal connection]] (on a [[circle group]]-[[principal bundle]]). Specifically we indicate how the general abstract definition in terms of [[cohesion]] reproduces in the context of [[smooth ∞-groupoid|smooth cohesion]] to the representation of circle $n$-connections by [[cocycles]] in smooth _[[Deligne cohomology]]_ ([dcct](#dcct)).

In every [[cohesive (∞,1)-topos]] $\mathbf{H}$ there is an intrinsic notion of differential cohomology with coefficients in an [[abelian group|abelian]] [[group object]] $A \in \mathbf{H}$ that classifies $\mathbf{B}^{n-1}A$-[[principal ∞-bundles]] with [[connection on an ∞-bundle|∞-connection]].

Here we discuss the specific realization for $\mathbf{H} =$ [[Smooth∞Grpd]] the [[(∞,1)-topos]] of [[smooth ∞-groupoids]] and $A = U(1)$ the [[circle group]].

In this case the intrinsic differential cohomology reproduces [[ordinary differential cohomology]] and generalizes it to base spaces that may be [[smooth manifold]]s, [[diffeological space]]s, [[orbifold]]s and generally [[smooth ∞-groupoid]]s such as [[delooping]]s $\mathbf{B}G$ of
[[smooth ∞-groupoid|smooth ∞-group]]s $G$. Differential cocycles on the latter support the [[∞-Chern-Weil homomorphism]] that sends [[nonabelian cohomology|nonabelian]] [[connection on an ∞-bundle|∞-connections]] to circle $n$-bundles whose [[curvature]] form realizes a [[characteristic class]] in [[de Rham cohomology]].

## The ambient context

Let $\mathbf{H} := $ [[Smooth∞Grpd]] be the [[cohesive (∞,1)-topos]] of [[smooth ∞-groupoid]].  As usual, write

$$
  (\Pi \dashv Disc \dashv \Gamma \dashv coDisc) :
  Smooth \infty Grpd
   \stackrel{\overset{\Pi}{\to}}{\stackrel{\overset{Disc}{\leftarrow}}{\stackrel{\overset{\Gamma}{\to}}{\underset{coDisc}{\leftarrow}}}}
  \infty Grpd
$$

for the terminal [[global section]] [[(∞,1)-geometric morphism]] with its extra [[left adjoint]], the [[fundamental ∞-groupoid in a locally ∞-connected (∞,1)-topos|intrinsic fundamental ∞-groupoid]] functor $\Pi$.

From this induced is the path ∞-groupoid adjunction

$$
  (\mathbf{\Pi} \dashv \mathbf{\flat}) :
   Smooth \infty Grpd \stackrel{\leftarrow}{\to}
   Smooth \infty Grpd
$$

and the intrinsic de Rham cohomology adjunction

$$
  (\mathbf{\Pi}_{dR} \dashv \mathbf{\flat}_{dR}) :
   */Smooth \infty Grpd \stackrel{\overset{\mathbf{\Pi}_{dR}}{\leftarrow}}{\underset{\mathbf{\flat}_{dR}}{\to}}
   Smooth \infty Grpd
  \,.
$$

For $A$ an abelian group object there for each [[integer]] $n$ is the universal curvature characteristic form, given by a [[cocycle]]-morphism

$$
  curv : \mathbf{B}^n A \to \mathbf{\flat}_{dR}\mathbf{B}^{n+1}A
  \,.
$$

The [[cocycle]]s for _differential cohomology_ in degree $n$ with coefficients in $A$ are the points in the [[homotopy fiber]] $\mathbf{H}_{diff}(-, \mathbf{B}^n A)$ of the morphism on [[cohomology]]

$$
  curv_* : \mathbf{H}(-, \mathbf{B}^n A) \to
   \mathbf{H}_{dR}(-, \mathbf{B}^{n+1}A)
$$

induced by this. Every such cocycle $\nabla \in \mathbf{H}_{diff}(X,\mathbf{B}^n A)$ we may think of as an [[connection on an ∞-bundle|∞-connection]] on the $\mathbf{B}^{n-1}A$-[[principal ∞-bundle]] classified by the underlying cocycle in $\mathbf{H}(X, \mathbf{B}^n A)$.

We consider these constructions in the model $\mathbf{H} = $ [[Smooth∞Grpd]]. This is the [[(∞,1)-category of (∞,1)-sheaves]]

$$
  Smooth\infty Grpd :=
  Sh_{(\infty,1)}(CartSp_{smooth})
$$

on the [[site]] [[CartSp]]${}_{smooth}$ of [[Cartesian space]]s and [[smooth function]]s between them. This is a general [[higher geometry]] context for [[differential geometry]]. For computations we can explicitly [[presentable (∞,1)-category|present]] this [[(∞,1)-category]] by a local [[model structure on simplicial presheaves]] $[CartSp^{op}, sSet]_{proj,loc}$

$$
  Smooth \infty Grpd \simeq ([CartSp^{op}, sSet]_{proj, loc})^\circ
$$

as described at [[presentations of (∞,1)-sheaf (∞,1)-toposes]].

In $\mathbf{H} = $ [[Smooth∞Grpd]] a canonical choice for $A$ is the [[circle group]]

$$
  A := U(1) = \mathbb{R}/\mathbb{Z}
  \,.
$$

We show how the notion of _smooth circle $n$-bundles with connection_ obtained by  applying the general setup above to this case reproduces [[ordinary differential cohomology]]:

* a _circle 1-bundle with connection_ is an ordinary $U(1)$ [[principal bundle]] [[connection on a bundle|with connection]];

* a _circle 2-bundle with connection_ is a $\mathbf{B}U(1)$-[[principal 2-bundle]] [[connection on a 2-bundle|with connection]], equivalently a $U(1)$-[[bundle gerbe]] [[connection on a bundle gerbe|with connection]];

* a _circle 3-bundle_ with connection if a $\mathbf{B}^2 U(1)$-[[principal ∞-bundle|principal 3-bundle]] [[connection on an ∞-bundle|with connection]], equivalently a $U(1)$-[[bundle 2-gerbe]] with connection;

* generally, a _circle $n$-bundle with connection_ is a $\mathbf{B}^{n-1}U(1)$-[[principal ∞-bundle|principal n-bundle]] [[connection on an ∞-bundle|with connection]], equivalently a [[cocycle]] in [[Deligne cohomology]] in degree $n+1$, equivalently a [[Cheeger-Simons differential character]] in that degree.

We assume in the following that the reader is familiar with basics of [[smooth ∞-groupoid]]s.

## Flat differential cohomology {#FlatCircleCohomology}

The coefficient object for flat differential cohomology in $\mathbf{H} = $ [[Smooth∞Grpd]] with values in $\mathbf{B}^n U(1)$ is $\mathbf{\flat} \mathbf{B}^n U(1) = LConst \Gamma \mathbf{B}^n U(1)$.

The coefficient object for intrinsic de Rham cohomology is $\mathbf{\flat}_{dR} \mathbf{B}^n U(1)$, defined by the [[(∞,1)-pullback]]

$$
  \array{
    \mathbf{\flat}_{dR} \mathbf{B}^n U(1) &\to&
    \mathbf{\flat} \mathbf{B}^n U(1)
    \\
    \downarrow && \downarrow
    \\
    * &\to& \mathbf{B}^n U(1)
  }
  \,.
$$

The following proposition provides models for these objects in in terms of ordinary [[differential form]]s.

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

A fibrant representative in $[CartSp^{op}, sSet]_{proj,cov}$ of $\mathbf{\flat} \mathbf{B}^n U(1)$ is

$$
  \mathbf{\flat}\mathbf{B}^n U(1)_{chn}
  :=
  \Xi[C^\infty(-,U(1)) \stackrel{d_{dR}}{\to} \Omega^1(-) \stackrel{d_{dR}}{\to} \cdots \stackrel{d_{dR}}{\to} \Omega^n_{cl}(-)]
  \,,
$$

and a fibrant representative of $\mathbf{\flat}_{dR} \mathbf{B}^n U(1)$ is

$$
  \mathbf{\flat}_{dR}\mathbf{B}^n U(1)_{chn}
  :=
  \Xi[0 \to \Omega^1(-) \stackrel{d_{dR}}{\to} \cdots \stackrel{d_{dR}}{\to} \Omega^n_{cl}(-)]
  \,.
$$

=--

Notice that the complex of sheaves $\mathbf{\flat}\mathbf{B}^n U(1)$ is that which defines _flat_ [[Deligne cohomology]], while that of $\mathbf{\flat}_{dR} \mathbf{B}^n U(1)$ is essentially that which defines [[de Rham cohomology]] in degree $n \gt 1$ (see [below](#OrdinaryDeRham)). Also notice that we denoted by $d_{dR}$ also the differential $C^\infty(-,U(1)) \stackrel{d_{dR} log}{\to} \Omega^1(-)$; this is to stress that we are looking at $U(1)$ as the quotient $\mathbb{R}/\mathbb{Z}$.

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

Since the [[global section]] functor $\Gamma$ amounts to evaluation on the  point $\mathbb{R}^0$ and since constant simplicial presheaves on [[CartSp]] satisfy [[nLab:descent]] (on objects in $CartSp$!), we have that $\mathbf{\flat} \mathbf{B}^n U(1)$ is represented by the complex of sheaves $\Xi[const U(1) \to 0 \to \cdots \to 0]$. This is weakly equivalent to $\Xi[C^\infty(-,U(1)) \stackrel{d_{dR}}{\to} \Omega^1(-) \stackrel{d_{dR}}{\to} \cdots \stackrel{d_{dR}}{\to} \Omega^n_{cl}(-)]$ by the [[Poincare lemma]] applied to each [[Cartesian space]] (using the same standard logic that proves the [[de Rham theorem]]) in that the degreewise inclusion

$$
  \array{
    const U(1) &\to& 0 &\to& \cdots &\to& 0
    \\
    \downarrow && \downarrow && && \downarrow
    \\
    C^\infty(-,U(1)) &\stackrel{d_{dR}}{\to}&
    \Omega^1(-) &\to& \cdots &\to& \Omega^n_{cl}(-)
  }
$$

is objectwise a [[nLab:quasi-isomorphism]].

Therefore a fibration in $[CartSp^{op}, sSet]_{proj}$ representing the counit $\mathbf{\flat} \mathbf{B}^n U(1) \to \mathbf{B}^n U(1)$ is the image under $\Xi$ of

$$
  \array{
    C^\infty(-,U(1)) &\to& \Omega^1(-) &\to & \cdots &\to& \Omega^n_{cl}(-)
    \\
    \downarrow^{\mathrlap{=}} && \downarrow && && \downarrow
    \\
    C^\infty(-, U(1)) &\to& 0 &\to& \cdots &\to& 0
  }
  \,.
$$

We observe that the [[pullback]] of this morphism to the point

$$
  \array{
    \Xi[0 \to \Omega^1(-) \to \cdots \to \Omega^n_{cl}(-)]
    &\to&
    \Xi[C^\infty(-,U(1)) \to \Omega^1(-) \to \cdots \to \Omega^n_{cl}(-)]
    \\
    \downarrow && \downarrow
    \\
    \Xi[0 \to 0 \to \cdots \to 0]
    &\to&
    \Xi[C^\infty(-,U(1)) \to 0 \to \cdots \to 0]
  }
$$

is the pullback over a cospan all whose objects are fibrant and one of whose morphisms is a fibration. Therefore this is a [[homotopy pullback]] diagram in $[CartSp^{op}, sSet]_{proj}$ which models the [[(∞,1)-limit]] over $* \to \mathbf{B}^n U(1) \leftarrow \mathbf{\flat}\mathbf{B}^n U(1)$ in $PSh_{(\infty,1)}(CartSp)$. Since [[nLab:∞-stackification]] preserves finite $(\infty,1)$-limits this models also the corresponding $(\infty,1)$-limit in $\mathbf{H}$. Therefore the top left object is indeed a model for $\mathbf{\flat}_{dR} \mathbf{B}^n U(1)$.

=--

## de Rham cohomology {#OrdinaryDeRham}

The intrinsic de Rham cohomology of [[Smooth∞Grpd]] with coefficients in $\mathbb{R}$ or $U(1) = \mathbb{R}/\mathbb{Z}$ coincides with the ordinary [[de Rham cohomology]] of [[smooth manifold]]s and smooth [[simplicial manifold]]s in degree greater than 1. This we discuss here. The meaning of the discrepancy in degee 1 and lower is discussed [below](#U1GroupoidBundle).

So for this section let $n \in \mathbb{N}$ with $n \geq 2$.

Above in _[Flat U(1)-valued differential cohomology](#FlatCircleCohomology)_ we found a fibrant representative of $\mathbf{\flat}_{dR} \mathbf{B}^n U(1) \in Smooth\infty Grpd$ to be given by

$$
  \Xi[\Omega^1(-) \stackrel{d_{dR}}{\to} \Omega^2(-)
  \stackrel{d_{dR}}{\to} \cdots \to \Omega^n_{cl}(-)]
$$

in $[CartSp^{op}, sSet]_{proj, cov}$.

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

For $X \in Smooth\infty Grpd$ a [[paracompact space|paracompact]] [[smooth manifold]] we have in for $\mathbf{H} = Smooth \infty Grpd$ a
[[natural isomorphism]]

$$
  H_{dR}(X,\mathbf{B}^n U(1))
  :=
  \pi_0 \mathbf{H}(X,\mathbf{\flat}_{dR} \mathbf{B}^n U(1))
  \simeq
  H_{dR}^n(X)
  \,,
$$

where on the left we have the intrinsic (∞,1)-topos theoretic notion of de Rham cohomology, and on the right the ordinary  notion of [[nLab:de Rham cohomology]] of a smooth manifold.

=--

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

Let $\{U_i \to X\}$ be a [[nLab:good open cover]]. At [[Smooth∞Grpd]] is discussed that then the [[nLab:Cech nerve]] $C(\{U_i\}) \to X$ is a cofibrant [[nLab:resolution]] of $X$ in $[CartSp^{op}, sSet]_{proj,cov}$. Therefore we have

$$
  \mathbf{H}(X,\mathbf{\flat}_{dR} \mathbf{B}^n U(1))
  \simeq
  [CartSp^{op}, sSet](C(\{U_i\}), \Xi[\Omega^1(-) \stackrel{d_{dR}}{\to} \cdots \to \Omega^n_{cl}(-)])
  \,.
$$

The right hand is the $\infty$-groupoid of cocylces in the [[nLab:Cech cohomology|Cech]] [[nLab:hypercohomology]] of the complex of sheaves of differential forms. A cocycle is given by a collection

$$
  (C_i, B_{i j}, A_{i j k}, \cdots  , Z_{i_0, \cdots, i_n})
$$

of differential forms, with $C_i \in \Omega^n_{cl}(U_i)$, $B_{i j} \in \Omega^{n-1}(U_i \cap U_j)$, etc. , such that this collection is annihilated by the total differentoal $D = d_{dR} \pm \delta$, where $d_{dR}$ is the de Rham differential and $\delta$ the alternating sum of the pullbacks along the face maps of the [[nLab:Cech nerve]].

It is a standard result of [[nLab:abelian sheaf cohomology]] that such cocycles represent classes in de Rham cohomology.

But for the record and since the details of this computation will show up again at some mildly subtle points in further discussion below, we spell this out in some detail.

We can explicitly construct coboundaries connecting such a generic cocycle to one of the form

$$
  (F_i, 0, 0, \cdots, 0)
$$

by using a [[nLab:partition of unity]] $(\rho_i \in C^\infty(X))$ subordinate to the cover $\{U_i \to X\}$, i.e. $x \in U_i \Rightarrow \rho_i(x) = 0$ and $\sum_i \rho_i = 1$.

For consider

$$
  \begin{aligned}
     & (C_i, B_{i j}, A_{i j k}, \cdots  , Y_{i_1, \cdots, i_{n}}, Z_{i_1, \cdots, i_{n+1}})
     \\
     + & D (0, 0, \cdots, \sum_{i_0} \rho_{i_0} Z_{i_0, i_1, \cdots, i_{n}},0)
     \\
     =
     & (C_i, B_{i j}, A_{i j k}, \cdots  , Y_{i_1, \cdots, i_{n}}
     + d_{dR}\sum_{i_0} \rho_{i_0} Z_{i_0, i_1, \cdots, i_{n}}, 0)
  \end{aligned}
  \,,
$$

where we use that from $(\delta Z)_{i_1, \cdots, i_{n+2}} = 0$ it follows that

$$
  \begin{aligned}
    (\delta \sum \rho Z)_{i_1, \cdots, i_{n+1}}
    &=
    \sum_{i_0} \rho_{i_0}
    \sum_{k = 1}^{n+1}
    (-1)^k
    Z_{i_0, i_1 \cdots, \hat i_k, \cdots, i_{n+1}}
    \\
    & =
    \sum_{i_0} \rho_{i_0}
    Z_{i_1 ,\cdots, i_{n+1}}
    \\
    & =
    Z_{i_1 ,\cdots, i_{n+1}}
  \end{aligned}
  \,.
$$

> where I am suppressing some evident signs...

By recurseively adding coboundaries this way, we can annihilate all the higher Cech-components of the original cocycle and arrive at a cocycle of the form $(F_i, 0, \cdots, 0)$.

Such a cocycle being $D$-closed says precisely that $F_i = F|_{U_i}$ for $F \in \Omega^n_{cl}(X)$ a globally defined closed differential form. Moreover, coboundaries between two cocycles both of this form

$$
  (F_i, 0, \cdots , 0) = (F'_i, 0, \cdots, 0) + D(\lambda_i, \lambda_{i j}, \cdots)
$$

are necessarily themselves of the form $(\lambda_i, \lambda_{i j}, \cdots) =  (\lambda_i, 0 ,\cdots, 0)$ with $\lambda_i = \lambda|_{U_i}$ for $\lambda \in \Omega^{n-1}(X)$ a globally defined differential $n$-form and $F = F' + d_{dR} \lambda$.

=--

## Differential cohomology
 {#DifferentialCohomology}

The intrinsic definition of the ∞-groupoid of cocycles for the intrinsic  differential cohomology in $\mathbf{H} = Smooth\infty Grpd$ with coefficients $\mathbf{B}^n U(1)$ is the object $\mathbf{H}_{diff}(X,\mathbf{B}^n U(1))$ in the [[(∞,1)-pullback]]

$$
  \array{
    \mathbf{H}_{diff}(X,\mathbf{B}^n U(1)) &\to & H_{dR}(X,\mathbf{B}^{n+1} U(1))
    \\
    \downarrow && \downarrow
    \\
    \mathbf{H}(X,\mathbf{B}^n U(1))
    &\stackrel{curv}{\to}&
    \mathbf{H}_{dR}(X, \mathbf{B}^{n+1} U(1))
  }
$$

in [[∞Grpd]].

We show now that for $n \geq 1$ this reproduces the [[nLab:Deligne cohomology]] $H(X,\mathbb{Z}(n+1)_D^\infty)$ of $X$:

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

For $X$ a [[nLab:paracompact space|paracompact]] [[nLab:smooth manifold]] we have

$$
  H_{diff}(X,\mathbf{B}^n U(1))
  \simeq
  \left(
     \;\; H(X,\mathbb{Z}(n+1)_D^\infty) \;\;
  \right)
  \times_{\Omega_{cl}^{n+1}(X)} H_{dR}^{n+1}_{int}(X)
  \,.
$$

=--

Here on the right we have the subset of Deligne cocycles that picks for each integral de Rham cohomology class of $X$ only one curvature form representative.

We give the proof [below](#ProofOfDeligneTheorem) after some preliminary expositional discussion.

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

The restriction to single representatives in each de Rham class is a reflection of the fact that in the above $(\infty,1)$-pullback diagram the morphism $H_{dR}(X,\mathbf{B}^{n+1}U(1)) \to \mathbf{H}_{dR}(X,\mathbf{B}^{n+1}U(1))$ by definition picks one representative in each connected component. Using the [above model](#OrdinaryDeRham) of the intrinsic de Rham cohomology in terms of globally defined differential froms, we could easily get rid of this restriction by considering instead of the above $(\infty,1)$-pullback the homotopy pullback

$$
  \array{
    \mathbf{H}'_{diff}(X,\mathbf{B}^n U(1)) &\to & \Omega_{cl}^{n+1}(X)
    \\
    \downarrow && \downarrow
    \\
    \mathbf{H}(X,\mathbf{B}^n U(1))
    &\stackrel{curv}{\to}&
    \mathbf{H}_{dR}(X, \mathbf{B}^{n+1} U(1))
  }
$$

where now the right vertical morphism is the inclusion of the set of objects of our concrete model for the  $\infty$-groupoid $\mathbf{H}_{dR}(X, \mathbf{B}^{n+1} U(1))$. With this definition we get the isomorphism

$$
  H'_{diff}(X,\mathbf{B}^n U(1))
  \simeq
  H(X,\mathbb{Z}(n+1)_D^\infty)
  \,.
$$

From the tradtional point of view of differential cohomology this may be what one expects to see, but from the intrinsic $(\infty,1)$-topos theoretic point of view it is quite unnatural -- and in fact "[[nLab:evil]]" -- to fix that set of objects of the $\infty$-groupoid. Of intrinsic meaning is only the set of their equivalences classes.

=--

### Circle bundles with connection {#CircleBundlesConnection}

Before discussing [the full theorem](#DeligneCohomologyTheorem), it is instructive to start by looking at the special case $n=1$ in some detail, which is about ordinary  $U(1)$-[[nLab:principal bundle]]s [[nLab:connection on a bundle|with connection]].

This contains in it already all the relevant structure of the general case, but the low categorical degree is more transparently written out and will allow us to pause to highlight some maybe noteworthy aspects of the situation, such as the phenomenon of _pseudo-connections_ [below](#CircleBunlePseudoConnection).

In terms of the [Dold-Kan correspondence](#DoldKan) the object $\mathbf{B}U(1) \in \mathbf{H}$ is modeled in $[CartSp^{op}, sSet]$ by

$$
  \mathbf{B}U(1)
  =
  \Xi(\;
    C^\infty(-,U(1)) \to 0
  \;)
  \,.
$$

Accordingly we have for the double [[nLab:delooping]] the model

$$
  \mathbf{B}^2 U(1)
  =
  \Xi(
    \;
    C^\infty(-,U(1)) \to 0  \to 0
  \;)
$$

and for the [[nLab:universal principal ∞-bundle|universal principal 2-bundle]]

$$
  \mathbf{E}\mathbf{B}U(1)
  =
  \Xi(
    \;
    C^\infty(-,U(1)) \stackrel{Id}{\to} C^\infty(-, U(1)) \to 0
    \;
  )
  \,.
$$

In this notation we have also the constant presheaf

$$
  \mathbf{\flat} \mathbf{B}^2 U(1)
  =
  \Xi(
    \;
     const U(1) \to 0 \to 0
    \;
  )
  \,.
$$

Above we already found the model

$$
  \mathbf{\flat}_{dR} \mathbf{B}^2 U(1)
  =
  \Xi(0 \to \Omega^1(-) \stackrel{d_{dR}}{\to} \Omega^2_{cl}(-))
  \,.
$$

In order to compute the differential cohomology
$\mathbf{H}_{diff}(-,\mathbf{B}U(1))$ by an ordinary
pullback in [[nLab:sSet]] we also want to resolve the curvature characteristic morphism $\mathbf{B}U(1) \to \mathbf{\flat}_{dR} \mathbf{B}^2 U(1)$ by a fibration. We claim that this may be obtained by choosing the resolution
$\mathbf{B}U(1) \stackrel{\simeq}{\leftarrow} \mathbf{B} U(1)_{diff,chn}$ given by

$$
  \mathbf{B}U(1)_{diff}
  :=
  \Xi(
    \;
    C^\infty(-,U(1))
    \oplus
    \Omega^1(-)
    \stackrel{d_{dR} \oplus Id}{\to}
    \Omega^1(-)
    \;
  )
$$

with the morphism $curv : \mathbf{B}_{diff}U(1) \to \mathbf{\flat}_{dR}\mathbf{B}^2 U(1)$ given by

$$
  \array{
    C^\infty(-,U(1))
    \oplus
    \Omega^1(-)
    &\stackrel{d_{dR} + Id}{\to}&
    \Omega^1(-)
    \\
    \downarrow^{\mathrlap{p_2}} && \downarrow^{\mathrlap{d_{dR}}}
    \\
    \Omega^1(-) &\stackrel{d_{dR}}{\to}&
    \Omega^2_{cl}(-)
  }
  \,.
$$

By the [[nLab:Poincare lemma]] applied to each [[nLab:Cartesian space]], this is indeed a fibration.

In the [next section](#AbGerbesConnection) we give the proof of this (simple) claim. Here in the warmup phase we instead want to discuss the geometric interpretation of this resolution, along the lines of the section curvature characteristics of 1-bundles in the [[schreiber:differential cohomology in an (∞,1)-topos -- survey|survey-part]].

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

We have the following geometric interpretation of the above models:

$$
  \mathbf{\flat}_{dR} \mathbf{B}^2 U(1)
  :
  U \mapsto
  \left\{
    \array{
      U &\to& *
      \\
      \downarrow && \downarrow
      \\
      \mathbf{\Pi}_2(U) &\to& \mathbf{B}^2 U(1)
    }
  \right\}
  =
  \left\{
    \mathbf{\Pi}_2(U) \to \mathbf{B}^2 U(1)
  \right\}
$$

and

$$
  \mathbf{B}U(1)_{diff}
  :
  U \mapsto
  \left\{
    \array{
      U &\to& \mathbf{B}U(1)
      \\
      \downarrow && \downarrow
      \\
      \mathbf{\Pi}_2(U) &\to& \mathbf{B}INN(U(1))
    }
  \right\}
  \,.
$$

And in this presentation the morphism $curv : \mathbf{B}_{diff}U(1) \to \mathbf{B}^2 U(1)$ is given over $U \in CartSp$ by forming the pasting composite

$$
  \array{
    U &\to& \mathbf{B}U(1) &&& underlying\;cocycle
    \\
    \downarrow && \downarrow
    \\
    \mathbf{\Pi}_2(U) &\to& \mathbf{B}INN(U(1)) &&& connection
    \\
    \downarrow && \downarrow
    \\
    \mathbf{\Pi}_2(U) &\to& \mathbf{B}^2 U(1) &&& curvature
  }
$$

and picking the lowest horizontal morphism.

=--

Here the terms mean the following:

* $INN(U(1))$ is the [[nLab:2-group]] $\Xi(U(1) \to U(1))$, which is a
  [[nLab:groupal model for universal principal ∞-bundles|groupal model for the universal U(1)-principal bundle]] $\mathbf{E}U(1)$;

* $\mathbf{\Pi}_2(U)$ is the [[nLab:path 2-groupoid]] with homotopy class of 2-dimensional paths as 2-morphisms

* the groupoids of diagrams in braces have as objects commuting diagrams in $[CartSp^{op}, sSet]$ as indicated, and horizontal 2-morphisms fitting into such diagrams as morphisms.

Using the discussion at [[nLab:2-groupoid of Lie 2-algebra valued forms]] (SchrWalII) we have the following:

1. For $X$ a [[nLab:smooth manifold]], morphisms in $[CartSp^{op}, 2Grpd]$ of the form $tra_A : \Pi_2(X) \to \mathbf{E}\mathbf{B}U(1)$ are in bijection with smooth 1-forms $A \in \Omega^1(X)$: the 2-functor sends a path in $X$ to the the [[nLab:parallel transport]] of $A$ along that path, and sends a surface in $X$ to the exponentiated integral of the curvature 2-form $F_A = d A$ over that surface. The [[nLab:Bianchi identity]] $d F_A = 0$ says precisely that this assignment indeed descends to homotopy classes of surfaces, which are the 2-morphisms in $\Pi_2(X)$.

1. Moreover [[nLab:k-morphism|2-morphisms]] of the form $(\lambda,\alpha) : tra_A \to \tra_{A'}$ in $[CartSp^{op}, 2Grpd]$ are in bijection with pairs consisting of a $\lambda \in C^\infty(X,U(1))$ and a 1-form $\alpha \in \Omega^1(X)$ such that $A' = A + d_{dR} \lambda - \alpha$.

1. And finally [[nLab:k-morphism|3-morphisms]] $h : (\lambda, \alpha) \to (\lambda', \alpha')$ are in bijection with $h \in C^\infty(X,U(1))$ such that $\lambda' = \lambda \cdot h$ and $\alpha' = \alpha + d_{dR} h$.

By the same reasoning we find that the coefficient object for flat $\mathbf{B}^2 U(1)$-valued differential cohomology is

$$
  \mathbf{\flat}\mathbf{B}^2 U(1)
  =
  [\Pi_2(-), \mathbf{B}^2U(1)]
  =
  \Xi(
    C^\infty(-,U(1))
    \stackrel{d_{dR}}{\to}
    \Omega^1(-) \stackrel{d_{dR}}{\to} \Omega^2_{cl}(-)
  )
  \,.
$$

So by the above definition of differential cohomology in $\mathbf{H}$ we find that $\mathbf{B}U(1)$-differential cohomology of a [[nLab:paracompact space|paracompact]] [[nLab:smooth manifold]] $X$ is given by choosing any [[nLab:good open cover]] $\{U_i \to X\}$, taking $C(\{U_i\})$ to be the [[nLab:Cech nerve]], which is then a cofibrant replacement of $X$ in $[CartSp^{op}, sSet]_{proj,cov}$ and forming the ordinary pullback

$$
  \array{
    \mathbf{H}_{diff}(X,\mathbf{B}U(1)) &\to& H^2_{dR}(X)
    \\
    \downarrow && \downarrow
    \\
    [CartSp^{op},sSet](C(\{U_i\}), \mathbf{B}_{diff}U(1))
    &\stackrel{curv}{\to}&
    [CartSp^{op},sSet](C(\{U_i\}), \flat_{dR}\mathbf{B}^2 U(1))
  }
$$

(because the bottom vertical morphism is a fibration, by the fact that our model for $\mathbf{B}_{diff} U(1) \to \flat_{dR}\mathbf{B}^2 U(1)$ is a fibration, that $C(\{U_i\})$ is cofibrant and using the axioms of the [[nLab:sSet]]-[[nLab:enriched model category]] $[CartSp^{op}, sSet]_{proj}$).

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

A cocycle in $[CartSp^{op},sSet](C(\{U_i\}), \mathbf{B}_{diff}U(1))$ is

1. a collection of functions

   $$
     (g_{i j } \in C^\infty(U_i \cap U_j, U(1)))
    $$

    satsifying $g_{i j} g_{j k} = g_{i k}$ on $U_i \cap U_j \cap U_k$;

1. a collection of 1-forms

   $$
      (A_i \in \Omega^1(U_i))
   $$

1. a collection of 1-forms

    $$
      (a_{i j} \in \Omega^1(U_i \cap U_j))
    $$

    such that

    $$
      A_j = A_i + d_{dR} g_{i j} + a_{i j}
    $$

    on $U_i \cap U_j$ and

    $$
      a_{i j} + a_{j k} = a_{i k}
    $$

    on $U_i \cap U_j \cap U_k$.

The curvature-morphism takes such a cocycle to the cocycle

$$
  (d A_i, a_{i j}, )
$$

in the [above model](#OrdinaryDeRham) $[CartSp^{op},sSet](C(\{U_i\}), \mathbf{\flat}_{dR}\mathbf{B}^2 U(1))$ for intrinsic de Rham cohomology.

Every cocycle with nonvanishing $(a_{i j})$ is in $[C(\{U_i\}), \mathbf{B}_{diff}U(1)]$ coboundant to one with vanishing $(a_{i j})$

=--

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

The first statements are effectively the definition and the construction of the above models. The last statement is as in the [above discussion](#OrdinaryDeRham) of our model for ordinary de Rham cohomology: given a cocycle with non-vanishing closed $a_{i j}$, pick a partition of unity $(\rho_i \in C^\infty(X))$ subordinate to the chosen cover and the coboundary given by $(\sum_{i_0} \rho_{i_0} a_{i_0 i})$. This connects $(A_i,a_{i j}, g_{i j})$ with the cocycle $(A'_i, a'_{i j}, g_{i j})$ where

$$
  A'_i = A_i + \sum_{i_0} \rho_{i_0} a_{i_0 i}
$$

and

$$
  \begin{aligned}
    a'_{i j}
      & = A'_j - A'_i - d g_{i j}
     \\
      & = a_{i j} +
      - \sum_{i_0}( a_{i_0 i} - a_{i_0 j} )
    \\
    & = 0
  \end{aligned}
  \,.
$$

=--

So in total we have found the following story:

1. In order to compute the curvature characteristic form of a [[nLab:Cech cohomology]] cocycle $g : C(\{U_i\}) \to \mathbf{B}U(1)$ of a $U(1)$-principal bundle, we first lift it

   $$
     \array{
       && \mathbf{B}_{diff}U(1)
       \\
       & {}^{\mathllap{(g,\nabla)}}\nearrow & \downarrow
       \\
       C(\{U_i\}) &\stackrel{g}{\to}& \mathbf{B}U(1)
     }
   $$

   to an equivalent $\mathbf{B}_{diff}U(1)$-cocycle, and this amounts to putting (the Cech-representatitve of) a _pseudo-connection_ on the $U(1)$-principal bundle.

1. From that lift the desired curvature characteristic is simply projected out

   $$
     \array{
       && \mathbf{B}_{diff}U(1) &\stackrel{curv}{\to}& \mathbf{\flat}_{dR}\mathbf{B}^2 U(1)
       \\
       & {}^{\mathllap{(g,\nabla)}}\nearrow & \downarrow
       \\
       C(\{U_i\}) &\stackrel{g}{\to}& \mathbf{B}U(1)
     }
     \,,
   $$

   and we find that it lives in the sheaf [[nLab:hypercohomology]] that models ordinary de Rham cohomology.

1. Therefore we find that in each _cohomology class_ of curvatures, there is at least one representative which is an ordinary globally defined 2-form. Moreover, the pseudo-connections that map to such a representative are precisely the _genuine_ connections, those for which the $(a_{i j})$-part of the cocycle vaishes.

So we see that ordinary connections on ordinary circle bundles are a means to model the homotopy pullback

$$
  \array{
    \mathbf{H}_{diff}(X,\mathbf{B}U(1)) &\to& H_{dR}^2(X)
    \\
    \downarrow && \downarrow
    \\
    \mathbf{H}(X,\mathbf{B}U(1)) &\to& \mathbf{H}_{dR}(X,\mathbf{B}U(1))
  }
$$

in a 2-step process: first the choice of a pseudo-connection realizes the bottom horizontal morphism as an [[nLab:anafunctor]], and then second the restriction imposed by forming the ordinary pullback chooses from all pseudo-connections precisely the genuine connections.

The general version of this story is discussed in detail at differential cohomology in an (∞,1)-topos -- Local (pseudo-)connections.

### Circle bundles with pseudo-connection
 {#CircleBunlePseudoConnection}

In the above discussion of extracting ordinary connections on ordinary $U(1)$-principal bundles from the abstract topos-theoretic definition of differential cohomology, we argued that a certain homotopy pullback may be computed by choosing in the Cech-hypercohomology of the complex of sheaves $(\Omega^1(-) \stackrel{d_{dR}}{\to} \Omega^2_{cl}(-))$ over a manifold $X$ those cohomology representatives that happen to be represented by globally defined 2-forms on $X$. We saw that the homotopy fiber of _pseudo-connections_ over these 2-forms happened to have connected components indexed by _genuine_ connections.

But by the general abstract theory, up to isomorphism the differential cohomology computed this way is guaranteed to be independent of all such choices, which only help us to compute things.

To get a feeling for what is going on, it may therefore be useful to re-tell the analgous story with pseudo-connections that are not genuine connections.

By the very fact that $\mathbf{B}U(1) \stackrel{\simeq}{\leftarrow} \mathbf{B}_{diff}U(1)$ is a weak equivalence, it follows that every pseudo-connection is equivalent to an ordinary connection as cocoycles in $[CartSp^{op}, sSet](C(\{U_i\}), \mathbf{B}_{diff}(G))$.

If we choose a [[nLab:partition of unity]] $(\rho_i \in C^\infty(X,\mathbb{R}))$ subordinate to the cover $\{U_i \to X\}$, then we can construct the corresponding coboundary explicitly:

let $(A_i g_{ij}, a_{i j})$ be an arbitrary pseudo-connection cocycle. Consider the Cech-hypercohomology coboundary given by $(\sum_{i_0} \rho_{i_0} a_{i_0 i}, 0)$. This lands in the shifted cocycle

$$
  (A'_i := A_i + \sum_{i_0} \rho_{i_0} a_{i_0 i}, g_{i j}, a'_{i j})
  \,,
$$

and we can find the new pseudo-components $a'_{i j}$ by

$$
  a'_{i j} = A'_j - A'_i - d_{dR} g_{i j}
  \,.
$$

Using the computation

$$
  \begin{aligned}
    \sum_{i_0} \rho_{i_0} (a_{i_0 i} - a_{i_0 j}
    &=
    - \sum_{i_0} \rho_{i_0} (a_{i i_0} + a_{i_0 j}
    \\
    & = \sum_{i_0} \rho_{i_0} a_{i j}
    \\
    & = a_{i j}
  \end{aligned}
$$

we find that these indeed vanish.

The most drastic example for this is a lift $\nabla$ of a cocycle $g = (g_{i j})$ in

$$
  \array{
    && \mathbf{B}_{diff} U(1)
    \\
    & {}^{\mathllap{\nabla}}\nearrow & \downarrow
    \\
    C(\{U_i\}) &\stackrel{g}{\to}& \mathbf{B}U(1)
  }
$$

is one which takes all the ordinary curvature forms to vanish identically

$$
  \nabla = (A_i := 0, g_{i j}, a_{i j})
  \,.
$$

This fixes the pseudo-components to be $a_{i j} = - d g_{i j}$. By the above discussion, this pseudo-connection with vanishing connection 1-forms is equivalent, as a pseudo-connection, to the ordinary connection cocycle with connection forms $(A_i := \sum_{i_0} \rho_{i_0} d g_{i_0 i})$. This is a standard formula for equipping $U(1)$-principal bundles with Cech cocycle $(g_{i j})$ with a connection.

### $U(1)_0$-groupoid bundles {#U1GroupoidBundle}

We saw above that the intrinsic coefficient object $\mathbf{\flat}_{dR} \mathbf{B}^n U(1)$ yields ordinary de Rham cohomology in degree $n \gt 1$. For $n = 1$ we [have](#FlatCircleCohomology) that $\mathbf{\flat}_{dR} \mathbf{B}U(1)$ is given simply by the 0-[[nLab:truncated]] sheaf of 1-forms, $\Omega^1(-) : CartSp^{op} \to Set \hookrightarrow sSet$. Accordingly we have for $X$ a paracompact smooth manifold

$$
  \mathbf{H}(X, \mathbf{\flat}_{dR}\mathbf{B}U(1)) =
  \Omega^1_{cl}(X)
$$

instead of $H^1_{dR}(X)$.

There is a good reason for this discrepancy: for $n \geq 1$ the object $\mathbf{\flat}_{dR} \mathbf{B}^n U(1)$ is the recipient of the intrinsic curvature characteristic morphism

$$
  curv_{\mathbf{B}^{n-1} U(1)} : \mathbf{B}^{n-1} U(1) \to
   \mathbf{\flat}_{dR} \mathbf{B}^n U(1)
  \,.
$$

For $X \to \mathbf{B}^{n-1} U(1)$ a [[nLab:cocycle]] (an $(n-2)$-gerbe without connection), the cohomology class of the composite $X \to \mathbf{B}^{n-1} U(1) \to \mathbf{\flat}_{dR} \mathbf{B}^n U(1)$ is precisely the obstruction to the existence of a flat extension $X \to \mathbf{\flat} \mathbf{B}^{n-1} U(1) \to \mathbf{B}^{n-1} U(1)$ for the original cocycle.

For $n = 2$ this is the usual [[nLab:curvature]] 2-form of a [[nLab:line bundle]], for $n = 3$ it is curvature 3-form of a [[nLab:bundle gerbe]], etc. But for $n = 1$ we have that the original cocycle is just a map of spaces

$$
  f : X \to U(1)
  \,.
$$

This may be understood as a cocycle for a _groupoid_ principal bundle, for the 0-truncated groupoid with $U(1)$ as its space of objects. Such a cocycle extends to a flat cocycle precisely if $f$ is _constant_ as a function. The corresponding **curvature 1-form** is $d_{dR} f$ and this is precisely the obstruction to constancy of $f$ already, in that $f$ is constant if and only if $d_{dR} f$ vanishes. _Not_ (necessarily) if it vanishes _in de Rham cohomology_ .

This is the simplest example of a general statement about curvatures of higher bundles: the curvature 1-form is not subject to gauge transformations.

### Circle $n$-bundles with connection {#AbGerbesConnection}

We now generalize the [above discussion](#CircleBundlesConnection) on the derivation of the notion of [[nLab:connection on a bundle|connections on circle bundles]] from abstract topos-theory to a proof of the full [theorem above](#DeligneCohomologyTheorem) on the derivation of general [[nLab:Deligne cohomology]].

The main step is to model the double [[nLab:(∞,1)-pullback]]

$$
  \array{
    \mathbf{B}^n U(1) &\to& *
    \\
    \downarrow && \downarrow
    \\
    \mathbf{\flat}_{dR} \mathbf{B}^{n+1}U(1) &\to&
    \mathbf{\flat} \mathbf{B}^{n+1} U(1)
    \\
    \downarrow && \downarrow
    \\
    * &\to& \mathbf{B}^{n+1} U(1)
  }
$$

in $\mathbf{H} = $ [[Smooth∞Grpd]] that gives the  [[fiber sequence]] $\mathbf{\flat} \mathbf{B}^n U(1) \to \mathbf{B}^{n} U(1) \to \mathbf{\flat}_{dR} \mathbf{B}^{n+1} U(1)$ which controls the obstruction theory for flat connections by a [[nLab:homotopy pullback]] realized suitably as an ordinary pullback of fibrations in $[CartSp^{op}, Ch_\bullet] \stackrel{\Xi}{\hookrightarrow} [CartSp^{op}, sSet]_{proj}$.

+-- {: .num_observation}
###### Observation

We have commuting diagrams

$$
  \array{
    \Xi(\;
     0\stackrel{}{\to}
    {C^\infty(-,U(1)) \atop \oplus \Omega^1(-)}
    \stackrel{d_{dR} - Id}{\to}
    {\Omega^1(-) \atop \oplus \Omega^2(-)}
    \stackrel{d_{dR} + Id}{\to}
    \cdots
    \stackrel{d_{dR} \pm Id}{\to}
    \Omega^n(-)
    \;)
    &\to&
    \Xi(\;
     C^\infty(-,U(1)) \stackrel{Id + d_{dR}}{\to}
    {C^\infty(-,U(1)) \atop \oplus \Omega^1(-)}
    \stackrel{d_{dR} - Id}{\to}
    {\Omega^1(-) \atop \oplus \Omega^2(-)}
    \stackrel{d_{dR} + Id}{\to}
    \cdots
    { \Omega^{n-1}(-) \atop \oplus \Omega^n(-)}
    \stackrel{d_{dR} \pm Id}{\to}
    \Omega^n(-)
    \;)
    \\
    \downarrow && \downarrow^{\mathrlap{(Id, p_2, p_2, \cdots, p_2,d_{dR})}}
    \\
    \Xi(
    \; 0 \stackrel{}{\to} \Omega^1(-)
    \stackrel{d_{dR}}{\to}
    \Omega^2(-)
    \stackrel{d_{dR}}{\to} \cdots \stackrel{d_{dR}}{\to}
    \Omega_{cl}^{n+1}(-))
    &\to&
    (C^\infty(-,U(1)) \stackrel{d_{dR}}{\to} \Omega^1(-)
    \stackrel{d_{dR}}{\to}
    \Omega^2(-)
    \stackrel{d_{dR}}{\to}
    \cdots \stackrel{d_{dR}}{\to}
    \Omega_{cl}^{n+1}(-)
    \;)
    \\
    \downarrow && \downarrow
    \\
    \Xi(
     \; 0 \to 0 \to 0 \to \cdots \to 0\;)
    &\to&
    \Xi(
     \; C^\infty(-,U(1)) \to 0 \to 0 \to \cdots \to 0
     \;)
  }
$$

in $[CartSp^{op},sSet]_{proj}$ where

* the objects are fibrant models for the corresponding objects in the
  above $(\infty,1)$-pullback diagram;

* the two right vertical morphisms are fibrations;

* the two squares are [[nLab:pullback]] squares.

Therefore this is a [[nLab:homotopy pullback]] in $[CartSp^{op}, sSet]_{proj}$ that realizes the  $(\infty,1)$-pullback in question in the [[nLab:(∞,1)-category of (∞,1)-presheaves]] $PSh_{(\infty,1)}(CartSp)$. Since [[nLab:∞-stackification]] preserves finite [[nLab:(∞,1)-limit]]s, it therefore also presents the above $(\infty,1)$-pullback in $\mathbf{H} = Sh_{(\infty,1)}(CartSp)$.

=--

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

For the lower square we had discussed this already [above](#OrdinaryDeRham). For the upper square the same type of reasoning applies. The main point is to find the chain complex in the top right such that it is a [[nLab:resolution]] of the point and maps by a fibration onto our model for $\mathbf{\flat}\mathbf{B}^n U(1)$. The top right complex is

$$
  \array{
    &&
    C^\infty(-,U(1))
     &\stackrel{d_{dR}}{\to}&
    \Omega^1(-)
     &\stackrel{d_{dR}}{\to}&
    \Omega^2(-)
     &\stackrel{d_{dR}}{\to}&
     \cdots
     &\stackrel{d_{dR}}{\to}&
      \Omega^n(-)
     \\
     &{}^{\mathrlap{id}}\nearrow& \oplus
     &{}^{\mathrlap{id}}\nearrow& \oplus
     &{}^{\mathrlap{id}}\nearrow& \oplus
     &{}^{\mathrlap{id}}\nearrow& \cdots  &
     {}^{\mathrlap{id}}\nearrow&
     \\
    C^\infty(-,U(1))
     &\stackrel{d_{dR}}{\to}&
    \Omega^1(-)
     &\stackrel{d_{dR}}{\to}&
    \Omega^2(-)
     &\stackrel{d_{dR}}{\to}&
     \cdots
     &\stackrel{d_{dR}}{\to}&
      \Omega^n(-)
  }
$$

and the vertical map out of it into $C^\infty(-,U(1)) \stackrel{d_{dR}}{\to} \Omega^1(-) \stackrel{}{\to} \cdots \stackrel{d_{dR}}{\to} \Omega^n(-) \stackrel{d_{dR}}{\to} \Omega^{n+1}_{cl}(-)$ is in positive degree the projection onto the lower row and in degree 0 the de Rham differential. This is manifestly surjective (by the [[nLab:Poincare lemma]] applied to each object $U \in $ [[nLab:CartSp]]) hence this is a fibration.

The pullback object in the top left is in this notation

$$
  \mathbf{B}^n U(1)_{diff,chn}
  :=
  \Xi
  \left(
    \array{
      C^\infty(-,U(1))
       &\stackrel{d_{dR}}{\to}&
      \Omega^1(-)
       &\stackrel{d_{dR}}{\to}&
       \cdots
       &\stackrel{d_{dR}}{\to}&
        \Omega^n(-)
       \\
       \oplus
       &{}^{\mathrlap{id}}\nearrow& \oplus
       &{}^{\mathrlap{id}}\nearrow& \cdots  &
       {}^{\mathrlap{id}}\nearrow&
       \\
      \Omega^1(-)
       &\stackrel{d_{dR}}{\to}&
       \cdots
       &\stackrel{d_{dR}}{\to}&
        \Omega^n(-)
    }
  \right)
$$

and in turn the top left vertical morphism $curv : \mathbf{B}_{diff}^n U(1) \to \mathbf{\flat}_{dR} \mathbf{B}^{n+1}U(1)$ is in positive degree the projection on the lower row and in degree 0 the de Rham differential.

=--

Notice that the evident forgetful morphism $\mathbf{B}^n U(1) \stackrel{}{\leftarrow} \mathbf{B}^n_{diff} U(1)$ is indeed a weak equivalence.

With this description we now have the proof of the [above theorem](#DeligneCohomologyTheorem)

{#ProofOfDeligneTheorem}
+-- {: .proof}
###### Proof (equivalence of with Deligne cohomology)

Since the above model for $curv : \mathbf{B}_{diff}^n U(1) \to \mathbf{\flat}_{dR} \mathbf{B}^{n+1}U(1)$ is a fibration and $C(\{U_i\})$ is cofibrant, also

$$
  [Cartp^{op}, sSet](C(\{U_i\}), \mathbf{B}^n_{diff}U(1))
  \to
  [Cartp^{op}, sSet](C(\{U_i\}), \mathbf{\flat}_{dR}\mathbf{B}^n U(1))
$$

is a [[nLab:Kan fibration]] by the fact that $[CartSp^{op}, sSet]_{proj}$ is an $sSet_{Quillen}$-[[nLab:enriched model category]]. Therefore the homotopy pullback is computed as an ordinary pullback.

By the [above discussion of de Rham cohomology](#OrdinaryDeRham) we have that we can assume the morphism $H_{dR}^{n+1}(X) \to [CartSp^{op}, sSet](C(\{U_i\}), \mathbf{\flat}_{dR}\mathbf{B}^{n+1})$ picks only cocylces represented by globally defined closed differential forms $F \in \Omega^{n+1}(X)$.

By the nature of the chain complexes apearing in the above proof, we see that the elements inm the fiber over such a globally defined form are precisely the cocycles with values only in the "upper row complex"

$$
  C^\infty(-,U(1)) \stackrel{d_{dR}}{\to} \Omega^1(-)
  \stackrel{d_{dR}}{\to}
  \cdots
  \stackrel{d_{dR}}{\to}
  \Omega^n(-)
  \,.
$$

This is precisely the complex of sheaves that defines [[nLab:Deligne cohomology]] in degree $(n+1)$.

=--

### Models from $\infty$-Lie integration {#U1FromLieIntegration}

In the [previous section](#FlatCircleCohomology) we discussed a model

$$
  \array{
    \mathbf{B}^n U(1)_{diff,chn}
    &\stackrel{curv_{chn}}{\to}&
    \mathbf{\flat}_{dR}\mathbf{B}^{n+1} U(1)_{chn}
    \\
    \downarrow^{\mathrlap{\simeq}}
    \\
    \mathbf{B}^n U(1)
  }
$$

in $[CartSp^{op}, sSet]$ for the canoncal curvature characteristic class $curv : \mathbf{B}^n U(1) \to \mathbf{\flat}_{dR} \mathbf{B}^{n+1}U(1)$ in [[Smooth∞Grpd]] with the special property that it did model the abstract [[(∞,1)-topos]]-theoretic class under the [[Dold-Kan correspondence]] precisely in terms of the familiar [[Deligne cohomology]] coefficient complex.

There is another model for the curvature class in $[CartSp^{op}, sSet]$, one that is useful for constructing the [∞-Chern-Weil homomorphism](#InfChernWeil) that maps from [[nonabelian cohomology]] in $Smooth \infty Grpd$ to $U(1)$-valued differential cohomology. This second model is the one naturally adapted to the construction of the object $\mathbf{B}^n U(1)$ by [[Lie integration]] from its [[∞-Lie algebra]] $b^{n-1} \mathbb{R}$. This is described at ∞-Lie groupoid -- Lie integration.

For distinguishing the two models, we will indicate the former one by the subscript ${}_{chn}$ and the one described now by the subscript ${}_{simp}$.

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

Here and in the following we adopt for differential forms on simplices the following notational convention:

* by $\Omega^\bullet(\Delta^n)$ we denote the complex of smooth differential forms on the standard smooth $n$-simplex with **sitting instants**: for every $k \in \mathbb{N}$ every $k$-face of $\Delta^n$ has a neighbourhood of its boundary such that the form restricted to that neighbourhood is constant in the direction perpendicular to that boundary.

* for $U \in CartSp$ we write $\Omega^\bullet(U \times \Delta^k)_{vert}$ for the complex of [[vertical differential form]]s with respect to the trivial simplex bundle $U \times \Delta^k \to U$.

=--

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

For $n \in \mathbb{N}$, define the [[simplicial presheaf]] $\mathbf{B}^n U(1)_{simp} \in [CartSp^{op}, sSet]$ by

$$
  \mathbf{B}^n U(1)_{simp}
  :=
  \mathbf{cosk}_{n+1}
  (
  (U, [k]) \mapsto
   Hom_{dgAlg}(
     CE(b^{n-1}\mathbb{R}),
     \Omega^\bullet(U \times \Delta^k)_{vert})
  )
  /\mathbf{B}^n \mathbb{Z}
  \,.
$$

Here $CE(b^{n-1}\mathbb{R})$ is the [[Chevalley-Eilenberg algebra]] of $b^{n-1}\mathbb{R}$, which is simply the graded-commutative [[dg-algebra]] (over $\mathbb{R}$) on a single generator in degree $n$ with vanishing differential.

Moreover, $\mathbf{cosk}_{n+1}(-)$ is the [[coskeleton]]-operation and the quotient is by constant $n$-forms $\omega \in \Omega^n_{cl}(U \times \Delta^k)_{vert}$ such that $\int_{\Delta^n}\omega \in \mathbb{Z}$. We take the quotient as a quotient of abelian [[simplicial group]]s (the group operation is the addition of differential forms).

=--

+-- {: .num_lemma }
###### Observation

Under the [Dold-Kan correspondence](#DoldKan) the [[Moore complex|normalized chain complex]] of $\mathbf{B}^n U(1)_{sim}$ is

$$
  N_\bullet(\mathbf{B}^n U(1)_{simp})
  =
  (
  \cdots
  \to
  \Omega^n_{cl}((-)\times \Delta^{n+1})_{vert}/\sim
  \stackrel{\sum_k (-1)^k \partial_k^* }{\to}
  \Omega^n_{cl}((-)\times \Delta^{n})_{vert}/\sim
  \to
  0
  \to
  \cdots
  \to
  0
)
  \,,
$$

where $\partial_k : \Delta^n \to \Delta^{n+1}$ denotes the embedding of the $k$th face of the smooth $(n+1)$-[[simplex]].

=--

Here and in the following we indicate the homologically trivial part of the normalized chain complex of an $(n+1)$-coskeletal simplicial abelian group just by ellipses.

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

The evident [[fiber integration]] of differential forms over simplices

$$
  \int_{\Delta^n} : \Omega^\bullet(U \times \Delta^n)
   \to \Omega^\bullet(U)
$$

yields a morphism

$$
  \int_{\Delta^\bullet} : \mathbf{B}^n U(1)_{simp} \stackrel{\simeq}{\to}
  \mathbf{B}^n U(1)
$$

in $[CartSp^{op}, sSet]_{proj}$, which is a weak equivalence.

=--

This is discussed at [[Lie integration]].

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

Write

$\mathbf{\flat} \mathbf{B}^n \mathbb{R}_{simp}$ for the [[simplicial presheaf]]

$$
  \mathbf{\flat}\mathbf{B}^n \mathbb{R}_{simp}
  :=
  \mathbf{cosk}_{n+1}
  (
    (U,[k])
    \mapsto
    Hom_{dgAlg}(
      CE(b^{n-1}\mathbb{R}),
      \Omega^{\bullet}(U \times\Delta^k)
    )
  )
$$

and write $\mathbf{\flat}_{dR}\mathbf{B}^n \mathbb{R}_{simp} \in [CartSp^{op}, sSet]$ for the simplicial presheaf

$$
  \mathbf{\flat}_{dR}\mathbf{B}^n \mathbb{R}_{simp}
  :=
  \mathbf{cosk}_{n+1}
  (
    (U,[k])
    \mapsto
    Hom_{dgAlg}(
      CE(b^{n-1}\mathbb{R}),
      \Omega^{\bullet \geq 1, \bullet}(U \times\Delta^k)
    )
  )
  \,,
$$

where on the right we have the subcomplex of $\Omega^\bullet(U \times \Delta^k)$ on those forms that are non-vanishing on some vector field tangent to $U$.

=--

At ∞-Lie groupoid -- Lie integrated ∞-groups -- Differential coefficients the following is shown:

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

The evident [[fiber integration]] over simplices induces morphisms of simplicial presheaves

$$
  \int_{\Delta^\bullet} :
  \mathbf{\flat} \mathbf{B}^n U(1)_{simp}
  \to
  \mathbf{\flat} \mathbf{B}^n U(1)_{chn}
$$

and

$$
  \int_{\Delta^\bullet} :
  \mathbf{\flat}_{dR} \mathbf{B}^n U(1)_{simp}
  \to
  \mathbf{\flat}_{dR} \mathbf{B}^n U(1)_{chn}
$$

that are weak equivalences in $[CartSp^{op}, sSet]_{proj}$.

=--

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

Write $\mathbf{B}^n  U(1)_{diff,simp}$ for the simplicial presheaf given by

$$
  \mathbf{B}^n  U(1)_{diff,simp}
  :=
  \mathbf{cosk}_{n+1}
  (
    U,[k]
    \mapsto
   \left\{
     \array{
        \Omega^\bullet(U \times \Delta^k)_{vert}
        &\leftarrow&
        CE(b^{n-1}\mathbb{R})
        \\
        \uparrow && \uparrow
        \\
        \Omega^\bullet(U \times \Delta^k )
        &\leftarrow&
        W(b^{n-1} \mathbb{R})
     }
   \right\}
  ) / \mathbf{B}^n \mathbb{Z}
  \,.
$$

Let the morphism

$$
  curv_{simp}
  :
  \mathbf{B}^n U(1)_{diff,simp}
  \to
  \mathbf{\flat}_{dR}\mathbf{B}^{n+1} U(1)_{simp}
$$

be the one given by postcomposition with the square of [[nLab:dg-algebra]]s

$$
  \array{
     CE(b^{n-1}\mathbb{R}) &\leftarrow& 0
     \\
     \uparrow && \uparrow
     \\
     W(b^{n-1}\mathbb{R}) &\leftarrow& CE(b^n \mathbb{R})
  }
$$

described at [[nLab:∞-Lie algebra cohomology]].

=--

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

The set of square diagrams of [[dg-algebra]]s above is over $(U,[k])$ the set of $n$-forms $\omega$ on $U \times \Delta^k$ whose [[curvature]] $(n+1)$-form $d \omega$ has no component with all legs along $\Delta^k$.

=--

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

The morphism given by [[fiber integration]] of differential forms over the simplex factor fits into a diagram

$$
  \array{
    \mathbf{B}^n U(1)_{diff,simp} &\stackrel{curv_{simp}}{\to}& \mathbf{\flat}_{dR}\mathbf{B}^{n+1}U(1)_{simp}
    \\
    \downarrow^{\mathrlap{\int}_{\Delta^\bullet}} &&
    \downarrow^{\mathrlap{\int}_{\Delta^\bullet}}
    \\
    \mathbf{B}^n U(1)_{diff,chn} &\stackrel{curv_{chn}}{\to}& \mathbf{\flat}_{dR}\mathbf{B}^{n+1}U(1)_{chn}
  }
  \,,
$$

where the vertical morphisms are weak equivalences.

=--

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

Fiber integration induces a weak equivalence

$$
  \int_{\Delta^\bullet} :
  \mathbf{B}^n \mathbb{R}_{diff,simp}
  \stackrel{\simeq}{\to}
  \mathbf{B}^n \mathbb{R}_{diff, chn}
$$

=--

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

Observe that $\mathbf{B}^n \mathbb{R}_{diff,simp}$ is the [[pullback]] of $\mathbf{\flat}_{dR} \mathbf{B}^{n+1}\mathbb{R}_{simp} \to \mathbf{\flat}\mathbf{B}^{n+1} \mathbb{R}_{simp}$ along the evident forgetful morphism from

$$
  (U,[k]) \mapsto \{\Omega^\bullet(U \times \Delta^k) \leftarrow
  W(b^{n-1} \mathbb{R})\}
  \,.
$$

This forgetful morphism is evidently a fibration (because it is a degreewise surjection under Dold-Kan), hence this pullback models the [[homotopy fiber]] of $\mathbf{\flat}_{dR} \mathbf{B}^{n+1} \mathbb{R} \to \mathbf{\flat} \mathbf{B}^{n+1} \mathbb{R}$. Since by the above fiber integration gives a weak equivalence of pulback diagrams the claim follows.

=--

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

Write $\mathbf{B}^n U(1)_{conn,simp} \hookrightarrow \mathbf{B}^n U(1)_{diff,simp}$ for the sub-presheaf which over $(U,[k])$ is the set of those forms $\omega$ on $U \times \Delta^k$ such that the [[curvature]] $d \omega$ has no leg along $\Delta^k$.

=--

+-- {: .num_cor }
###### Corollary

Under fiber integration over simplices, $\mathbf{B}^n U(1)_{conn,simp}$ is [[quasi-isomorphism|quasi-isomorphic]] to the [[Deligne cohomology]]-complex.

=--

$$
  \array{
     &&
     \mathbf{B}^n U(1)_{conn,simp}
     &\stackrel{\int_{\Delta^\bullet}}{\to^\simeq}&
     U(1)(n)_D^\infty
     &&&
     connection
     \\
     & \nearrow & \downarrow && \downarrow
     \\
     \hat X &\to&
     \mathbf{B}^n U(1)_{diff,simp}
     &\stackrel{\int_{\Delta^\bullet}}{\to^\simeq}&
     \mathbf{B}^n U(1)_{diff,chn}
     &&&
     pseudo-connection
     \\
     & \searrow & \downarrow && \downarrow
     \\
     && \mathbf{\flat}_{dR}\mathbf{B}^{n+1}U(1)_{simp}
      &\stackrel{\int_{\Delta^\bullet}}{\to^\simeq}&
     \mathbf{\flat}_{dR}\mathbf{B}^{n+1}U(1)_{chn}
     &&&
     curvature
  }
  \,.
$$

In summary this gives us the following alternative perspective on connections on $\mathbf{B}^{n-1}U(1)$-[[principal ∞-bundle]]s: such a connection is a cocycle with values in the $\mathbf{B}^n \mathbb{Z}$-quotient of the  $(n+1)$-coskeleton of the simplicial presheaf which over $(U,[k])$ is the set of diagrams of [[dg-algebra]]s

$$
  \array{
    C^\infty(U)\otimes \Omega^\bullet(\Delta^k)
     &\leftarrow&
    CE(b^{n-1}\mathbb{R})
    &&&
    underlying\;cocycle
    \\
    \uparrow && \uparrow
    \\
    \Omega^\bullet(U) \otimes \Omega^\bullet(\Delta^k)
     &\leftarrow&
    W(b^{n-1}\mathbb{R})
    &&&
    connection
    \\
    \uparrow && \uparrow
    \\
    \Omega^\bullet(U)\otimes C^\infty(\Delta^k) &\leftarrow& CE(b^n \mathbb{R})
    &&&
    curvature
  }
$$

where the restriction to the top morphism is the underlying cocycle and the restriction to the bottom morphism the curvature form.

The generalization to such diagram cocycles from $b^{n-1}\mathbb{R}$ to general [[nLab:∞-Lie algebra]]s $\mathfrak{g}$ we discuss below in [∞-Lie algebra valued connections](#InfinityLieAlgebraConnection).

### In homotopy type theory
 {#InHomtopyTypeTheory}

We discuss the formulation of the above in the [[homotopy type theory]]-[[internal language]] of the [[(∞,1)-topos]] $\mathbf{H} = $ [[Smooth∞Grpd]].

Given the two functions

$$
  \Omega^{(n+1)}_{cl} \to \mathbf{\flat}_{dR}\mathbf{B}^{n+1} U(1)
$$

(inclusion of the set of closed $(n+1)$-forms into the $(n+1)$-groupoid of de Rham cocycles)

and

$$
  curv : \mathbf{B}^n U(1) \to \mathbf{\flat}_{dR} \mathbf{B}^{n+1} U(1)
$$

(the universal curvature class / Maurer-Cartan form of the circle $(n-1)$-group)

the [[smooth ∞-groupoid|smooth]] [[moduli ∞-stack]] of circle $n$-bundles with connection from [above](#DifferentialCohomology) is expressed in [[homotopy type theory]] as

$$
  \mathbf{B}^n U(1)_{conn}
  \simeq
  \left\{
    P  : \mathbf{B}^n U(1) | \exists F \in \Omega^{n+1}_{cl} . curv(P) = F
  \right\}
  \,.
$$

Spelled out this expresses $\mathbf{B}^n U(1)_{conn}$  as

* the [[dependent sum]] over $\mathbf{B}^n U(1)$ of

* the [[dependent sum]] over $\Omega^n_{cl}$ of

* the [[substitution]] by $curv$ of

* the [[dependent type|dependent]] [[identity type]] on $\mathbf{\flat}_{dR} \mathbf{B}^{n+1} U(1)$.

See the discussion at _[[homotopy pullback]]_ for why this is indeed interpreted by the homotopy pullback $\mathbf{B}^n U(1) \times_{\mathbf{\flat}_{dR} \mathbf{B}^{n+1}U(1)} \Omega^{n+1}_{cl}$.

## Examples

For $n = 1$ a circle $n$-bundle with connection in the sense discussed here is indeed an ordinary hermitian [[line bundle]] or equivalently $U(1)$-[[principal bundle]] with connection.

For $n = 2$ a circle 2-bundle with connection is equivalent to a [[bundle gerbe]] with connection (at least over a smooth manifold. Over an [[orbifold]] the definition given here does produce the correct [[equivariant cohomology]], which is different from that of bundle gerbes that are equivariant in the ordinary sense.)

Classes of examples of higher circle bundles with connection are provided by [[∞-Chern-Weil theory]] which provides homomorphisms of the form

$$
  \mathbf{H}_{conn}(X,\mathbf{B}G) \to \mathbf{H}_{diff}(X, \mathbf{B}^n U(1))
  \,.
$$

See for instance

* [[Chern-Simons circle 3-bundle]]

for the class of circle 3-bundles that arise as differential refinements of degree 4 [[characteristic class]]es such as the [[Pontryagin class]].

* [[Chern-Simons circle 7-bundle]].

## Properties

### Moduli

[[!include moduli of higher lines -- table]]

## Related concepts

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

* [[contact geometry]]

* [[higher U(1)-gauge theory]]

* [[principal infinity-connection]]

* [[curving]]. [[higher Courant groupoid]]

* [[prequantum n-bundle]], [[quantomorphism n-group]]

* [[Super Gerbes]]

## References

The above discussion is from

* {#dcct} [[Urs Schreiber]], _[[schreiber:differential cohomology in a cohesive topos]]_ .

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

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

[[!redirects circle n-bundles with connection]]
[[!redirects circle n-bundles with connections]]

[[!redirects circle n-bundle]]
[[!redirects circle n-bundles]]

[[!redirects line bundle with connection]]
[[!redirects line bundles with connection]]
[[!redirects line n-bundle with connection]]
[[!redirects line n-bundles with connection]]

[[!redirects circle bundle with connection]]
[[!redirects circle bundles with connection]]
[[!redirects circle-bundle with connection]]
[[!redirects circle-bundles with connection]]

[[!redirects circle 2-bundle with connection]]
[[!redirects circle 2-bundles with connection]]

[[!redirects circle 3-bundle with connection]]
[[!redirects circle 3-bundles with connection]]

[[!redirects circle n-connection]]
[[!redirects circle n-connections]]


circle n-group

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Group Theory
+-- {: .hide}
[[!include group theory - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

For $\mathbf{H}$ a [[cohesive (∞,1)-topos]] such as [[ETop∞Grpd]] or [[Smooth∞Grpd]], both the [[natural number]]s $\mathbb{Z}$ and the [[real number]]s are naturally [[abelian group|abelian]] [[group object]]s in $\mathbf{H}$. Accordingly their [[quotient]]

$$
 U(1) := \mathbb{R}/\mathbb{Z}
$$

under the canonical embedding $\mathbb{Z} \hookrightarrow \mathbb{R}$ exists in $\mathbf{H}$ and is an abelian group object: the [[circle group]]. Therefore for all $n \in \mathbb{N}$ the [[delooping]]

$$
  \mathbf{B}^n U(1) \in \mathbf{H}
$$

exists and has the structure of an abelian [[n-group|(n+1)-group]] object. This is the topological or smooth, respectively, **circle $(n+1)$-group** .

## Definition

Details for the smooth case are at _[[smooth ∞-groupoid]]_ in the section [circle Lie n-group](smooth+infinity-groupoid#CircleLienGroup).

## Examples

For $n = 1$ the **circle 2-group** $\mathbf{B}U(1)$ can be identified with the [[strict 2-group]] whose corresponding [[crossed module]] of groups is simply $[U(1) \to 1]$.

Generally, for any $n$ $\mathbf{B}^{n-1}U(1)$ is an [[n-group]] that corresponds under the [[Dold-Kan correspondence]] to the [[chain complex]] or [[crossed complex]] of groups $U(1)[n]$ concentrated in degree $n$.

## Properties

The [[geometric realization]] of the circle $n$-group is the [[Eilenberg-MacLane space]]

$$
  |\mathbf{B}^{n-1} U(1)|
  \simeq
  B^{n-1} U(1)
  \simeq
  B^{n} \mathbb{Z}
  \simeq
  K(\mathbb{Z}, n)
 \,.
$$

## Related concepts

A circle $n$-group-[[principal ∞-bundle]] is a [[circle n-bundle]], equivalently an  $(n-1)$-[[bundle gerbe]].

[[!redirects circle n-groups]]

[[!redirects circle 2-group]]
[[!redirects circle 2-groups]]

[[!redirects circle 3-group]]
[[!redirects circle 3-groups]]

[[!redirects circle Lie n-group]]
[[!redirects circle Lie n-groups]]


circle object

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### 2-Category Theory
+--{: .hide}
[[!include higher category theory - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

The semantics of the [[circle type]] in (2,1)-category theory.

## Definition

In a weak (2,1)-category $C$ with [[terminal object]] $*$, a **circle object** is an object $S^1$ in $C$ with a [[global element]] $\beta:* \to S^1$ and an [[equivalence]] $\lambda:\beta \cong \beta$ such that for every other object $A$ in $C$ with a [[global element]] $\beta_A:* \to A$ and an [[equivalence]] $\lambda_A:\beta_A \cong \beta_A$, there is a [[functor]] $f:S^1 \cong A$ and a functor $f^{'}:(\beta \cong \beta) \to (\beta_A \cong \beta_A)$ and [[equivalences]] $p:f \circ \beta \cong \beta_A$ and $q:f^{'} \circ \lambda \cong \lambda_A \circ f^{'}$ satisfying [[coherence laws]].

## Properties

The [[loop space object]] of the object $S^1$ with global element $\beta$ the circle object is equivalent to an [[integers object]] $Z$.

## See also

* [[circle type]]

* [[integers object]]


circle principal bundle

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Bundles
+-- {: .hide}
[[!include bundles - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Definition

A [[circle group]]-[[principal bundle]].

## Properties

### Relation to complex line bundles

Under the standard [[linear representation]] of $U(1)$ on the [[complex numbers]] the [[associated bundle]] to a $U(1)$-principal bundle is a [[complex line bundle]].

## Related concepts

* [[circle action]]

* [[circle n-bundle with connection]]

* the [[classifying space]] is infinite [[complex projective space]]

* [[complex oriented cohomology]]

[[!redirects circle principal bundles]]


circle type

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Type theory
+-- {: .hide}
[[!include type theory - contents]]
=--
=--
=--

\tableofcontents

## Idea

The _circle type_ is an [[axiom|axiomatization]] of the [[homotopy type]] of the ([[shape modality|shape]] of) the [[circle]] in the context of [[homotopy type theory]].

## Definition

There are two ways to define the circle type: one can define it in terms of explicit [[inference rules]], or one can use the notion of [[higher inductive type]].  We will discuss both formulations.

* [with inference rules](#ExplicitDefinition).

* [in terms of higher inductive types](#InTermsOfHigherInductiveTypes)

### With inference rules
{#ExplicitDefinition}

Let $a =_A b$ denote the [[identification type]] between elements $a:A$ and $b:A$ of type $A$, and let $y =_{x:A.B(x)}^{(a, b, p)} z$ denote the [[heterogeneous identification type]] between elements $y:B(a)$ and $z:B(b)$ of type family $x:A \vdash B(x)$, given elements $a:A$ and $b:A$ and [[identification]] $p:a =_A b$. Let $\mathrm{apd}_f(a, b, p)$ denote the [[dependent function application]] of the [[dependent function]] $f:\prod_{x:A} B(x)$ to the [[identification]] $p:a =_A b$

In the [[natural deduction]] formulation of [[dependent type theory]], the circle type is given by the following [[inference rules]]:

First the rule that defines the circle type itself in some context $\Gamma$.

**[[type formation]]**
$$\frac{\Gamma \; \mathrm{ctx}}{\Gamma \vdash S^1 \; \mathrm{type}}$$

Now the basic “introduction” rule, which says that the circle type consists of a base point $\mathrm{base}:S^1$ and a loop identification $\mathrm{loop}:\mathrm{base} =_{S^1} \mathrm{base}$.

**[[term introduction]]**:
$$\frac{\Gamma \; \mathrm{ctx}}{\Gamma \vdash \mathrm{base}:S^1} \qquad \frac{\Gamma \; \mathrm{ctx}}{\Gamma \vdash \mathrm{loop}:\mathrm{base} =_{S^1} \mathrm{base}}$$

#### Induction principle

There are different induction principles associated with the [[circle type]], depending on whether [[judgmental equalities]] or [[identifications]] are used; if the former are used, this results in **strict circle types**, and if the latter are used, this results in **weak circle types**.

The induction principle for strict circle types states that if:

1. For all $x:S^1$ we have a type $C(x)$

1. For any term $c_\mathrm{base}:C(\mathrm{base})$ and any heterogeneous identification $c_\mathrm{loop}:c_\mathrm{base} =_{x:S^1.C(x)}^{(\mathrm{base}, \mathrm{base}, \mathrm{loop})} c_\mathrm{base}$ that $c_\mathrm{base}$ is equal to itself across $\mathrm{loop}$

then we have a dependent function $c:\prod_{x:S^1} C(x)$ such that evaluating $c$ at $\mathrm{base}$ results in $c_\mathrm{base}$ and applying $c$ across $\mathrm{loop}$ results in $c_\mathrm{loop}$.

$$c(\mathrm{base}) \equiv c_\mathrm{base} \qquad \mathrm{apd}_c(\mathrm{base}, \mathrm{base}, \mathrm{loop}) \equiv c_\mathrm{loop}$$

The induction principle for weak circle types states that if:

1. For all $x:S^1$ we have a type $C(x)$

1. For any term $c_\mathrm{base}:C(\mathrm{base})$ and any heterogeneous identification $c_\mathrm{loop}:c_\mathrm{base} =_{x:S^1.C(x)}^{(\mathrm{base}, \mathrm{base}, \mathrm{loop})} c_\mathrm{base}$ that $c_\mathrm{base}$ is equal to itself across $\mathrm{loop}$

there exists a dependent function $c:\prod_{x:S^1} C(x)$ and an identification $p:c(\mathrm{base}) =_{C(\mathrm{base})} c_\mathrm{base}$ such that applying $c$ across $\mathrm{loop}$ is equal to $c_\mathrm{loop}$ across $p$.

$$\mathrm{ap}_c(\mathrm{base}, \mathrm{base}, \mathrm{loop}) =_{z:C(\mathrm{base}).z =_{x:S^1.C(x)}^{(\mathrm{base}, \mathrm{base}, \mathrm{loop})} z}^{(c(\mathrm{base}), c_\mathrm{base}, p)} c_\mathrm{loop}$$

These are both formalized via the elimination and computation rules for the circle type:

Elimination rules for the circle type:
$$\frac{\Gamma, x:S^1 \vdash C(x) \; \mathrm{type} \quad \Gamma \vdash c_\mathrm{base}:C(\mathrm{base}) \quad \Gamma \vdash c_\mathrm{loop}:c_\mathrm{base} =_{x:S^1.C(x)}^{(\mathrm{base}, \mathrm{base}, \mathrm{loop})} c_\mathrm{base}}{\Gamma \vdash \mathrm{ind}_{S^1}(c_\mathrm{base}, c_\mathrm{loop}):\prod_{x:S^1} C(x)}$$

**Computation rules for the strict circle type**
$$\frac{\Gamma, x:S^1 \vdash C(x) \; \mathrm{type} \quad \Gamma \vdash c_\mathrm{base}:C(\mathrm{base}) \quad \Gamma \vdash c_\mathrm{loop}:c_\mathrm{base} =_{x:S^1.C(x)}^{(\mathrm{base}, \mathrm{base}, \mathrm{loop})} c_\mathrm{base}}{\Gamma \vdash \mathrm{ind}_{S^1}(c_\mathrm{base}, c_\mathrm{loop})(\mathrm{base}) \equiv c_\mathrm{base}:C(\mathrm{base})}$$

* Judgmental path constructors

$$\frac{\Gamma, x:S^1 \vdash C(x) \; \mathrm{type} \quad \Gamma \vdash c_\mathrm{base}:C(\mathrm{base}) \quad \Gamma \vdash c_\mathrm{loop}:c_\mathrm{base} =_{x:S^1.C(x)}^{(\mathrm{base}, \mathrm{base}, \mathrm{loop})} c_\mathrm{base}}{\Gamma \vdash \mathrm{apd}_{\mathrm{ind}_{S^1}(c_\mathrm{base}, c_\mathrm{loop})}(\mathrm{base}, \mathrm{base}, \mathrm{loop}) \equiv c_\mathrm{loop}:c_\mathrm{base} =_{x:S^1.C(x)}^{(\mathrm{base}, \mathrm{base}, \mathrm{loop})} c_\mathrm{base}}$$

* Propositional path constructors

$$\frac{\Gamma, x:S^1 \vdash C(x) \; \mathrm{type} \quad \Gamma \vdash c_\mathrm{base}:C(\mathrm{base}) \quad \Gamma \vdash c_\mathrm{loop}:c_\mathrm{base} =_{x:S^1.C(x)}^{(\mathrm{base}, \mathrm{base}, \mathrm{loop})} c_\mathrm{base}}{\Gamma \vdash \beta_{S^1}^{\mathrm{loop}}(c_\mathrm{base}, c_\mathrm{loop}):\mathrm{apd}_{\mathrm{ind}_{S^1}(c_\mathrm{base}, c_\mathrm{loop})}(\mathrm{base}, \mathrm{base}, \mathrm{loop}) =_{c_\mathrm{base} =_{x:S^1.C(x)}^{(\mathrm{base}, \mathrm{base}, \mathrm{loop})} c_\mathrm{base}} c_\mathrm{loop}}$$

**Computation rules for the weak circle type**
$$\frac{\Gamma, x:S^1 \vdash C(x) \; \mathrm{type} \quad \Gamma \vdash c_\mathrm{base}:C(\mathrm{base}) \quad \Gamma \vdash c_\mathrm{loop}:c_\mathrm{base} =_{x:S^1.C(x)}^{(\mathrm{base}, \mathrm{base}, \mathrm{loop})} c_\mathrm{base}}{\Gamma \vdash \beta_{S^1}^{\mathrm{base}}(c_\mathrm{base}, c_\mathrm{loop}):\mathrm{ind}_{S^1}(c_\mathrm{base}, c_\mathrm{loop})(\mathrm{base}) =_{C(\mathrm{base})} c_\mathrm{base}}$$

$$\frac{\Gamma, x:S^1 \vdash C(x) \; \mathrm{type} \quad \Gamma \vdash c_\mathrm{base}:C(\mathrm{base}) \quad \Gamma \vdash c_\mathrm{loop}:c_\mathrm{base} =_{x:S^1.C(x)}^{(\mathrm{base}, \mathrm{base}, \mathrm{loop})} c_\mathrm{base}}{\Gamma \vdash \beta_{S^1}^{\mathrm{loop}}(c_\mathrm{base}, c_\mathrm{loop}):\mathrm{apd}_{\mathrm{ind}_{S^1}(c_\mathrm{base}, c_\mathrm{loop})}(\mathrm{base}, \mathrm{base}, \mathrm{loop}) =_{z:C(\mathrm{base}).z =_{x:S^1.C(x)}^{(\mathrm{base}, \mathrm{base}, \mathrm{loop})} z}^{(\mathrm{ind}_{S^1}(c_\mathrm{base}, c_\mathrm{loop})(\mathrm{base}), c_\mathrm{base}, \beta_{S^1}^{\mathrm{base}}(c_\mathrm{base}, c_\mathrm{loop}))} c_\mathrm{loop}}$$

For weak circle types, we can package the elimination and propositional computation rules together using [[dependent sum types]] to get a single rule for the induction principle of the circle type:

$$\frac{\Gamma, x:S^1 \vdash C(x) \; \mathrm{type} \quad \Gamma \vdash c_\mathrm{base}:C(\mathrm{base}) \quad \Gamma \vdash c_\mathrm{loop}:c_\mathrm{base} =_{x:S^1.C(x)}^{(\mathrm{base}, \mathrm{base}, \mathrm{loop})} c_\mathrm{base}}{\Gamma \vdash \mathrm{ind}_{S^1}^{x:S^1.C(x)}(c_\mathrm{base}, c_\mathrm{loop}):\sum_{c:\prod_{x:S^1} C(x)} \sum_{p:c(\mathrm{base}) =_{C(\mathrm{base})} c_\mathrm{base}} \mathrm{apd}_c(\mathrm{base}, \mathrm{base}, \mathrm{loop}) =_{z:C(\mathrm{base}).z =_{x:S^1.C(x)}^{(\mathrm{base}, \mathrm{base}, \mathrm{loop})} z}^{(c(\mathrm{base}), c_\mathrm{base}, p)} c_\mathrm{loop}}$$

Since the [[circle type]] is a [[positive type]], it is not necessary to include a [[uniqueness rule]] for the induction principle circle types, since the propositional uniqueness rule can be proven from the the other inference rules for the circle type. Nevertheless, one can include the uniqueness rule in the combined single induction rule by turning the dependent sum type into a [[uniqueness quantifier]], resulting in the **dependent universal property of the circle type**.

$$\frac{\Gamma, x:S^1 \vdash C(x) \; \mathrm{type} \quad \Gamma \vdash c_\mathrm{base}:C(\mathrm{base}) \quad \Gamma \vdash c_\mathrm{loop}:c_\mathrm{base} =_{x:S^1.C(x)}^{(\mathrm{base}, \mathrm{base}, \mathrm{loop})} c_\mathrm{base}}{\Gamma \vdash \mathrm{ind}_{S^1}^{x:S^1.C(x)}(c_\mathrm{base}, c_\mathrm{loop}):\exists!c:\prod_{x:S^1} C(x).\sum_{p:c(\mathrm{base}) =_{C(\mathrm{base})} c_\mathrm{base}} \mathrm{apd}_c(\mathrm{base}, \mathrm{base}, \mathrm{loop}) =_{z:C(\mathrm{base}).z =_{x:S^1.C(x)}^{(\mathrm{base}, \mathrm{base}, \mathrm{loop})} z}^{(c(\mathrm{base}), c_\mathrm{base}, p)} c_\mathrm{loop}}$$

#### Recursion principle

In general, given a type $C$, by the [[weakening rule]] of [[dependent type theory]], one can form the constant type family $C$ indexed by $x:S^1$; then one can get the recursion principle of the circle type from the induction principle on $C$ regarded as a constant type family:

The recursion principle for strict circle types states that if:

1. We have a type $C$

1. For any term $c_\mathrm{base}:C$ and any identification $c_\mathrm{loop}:c_\mathrm{base} =_{C} c_\mathrm{base}$ that $c_\mathrm{base}$ is equal to itself

then we have a function $c:S^1 \to C$ such that evaluating $c$ at $\mathrm{base}$ results in $c_\mathrm{base}$ and applying $c$ across $\mathrm{loop}$ results in $c_\mathrm{loop}$.

$$c(\mathrm{base}) \equiv c_\mathrm{base} \qquad \mathrm{ap}_c(\mathrm{base}, \mathrm{base}, \mathrm{loop}) \equiv c_\mathrm{loop}$$

The recursion principle for weak circle types states that if:

1. We have a type $C$

1. For any term $c_\mathrm{base}:C$ and any identification $c_\mathrm{loop}:c_\mathrm{base} =_{C} c_\mathrm{base}$ that $c_\mathrm{base}$ is equal to itself

there exists a function $c:S^1 \to C$ and an identification $p:c(\mathrm{base}) =_{C(\mathrm{base})} c_\mathrm{base}$ such that applying $c$ across $\mathrm{loop}$ is equal to $c_\mathrm{loop}$ across $p$.

$$\mathrm{ap}_c(\mathrm{base}, \mathrm{base}, \mathrm{loop}) =_{z:C.z =_{C} z}^{(c(\mathrm{base}), c_\mathrm{base}, p)} c_\mathrm{loop}$$

The recursion principle for weak circle types can be presented as a single rule via the use of a [[dependent sum type]]:

$$\frac{\Gamma \vdash C \; \mathrm{type} \quad \Gamma \vdash c_\mathrm{base}:C \quad \Gamma \vdash c_\mathrm{loop}:c_\mathrm{base} =_{C} c_\mathrm{base}}{\Gamma \vdash \mathrm{rec}_{S^1}^{C}(c_\mathrm{base}, c_\mathrm{loop}):\sum_{c:S^1 \to C} \sum_{p:c(\mathrm{base}) =_{C(\mathrm{base})} c_\mathrm{base}} \mathrm{ap}_c(\mathrm{base}, \mathrm{base}, \mathrm{loop}) =_{z:C.z =_{C} z}^{(c(\mathrm{base}), c_\mathrm{base}, p)} c_\mathrm{loop}}$$

and similarly, the (non-dependent) **universal property of the circle type** is presented as a single rule by replacing the dependent sum type in the recursion principle with a uniqueness quantifier

$$\frac{\Gamma \vdash C \; \mathrm{type} \quad \Gamma \vdash c_\mathrm{base}:C \quad \Gamma \vdash c_\mathrm{loop}:c_\mathrm{base} =_{C} c_\mathrm{base}}{\Gamma \vdash \mathrm{rec}_{S^1}^{C}(c_\mathrm{base}, c_\mathrm{loop}):\exists!c:S^1 \to C.\sum_{p:c(\mathrm{base}) =_{C(\mathrm{base})} c_\mathrm{base}} \mathrm{ap}_c(\mathrm{base}, \mathrm{base}, \mathrm{loop}) =_{z:C.z =_{C} z}^{(c(\mathrm{base}), c_\mathrm{base}, p)} c_\mathrm{loop}}$$

### In terms of higher inductive types
{#InTermsOfHigherInductiveTypes}

As a [[higher inductive type]], the circle is given by

    Inductive Circle : Type
      | base : Circle
      | loop : Id Circle base base

This says that the type is inductively constructed from

1. a [[term]] of circle type whose [[semantics|interpretation]] is as the [[base point]] of the circle,

1. a term of the [[identification type]] between these two terms, whose interpretation is as the 1-cell of the circle

   $$
     base \stackrel{loop}{\to} base
     \,
   $$

   Hence as a non-constant path from the base point to itself.

## Properties
 {#Properties}

\begin{theorem}
Suppose that the strict circle type has an identification $K:\mathrm{refl}_{S^1}(\mathrm{base}) = \mathrm{loop}$. Then every type is a [[set]].
\end{theorem}

\begin{proof}
For all types $A$, terms $x:A$, and self-identifications $p:x =_A x$, by the recursion principle of the strict circle type, we have the function $\mathrm{rec}_{S^1}(x, p):S^1 \to A$ such that
$\mathrm{rec}_{S^1}(x, p)(\mathrm{base}) \equiv x$ and $\mathrm{ap}_{\mathrm{rec}_{S^1}^A(x, p)}(\mathrm{base}, \mathrm{base}, \mathrm{loop}) \equiv p$. We also have by definition of $\mathrm{ap}_{\mathrm{rec}_{S^1}(x, p)}$ that
$$\mathrm{ap}_{\mathrm{rec}_{S^1}(x, p)}(\mathrm{base}, \mathrm{base}, \mathrm{refl}_{S^1}(\mathrm{base})) \equiv \mathrm{refl}_{A}(\mathrm{rec}_{S^1}(x, p)(\mathrm{base})) \equiv \mathrm{refl}_A(x)$$
By applying $\mathrm{ap}_{\mathrm{rec}_{S^1}(x, p)}(\mathrm{base}, \mathrm{base})$ to $K$, we have identification
$$\mathrm{ap}_{\mathrm{ap}_{\mathrm{rec}_{S^1}(x, p)}(\mathrm{base}, \mathrm{base})}(\mathrm{refl}_{S^1}(\mathrm{base}), \mathrm{loop}, K):\mathrm{ap}_{\mathrm{rec}_{S^1}(x, p)}(\mathrm{base}, \mathrm{base}, \mathrm{refl}_{S^1}(\mathrm{base})) = \mathrm{ap}_{\mathrm{rec}_{S^1}(x, p)}(\mathrm{base}, \mathrm{base}, \mathrm{loop})$$
which reduces down to
$$\mathrm{ap}_{\mathrm{ap}_{\mathrm{rec}_{S^1}(x, p)}(\mathrm{base}, \mathrm{base})}(\mathrm{refl}_{S^1}(\mathrm{base}), \mathrm{loop}, K):\mathrm{refl}_{A}(x) = p$$
which is precisely [[axiom K]] for type $A$. Thus every type $A$ is a set.
\end{proof}

### Types equivalent to the circle type

The circle type is equivalent to the following types

* The [[suspension type]] $\Sigma \mathrm{Bool}$ of the [[boolean domain]].

* The [[coequalizer type]] of any two [[endofunctions]] on the [[unit type]]

$$\mathbf{1} \rightrightarrows \mathbf{1} \to S^1$$

* Following [[synthetic homotopy theory]], the [[coequalizer type]] of the pair of functions on the [[line type]]

$$
A^1\underoverset{\quad s \quad}{\mathrm{id}_{A^1}}{\rightrightarrows}A^1 \to S^1
$$

* Given a [[univalent universe]] $U$, the type of $U$-small $\mathbb{Z}$-torsors. One can then prove that this type satisfies the same induction principle (propositionally). This is due to [[Dan Grayson]].

### Induction principle without heterogeneous identifications

There is a version of the induction principle which uses a type $C$ and a function $f:C \to S^1$ instead of a type family $P(x)$ indexed by $x:S^1$. It has the benefit of not requiring that one has first defined [[heterogeneous identification types]], whether as an [[inductive family]] of types or by using [[transport]].

The induction principle of the circle type says that given a type $C$ and a function $f:C \to S^1$, as well as

* an element $c_\mathrm{base}:C$

* identifications $c_\mathrm{loop}:c_\mathrm{base} =_C c_\mathrm{base}$ and $p_\mathrm{base}:f(c_\mathrm{base}) =_{S^1} \mathrm{base}$ such that $\mathrm{ap}_f(c_\mathrm{loop})$, $p_\mathrm{base}$, $p_\mathrm{base}$, and $\mathrm{loop}$ form a square

$$
\begin{array}{c}
    & f(c_\mathrm{base}) & \overset{\mathrm{ap}_{f}(c_\mathrm{loop})}= & f(c_\mathrm{base}) &  \\
    p_\mathrm{base} & \Vert &  & \Vert & p_\mathrm{base}\\
    & \mathrm{base} & \underset{\mathrm{loop}}= & \mathrm{base} & \\
\end{array}
$$

* an identification saying that the square commutes

$$p_\mathrm{loop}:\mathrm{ap}_f(c_\mathrm{loop}) \bullet p_\mathrm{base} =_{f(c_\mathrm{base}) =_{S^1} \mathrm{loop}} p_\mathrm{base} \bullet \mathrm{loop}$$

one can construct

* a function

$$g:S^1 \to C$$

* a homotopy witnessing that $g$ is a [[section]] of $f$:

$$\mathrm{sec}_g:\prod_{x:S^1} f(g(x)) =_{S^1} x$$

such that

$$g(\mathrm{base}) \equiv c_\mathrm{base} \quad \mathrm{sec}_g(\mathrm{base}) \equiv p_\mathrm{base} \quad \mathrm{ap}_{g}(\mathrm{loop}) \equiv c_\mathrm{loop}$$

$$\mathrm{ind}_{=}\left(\lambda x:S^1.\mathrm{refl}_{f(g(x)) =_{S^1} x}(\mathrm{sec}_g(x)), \mathrm{base}, \mathrm{base}, \mathrm{loop}\right) \equiv p_\mathrm{loop}$$

The last condition needs some explanation. Since $g$ is a section of $f$, the composite $f \circ g$ is [[generalized the|the]] [[identity function]] on the [[circle type]], up to [[identification]]. Now, given any identity function on the circle type $i:S^1 \to S^1$ with homotopy

$$j:\prod_{x:S^1} i(x) =_{S^1} x$$

we have the following square for all $x:S^1$, $y:S^1$, and $q:x =_{S^1} y$:

$$
\begin{array}{c}
    & i(x) & \overset{\mathrm{ap}_{i}(q)}= & i(y) &  \\
    j(x) & \Vert &  & \Vert & j(y)\\
    & x & \underset{q}= & y & \\
\end{array}
$$

This [[commutative square|square commutes]] via the [[J rule]]: it suffices to construct an element of

$$\mathrm{ap}_{i}(\mathrm{refl}_{S^1}(x)) \bullet j(x) =_{i(x) =_{S^1} x} j(x) \bullet \mathrm{refl}_{S^1}(x)$$

But $\mathrm{ap}_{i}(\mathrm{refl}_{S^1}(x)) \bullet j(x)$ reduces down to $j(x)$ via
$$\mathrm{ap}_{i}(\mathrm{refl}_{S^1}(x)) \bullet j(x) \equiv \mathrm{refl}_{S^1}(i(x)) \bullet j(x) \equiv j(x)$$
and similarly $j(x) \bullet \mathrm{refl}_{S^1}(x)$ reduces down to $j(x)$, so just take reflexivity of $j(x)$.

So the naturality square is inductively defined by

$$\mathrm{ind}_{=}\left(\lambda x:S^1.\mathrm{refl}_{i(x) =_{S^1} x}(j(x)), x, y, q\right):\mathrm{ap}_{i}(q) \bullet j(y) =_{i(x) =_{S^1} y} j(x) \bullet q$$

When $i \coloneqq f \circ g$, $j \coloneqq \mathrm{sec}_g$, $x \coloneqq \mathrm{base}$, $y \coloneqq \mathrm{base}$, and $q \coloneqq \mathrm{loop}$, this results in the identification

$$\mathrm{ind}_{=}\left(\lambda x:S^1.\mathrm{refl}_{f(g(x)) =_{S^1} x}(\mathrm{sec}_g(x)), \mathrm{base}, \mathrm{base}, \mathrm{loop}\right)$$

which is of the same type as $p_\mathrm{loop}$ due to the [[judgmental equalities]] in the other computation rules.

One gets back the usual induction principle of the interval type when $C \equiv \sum_{x:S^1} P(x)$ and $f \equiv \pi_1$ the first projection function of the dependent sum type, and one gets back the recursion principle of the interval type when $C \equiv S^1 \times P$ and $f \equiv \pi_1$ the first projection function of the product type.

### Extensionality principle of the circle type

The extensionality principle of the circle type says that there is an equivalence of types between the identification type $\mathrm{base} =_{S^1} \mathrm{base}$ and the type of integers $\mathbb{Z}$:

$$\mathrm{ext}_{S^1}:(\mathrm{base} =_{S^1} \mathrm{base}) \simeq \mathbb{Z}$$

Equivalently, that the [[loop space type]] $\Omega(S^1, \mathrm{base})$ is equivalent to the [[integers]].

Thus, the extensionality principle of the circle type implies that the integers and thus the natural numbers are [[contractible types]] if [[axiom K]] or [[uniqueness of identity proofs]] hold for the entire type theory. If the extensionality principle of the natural numbers also hold in the type theory, then every type is contractible.

One can prove the extensionality principle of the circle type, given a [[univalent universe]] where the circle is small relative to the universe. The [[HoTT book]] provides a number of such proofs.

If one doesn't have a type of integers or a [[type universe]], then the extensionality principle of the [[circle type]] says that the [[identity types]] $\mathrm{base} =_{S^1} \mathrm{base}$ satisfy the [[dependent universal property of the integers type]] with respect to the element $\mathrm{refl}_{S^1}(\mathrm{base}):\mathrm{base} =_{S^1} \mathrm{base}$ and the equivalence
$$\lambda p:\mathrm{base} =_{S^1} \mathrm{base}.p \bullet \mathrm{loop}:(\mathrm{base} =_{S^1} \mathrm{base}) \simeq (\mathrm{base} =_{S^1} \mathrm{base})$$

This is a special case of the extensionality principle of [[coequalizer types]] and [[pushout types]] as detailed in [Kraus and Raumer (2019)](#KrausRaumer19).

### The circle type and infinity

The extensionality principle of the circle type says that the loop space of the circle at its base point $\Omega(S^1, \mathrm{base})$ is a [[denumerable set]]. However, there is a weaker axiom that one can add to the circle type: that $\Omega(S^1, \mathrm{base})$ is an [[infinite set|infinite]] type.

By the recursion principle of the natural numbers, there is a function

$$\mathrm{rec}_{\mathbb{N}}(\mathrm{refl}_{S^1}(\mathrm{base}), \lambda l:\Omega(S^1, \mathrm{base}).l \bullet \mathrm{loop}):\mathbb{N} \to \Omega(S^1, \mathrm{base})$$

which takes zero to reflexivity of $\mathrm{base}$ and the successor function to concatenation of a self-identification by $\mathrm{loop}$. That $\Omega(S^1, \mathrm{base})$ is an infinite set, states that the above function is an [[embedding of types]],
$$\mathrm{inf}_{S^1}:\mathrm{isEmbed}\left(\mathrm{rec}_{\mathbb{N}}(\mathrm{refl}_{S^1}(\mathrm{base}), \lambda l:\Omega(S^1, \mathrm{base}).l \bullet \mathrm{loop})\right)$$
or equivalently that application of the above function is an equivalence for all natural numbers $m:\mathbb{N}$, $n:\mathbb{N}$

$$\mathrm{inf}_{S^1}:\prod_{m:\mathbb{N}} \prod_{n:\mathbb{N}} \mathrm{isEquiv}\left(\mathrm{ap}_{\mathrm{rec}_{\mathbb{N}}(\mathrm{refl}_{S^1}(\mathrm{base}), \lambda l:\Omega(S^1, \mathrm{base}).l \bullet \mathrm{loop})}(m, n)\right)$$

If one doesn't have a [[natural numbers type]], the loop space $\Omega(S^1, \mathrm{base})$ of the circle type at its base point $\mathrm{base}:S^1$ is an [[infinite set]] if and only if $\Omega(S^1, \mathrm{base})$ is equivalent to the [[sum type]] $\Omega(S^1, \mathrm{base}) + \Omega(S^1, \mathrm{base})$.

$$\mathrm{inf}_{S^1}:\Omega(S^1, \mathrm{base}) \simeq (\Omega(S^1, \mathrm{base}) + \Omega(S^1, \mathrm{base}))$$

See [Rasekh 21](#Rasekh21) for proving this statement in the presence of the [[univalence axiom]].

### Kuratowski-finiteness

The circle type is [[Kuratowski-finite]]. (Cf [Frumin et al. 18](#FGGW))

### H-space structures on the circle type

The type of [[H-spaces]] on the circle type is a [[contractible type]].

## Related concepts

* [[circle]], [[unit circle]], [[circle group]], [[circle object]]

* [[interval type]]

* [[line type]]

* [[minimal simplicial circle]]

* [[suspension type]]

* [[sphere type]]

* [[homotopy groups of spheres]]

* [[Jordan curve]]

* [[lemniscate type]]

## References
 {#References}

The formalization of the [[shape modality|shape]] [[homotopy type]] $ʃ S^1 \simeq \mathbf{B}\mathbb{Z}$ of the circle as a [[higher inductive type]] in [[homotopy type theory]], along with a proof that $\Omega ʃS^1\simeq {\mathbb{Z}}$ (and hence $\pi_1(ʃS^1) \simeq \mathbb{Z}$):

* {#LicataShulman13} [[Dan Licata]], [[Mike Shulman]], *Calculating the Fundamental Group of the Circle in Homotopy Type Theory*, LICS '13: Proceedings of the 2013 28th Annual ACM/IEEE Symposium on Logic in Computer Science June 2013 ([arXiv:1301.3443](http://arxiv.org/abs/1301.3443), [doi:10.1109/LICS.2013.28](https://doi.org/10.1109/LICS.2013.28))

  > blog announcements:

  > [A formal proof that $\pi_1(S^1) = \mathbb{Z}$](https://homotopytypetheory.org/2011/04/29/a-formal-proof-that-pi1s1-is-z/)

  > [A simpler proof that $\pi_1(S^1) = \mathbb{Z}$](http://homotopytypetheory.org/2012/06/07/a-simpler-proof-that-%cf%80%e2%82%81s%c2%b9-is-z/)

Formalization in [[proof assistants]]:

in [[Coq]]:

* The HoTT Coq library: [theories/hit/Circle.v](https://github.com/HoTT/HoTT/blob/master/theories/hit/Circle.v)

in [[Agda]]:

* The HoTT Agda library: [homotopy/LoopSpaceCircle.agda](https://github.com/HoTT/HoTT-Agda/blob/2.0/homotopy/LoopSpaceCircle.agda)

Exposition and review:

* {#UFP13} [[Univalent Foundations Project]], §8.1 *[[Homotopy Type Theory -- Univalent Foundations of Mathematics]]* (2013) [[web](http://homotopytypetheory.org/book/), [pdf](http://hottheory.files.wordpress.com/2013/03/hott-online-323-g28e4374.pdf)]

* {#Rijke22} [[Egbert Rijke]], *[[Introduction to Homotopy Type Theory]]*, Cambridge Studies in Advanced Mathematics, Cambridge University Press ([arXiv:2212.11082](https://arxiv.org/abs/2212.11082))

Alternative construction of the circle type as the type of $\mathbb{Z}$-torsors:

* {#BezemBuchholtzGraysonShulman19} [[Marc Bezem]], [[Ulrik Buchholtz]], [[Daniel Grayson]], [[Michael Shulman]], _Construction of the Circle in UniMath_, Journal of Pure and Applied Algebra Volume 225, Issue 10, October 2021, 106687 ([arXiv:1910.01856](https://arxiv.org/abs/1910.01856))

Alternative construction of the circle type as a coequalizer:

* {#Shulman15} [[Mike Shulman]], Section 9 of  _Brouwer's fixed-point theorem in real-cohesive homotopy type theory_, Mathematical Structures in Computer Science Vol 28 (6) (2018): 856-941 ([arXiv:1509.07584](https://arxiv.org/abs/1509.07584), [doi:10.1017/S0960129517000147](https://doi.org/10.1017/S0960129517000147))

For the fact that the type of H-space structures on a circle type is contractible:

* [[Ulrik Buchholtz]], [[J. Daniel Christensen]], [[Jarl G. Taxerås Flaten]], [[Egbert Rijke]], *Central H-spaces and banded types* [[arXiv:2301.02636](https://arxiv.org/abs/2301.02636)]

For the fact that the circle type is Kuratowski-finite:

* {#FGGW} [[Dan Frumin]], [[Herman Geuvers]], [[Léon Gondelman]], [[Niels van der Weide]], _Finite Sets in Homotopy Type Theory_, in *CPP 2018: Proceedings of the 7th ACM SIGPLAN International Conference on Certified Programs and Proofs* (2018) 201–214 [[doi:10.1145/3167085](https://doi.org/10.1145/3167085), [pdf](http://cs.ru.nl/~nweide/FiniteSetsInHoTT.pdf)]

For the fact that one can construct a [[natural numbers type]] from the circle type:

* {#Rasekh21} [[Nima Rasekh]], _Every Elementary Higher Topos has a Natural Number Object_, Theory and Applications of Categories **37** 13 (2021) 337-377.  ([arXiv:1809.01734](https://arxiv.org/abs/1809.01734), [tac:37-13](http://www.tac.mta.ca/tac/volumes/37/13/37-13abs.html))

For the induction principle of the identity types of the circle type:

* {#KrausRaumer19} [[Nicolai Kraus]], [[Jakob von Raumer]], *Path Spaces of Higher Inductive Types in Homotopy Type Theory*. [[arXiv:1901.06022](https://arxiv.org/abs/1901.06022)]

[[!redirects circle type]]
[[!redirects circle types]]

[[!redirects strict circle type]]
[[!redirects strict circle types]]
[[!redirects weak circle type]]
[[!redirects weak circle types]]

[[!redirects dependent universal property of the circle type]]
[[!redirects dependent universal property of circle types]]

[[!redirects universal property of the circle type]]
[[!redirects universal property of circle types]]

[[!redirects dependent universal property of the identity types of the circle type]]
[[!redirects dependent universal property of the identity types of circle types]]
[[!redirects dependent universal property of identity types of the circle type]]
[[!redirects dependent universal property of identity types of circle types]]

[[!redirects universal property of the identity types of the  the circle type]]
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[[!redirects universal property of identity types of the  the circle type]]
[[!redirects universal property of identity types of circle types]]


circled set

+-- {: .rightHandSide}
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###Context###
#### Functional analysis
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[[!include functional analysis - contents]]
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A subset $C$ of a [[vector space]] $V$ over either $\mathbb{R}$ or $\mathbb{C}$ is **circled** if $\lambda C \subseteq C$ whenever ${|\lambda|} \le 1$.

category: functional analysis


Circuitoids > history

< [[Circuitoids]]


circumference of a circle

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###Context###
#### Analysis
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[[!include analysis - contents]]
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#### Geometry
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[[!include higher geometry - contents]]
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#### Differential geometry
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[[!include synthetic differential geometry - contents]]
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# Contents
* table of contents
{: toc}

## Idea ##

The circumference of a [[circle]] (the [[length]] of the [[curve]] that is the circle) as discussed in geometry.

## Definition/proposition and proof

+-- {: .num_prop}
###### Proposition
Depending on which [[circle constant]] you use, given a [[radius]] $r$ of a circle $\mathcal{C}$ in the [[Euclidean space|Euclidean]] [[plane]] $\mathbb{R}^2$, the circumference of a circle is either $C(r) = \tau r$ or $C(r) = 2 \pi r$.
=--

### Proof by integration

+-- {: .proof}
###### Proof
In this proof, we are using the circle constant $\tau = 2 \pi$.

Given any [[Euclidean space|Euclidean]] [[plane]] $\mathbb{R}^2$, one could select an [[orthonormal basis]] on $\mathbb{R}^2$ by postulating an origin $0$ at the center of the circle $\mathcal{C}$ and two [[orthonormal vectors]] $\hat{i}$ and $\hat{j}$. The circle $\mathcal{C}$ could be parameterized by a function $\overrightarrow{r}:[0, \tau] \to \mathbb{R}^2$ defined as
$$\overrightarrow{r}(\theta) \coloneqq r \cos(\theta) \hat{i} + r \sin(\theta) \hat{j}$$

Then the circumference of $\mathcal{C}$ is given by the following [[integral]]:

$$C(r) = \int_{0}^{\tau} \vert \overrightarrow{r}(\theta) \vert d \theta$$

which evaluates to

$$C(r) = \int_{0}^{\tau} \vert r \cos(\theta) \hat{i} + r \sin(\theta) \hat{j} \vert d \theta = \int_{0}^{\tau} r((\cos(\theta))^2 + (\sin(\theta))^2) d \theta = \int_{0}^{\tau} r d \theta = \tau r$$
=--

### Proof by limits of regular polygons

+-- {: .proof}
###### Proof
In this proof, we are using the circle constant $\tau = 2 \pi$.

The perimeter of a [[regular polygon]] $\mathcal{P}_n$ (strictly speaking, a [[regular polygonal line]]) with $n$ sides and [[circumradius]] $r$ is given by the sequence of functions $P_\mathcal{P}:\mathbb{N} \to (\mathbb{R} \to \mathbb{R})$

$$P_\mathcal{P}(n)(r) = r (2 n) \sin\left(\frac{\tau}{2n}\right)$$

which [[embedding|embeds]] in the $\mathbb{R}_+$-[[action]] $P_\mathcal{P}^\prime:\mathbb{R}_+ \to (\mathbb{R} \to \mathbb{R})$, defined as

$$P_\mathcal{P}^\prime(n)(r) = r (2 n) \sin\left(\frac{\tau}{2 n}\right)$$

The [[limit of a function|limit]] of $P^\prime$ as $n$ goes to infinity is the circumference of a circle with radius $r$:

$$C(r) = \lim_{n \to \infty} P_\mathcal{P}^\prime(n)(r) = \lim_{n \to \infty} r (2 n) \sin\left(\frac{\tau}{2 n}\right) = r \lim_{m \to 0} \frac{\sin(\tau m)}{m} = r \lim_{m \to 0} \frac{\partial_m \sin(\tau m)}{\partial_m m} = r \lim_{m \to 0} \frac{\tau \cos(\tau m)}{1} = \tau r$$
=--

## See also

* [[pi]]

* [[circle]]

* [[unit circle]]

* [[arc length]]

* [[area of a circle]]

* [[Euclidean space]]