(see also Chern-Weil theory, parameterized homotopy theory)
structures in a cohesive (∞,1)-topos
infinitesimal cohesion?
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 cohesion to the representation of circle $n$-connections by cocycles in smooth Deligne cohomology (dcct).
In every cohesive (∞,1)-topos $\mathbf{H}$ there is an intrinsic notion of differential cohomology with coefficients in an abelian group object $A \in \mathbf{H}$ that classifies $\mathbf{B}^{n-1}A$-principal ∞-bundles with ∞-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 manifolds, diffeological spaces, orbifolds and generally smooth ∞-groupoids such as deloopings $\mathbf{B}G$ of smooth ∞-groups $G$. Differential cocycles on the latter support the ∞-Chern-Weil homomorphism that sends nonabelian ∞-connections to circle $n$-bundles whose curvature form realizes a characteristic class in de Rham cohomology.
Let $\mathbf{H} :=$ Smooth∞Grpd be the cohesive (∞,1)-topos of smooth ∞-groupoid. As usual, write
for the terminal global section (∞,1)-geometric morphism with its extra left adjoint, the intrinsic fundamental ∞-groupoid functor $\Pi$.
From this induced is the path ∞-groupoid adjunction
and the intrinsic de Rham cohomology adjunction
For $A$ an abelian group object there for each integer $n$ is the universal curvature characteristic form, given by a cocycle-morphism
The cocycles 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
induced by this. Every such cocycle $\nabla \in \mathbf{H}_{diff}(X,\mathbf{B}^n A)$ we may think of as an ∞-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
on the site CartSp${}_{smooth}$ of Cartesian spaces and smooth functions between them. This is a general higher geometry context for differential geometry. For computations we can explicitly present this (∞,1)-category by a local model structure on simplicial presheaves $[CartSp^{op}, sSet]_{proj,loc}$
as described at presentations of (∞,1)-sheaf (∞,1)-toposes.
In $\mathbf{H} =$ Smooth∞Grpd a canonical choice for $A$ is the circle group
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 with connection;
a circle 2-bundle with connection is a $\mathbf{B}U(1)$-principal 2-bundle with connection, equivalently a $U(1)$-bundle gerbe with connection;
a circle 3-bundle with connection if a $\mathbf{B}^2 U(1)$-principal 3-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 n-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 ∞-groupoids.
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
The following proposition provides models for these objects in in terms of ordinary differential forms.
A fibrant representative in $[CartSp^{op}, sSet]_{proj,cov}$ of $\mathbf{\flat} \mathbf{B}^n U(1)$ is
and a fibrant representative of $\mathbf{\flat}_{dR} \mathbf{B}^n U(1)$ is
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). 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 $\mathbb{R}/\mathbb{Z}$.
Since the global section functor $\Gamma$ amounts to evaluation on the point $\mathbb{R}^0$ and since constant simplicial presheaves on CartSp satisfy 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
is objectwise a 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
We observe that the pullback of this morphism to the point
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 ∞-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 $\mathbf{\flat}_{dR} \mathbf{B}^n U(1)$.
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 manifolds and smooth simplicial manifolds in degree greater than 1. This we discuss here. The meaning of the discrepancy in degee 1 and lower is discussed below.
So for this section let $n \in \mathbb{N}$ with $n \geq 2$.
Above in Flat U(1)-valued differential cohomology we found a fibrant representative of $\mathbf{\flat}_{dR} \mathbf{B}^n U(1) \in Smooth\infty Grpd$ to be given by
in $[CartSp^{op}, sSet]_{proj, cov}$.
For $X \in Smooth\infty Grpd$ a paracompact smooth manifold we have in for $\mathbf{H} = Smooth \infty Grpd$ a natural isomorphism
where on the left we have the intrinsic (∞,1)-topos theoretic notion of de Rham cohomology, and on the right the ordinary notion of de Rham cohomology of a smooth manifold.
Let $\{U_i \to X\}$ be a good open cover. At Smooth∞Grpd is discussed that then the Cech nerve $C(\{U_i\}) \to X$ is a cofibrant resolution of $X$ in $[CartSp^{op}, sSet]_{proj,cov}$. Therefore we have
The right hand is the $\infty$-groupoid of cocylces in the Cech hypercohomology of the complex of sheaves of differential forms. A cocycle is given by a collection
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 Cech nerve.
It is a standard result of 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
by using a 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
where we use that from $(\delta Z)_{i_1, \cdots, i_{n+2}} = 0$ it follows that
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
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 $F = F' + d_{dR} \lambda$.
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
in ∞Grpd.
We show now that for $n \geq 1$ this reproduces the Deligne cohomology $H(X,\mathbb{Z}(n+1)_D^\infty)$ of $X$:
For $X$ a paracompact smooth manifold we have
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 after some preliminary expositional discussion.
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 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
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
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 “evil” – to fix that set of objects of the $\infty$-groupoid. Of intrinsic meaning is only the set of their equivalences classes.
Before discussing the full theorem, it is instructive to start by looking at the special case $n=1$ in some detail, which is about ordinary $U(1)$-principal bundles 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.
In terms of the Dold-Kan correspondence the object $\mathbf{B}U(1) \in \mathbf{H}$ is modeled in $[CartSp^{op}, sSet]$ by
Accordingly we have for the double delooping the model
and for the universal principal 2-bundle
In this notation we have also the constant presheaf
Above we already found the model
In order to compute the differential cohomology $\mathbf{H}_{diff}(-,\mathbf{B}U(1))$ by an ordinary pullback in 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
with the morphism $curv : \mathbf{B}_{diff}U(1) \to \mathbf{\flat}_{dR}\mathbf{B}^2 U(1)$ given by
By the Poincare lemma applied to each Cartesian space, this is indeed a fibration.
In the next section 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 survey-part.
We have the following geometric interpretation of the above models:
and
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
and picking the lowest horizontal morphism.
Here the terms mean the following:
$INN(U(1))$ is the 2-group $\Xi(U(1) \to U(1))$, which is a groupal model for the universal U(1)-principal bundle $\mathbf{E}U(1)$;
$\mathbf{\Pi}_2(U)$ is the 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 2-groupoid of Lie 2-algebra valued forms (SchrWalII) we have the following:
For $X$ a 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 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 Bianchi identity $d F_A = 0$ says precisely that this assignment indeed descends to homotopy classes of surfaces, which are the 2-morphisms in $\Pi_2(X)$.
Moreover 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 $A' = A + d_{dR} \lambda - \alpha$.
And finally 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
So by the above definition of differential cohomology in $\mathbf{H}$ we find that $\mathbf{B}U(1)$-differential cohomology of a paracompact smooth manifold $X$ is given by choosing any good open cover $\{U_i \to X\}$, taking $C(\{U_i\})$ to be the Cech nerve, which is then a cofibrant replacement of $X$ in $[CartSp^{op}, sSet]_{proj,cov}$ and forming the ordinary pullback
(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 sSet-enriched model category $[CartSp^{op}, sSet]_{proj}$).
A cocycle in $[CartSp^{op},sSet](C(\{U_i\}), \mathbf{B}_{diff}U(1))$ is
a collection of functions
satsifying $g_{i j} g_{j k} = g_{i k}$ on $U_i \cap U_j \cap U_k$;
a collection of 1-forms
a collection of 1-forms
such that
on $U_i \cap U_j$ and
on $U_i \cap U_j \cap U_k$.
The curvature-morphism takes such a cocycle to the cocycle
in the above model $[CartSp^{op},sSet](C(\{U_i\}), \mathbf{\flat}_{dR}\mathbf{B}^2 U(1))$ 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 $(a_{i j})$
The first statements are effectively the definition and the construction of the above models. The last statement is as in the above discussion 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
and
So in total we have found the following story:
In order to compute the curvature characteristic form of a Cech cohomology cocycle $g : C(\{U_i\}) \to \mathbf{B}U(1)$ of a $U(1)$-principal bundle, we first lift it
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.
From that lift the desired curvature characteristic is simply projected out
and we find that it lives in the sheaf hypercohomology that models ordinary de Rham cohomology.
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
in a 2-step process: first the choice of a pseudo-connection realizes the bottom horizontal morphism as an 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.
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 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
and we can find the new pseudo-components $a'_{i j}$ by
Using the computation
we find that these indeed vanish.
The most drastic example for this is a lift $\nabla$ of a cocycle $g = (g_{i j})$ in
is one which takes all the ordinary curvature forms to vanish identically
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.
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 that $\mathbf{\flat}_{dR} \mathbf{B}U(1)$ is given simply by the 0-truncated sheaf of 1-forms, $\Omega^1(-) : CartSp^{op} \to Set \hookrightarrow sSet$. Accordingly we have for $X$ a paracompact smooth manifold
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
For $X \to \mathbf{B}^{n-1} U(1)$ a 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 curvature 2-form of a line bundle, for $n = 3$ it is curvature 3-form of a bundle gerbe, etc. But for $n = 1$ we have that the original cocycle is just a map of spaces
This can be understoody 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.
We now generalize the above discussion on the derivation of the notion of connections on circle bundles from abstract topos-theory to a proof of the full theorem above on the derivation of general Deligne cohomology.
The main step is to model the double (∞,1)-pullback
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 homotopy pullback realized suitably as an ordinary pullback of fibrations in $[CartSp^{op}, Ch_\bullet] \stackrel{\Xi}{\hookrightarrow} [CartSp^{op}, sSet]_{proj}$.
We have commuting diagrams
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 pullback squares.
Therefore this is a homotopy pullback in $[CartSp^{op}, sSet]_{proj}$ that realizes the $(\infty,1)$-pullback in question in the (∞,1)-category of (∞,1)-presheaves $PSh_{(\infty,1)}(CartSp)$. Since ∞-stackification preserves finite (∞,1)-limits, it therefore also presents the above $(\infty,1)$-pullback in $\mathbf{H} = Sh_{(\infty,1)}(CartSp)$.
For the lower square we had discussed this already above. 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 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
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 Poincare lemma applied to each object $U \in$ CartSp) hence this is a fibration.
The pullback object in the top left is in this notation
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
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
is a Kan fibration by the fact that $[CartSp^{op}, sSet]_{proj}$ is an $sSet_{Quillen}$-enriched model category. Therefore the homotopy pullback is computed as an ordinary pullback.
By the above discussion of de Rham cohomology 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”
This is precisely the complex of sheaves that defines Deligne cohomology in degree $(n+1)$.
In the previous section we discussed a model
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 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 ${}_{simp}$.
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 forms with respect to the trivial simplex bundle $U \times \Delta^k \to U$.
For $n \in \mathbb{N}$, define the simplicial presheaf $\mathbf{B}^n U(1)_{simp} \in [CartSp^{op}, sSet]$ by
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 groups (the group operation is the addition of differential forms).
Under the Dold-Kan correspondence the normalized chain complex of $\mathbf{B}^n U(1)_{sim}$ is
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.
The evident fiber integration of differential forms over simplices
yields a morphism
in $[CartSp^{op}, sSet]_{proj}$, which is a weak equivalence.
This is discussed at Lie integration.
Write
$\mathbf{\flat} \mathbf{B}^n \mathbb{R}_{simp}$ for the simplicial presheaf
and write $\mathbf{\flat}_{dR}\mathbf{B}^n \mathbb{R}_{simp} \in [CartSp^{op}, sSet]$ for the simplicial presheaf
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:
The evident fiber integration over simplices induces morphisms of simplicial presheaves
and
that are weak equivalences in $[CartSp^{op}, sSet]_{proj}$.
Write $\mathbf{B}^n U(1)_{diff,simp}$ for the simplicial presheaf given by
Let the morphism
be the one given by postcomposition with the square of dg-algebras
described at ∞-Lie algebra cohomology.
The set of square diagrams of dg-algebras 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 $\Delta^k$.
The morphism given by fiber integration of differential forms over the simplex factor fits into a diagram
where the vertical morphisms are weak equivalences.
Fiber integration induces a weak equivalence
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
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.
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 $\Delta^k$.
Under fiber integration over simplices, $\mathbf{B}^n U(1)_{conn,simp}$ is quasi-isomorphic to the Deligne cohomology-complex.
In summary this gives us the following alternative perspective on connections on $\mathbf{B}^{n-1}U(1)$-principal ∞-bundles: 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-algebras
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 ∞-Lie algebras $\mathfrak{g}$ we discuss below in ∞-Lie algebra valued connections.
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
(inclusion of the set of closed $(n+1)$-forms into the $(n+1)$-groupoid of de Rham cocycles)
and
(the universal curvature class / Maurer-Cartan form of the circle $(n-1)$-group)
the smooth moduli ∞-stack of circle $n$-bundles with connection from above is expressed in homotopy type theory as
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 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 $\mathbf{B}^n U(1) \times_{\mathbf{\flat}_{dR} \mathbf{B}^{n+1}U(1)} \Omega^{n+1}_{cl}$.
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
See for instance
for the class of circle 3-bundles that arise as differential refinements of degree 4 characteristic classes such as the Pontryagin class.
moduli spaces of line n-bundles with connection on $n$-dimensional $X$
A general picture of bundle n-gerbe?s (with connection) as something very similar to th circle $n$-bundle with connection discussed here is in
The above discussion is from