This page is currently being reworked into the more comprehensive discussion at Smooth∞Grpd
structures in a cohesive (∞,1)-topos
infinitesimal cohesion?
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