This page is currently being reworked into the more comprehensive discussion at Smooth∞Grpd
Classes of bundles
Examples and Applications
Connections on bundles
Higher abelian differential cohomology
Higher nonabelian differential cohomology
Application to gauge theory
In every cohesive (∞,1)-topos there is an intrinsic notion of differential cohomology with coefficients in an abelian group object that classifies -principal ∞-bundles with ∞-connection.
Here we discuss the specific realization for Smooth∞Grpd the (∞,1)-topos of smooth ∞-groupoids and 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 of smooth ∞-groups . Differential cocycles on the latter support the ∞-Chern-Weil homomorphism that sends nonabelian ∞-connections to circle -bundles whose curvature form realizes a characteristic class in de Rham cohomology.
The ambient context
Let 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 .
From this induced is the path ∞-groupoid adjunction
and the intrinsic de Rham cohomology adjunction
For an abelian group object there for each integer is the universal curvature characteristic form, given by a cocycle-morphism
The cocycles for differential cohomology in degree with coefficients in are the points in the homotopy fiber of the morphism on cohomology
induced by this. Every such cocycle we may think of as an ∞-connection on the -principal ∞-bundle classified by the underlying cocycle in .
We consider these constructions in the model Smooth∞Grpd. This is the (∞,1)-category of (∞,1)-sheaves
on the site CartSp 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
as described at presentations of (∞,1)-sheaf (∞,1)-toposes.
In Smooth∞Grpd a canonical choice for is the circle group
We show how the notion of smooth circle -bundles with connection obtained by applying the general setup above to this case reproduces ordinary differential cohomology:
We assume in the following that the reader is familiar with basics of smooth ∞-groupoids.
Flat differential cohomology
The coefficient object for flat differential cohomology in Smooth∞Grpd with values in is .
The coefficient object for intrinsic de Rham cohomology is , defined by the (∞,1)-pullback
The following proposition provides models for these objects in in terms of ordinary differential forms.
A fibrant representative in of is
and a fibrant representative of is
Notice that the complex of sheaves is that which defines flat Deligne cohomology, while that of is essentially that which defines de Rham cohomology in degree (see below). Also notice that we denoted by also the differential ; this is to stress that we are looking at as the quotient .
Since the global section functor amounts to evaluation on the point and since constant simplicial presheaves on CartSp satisfy descent (on objects in !), we have that is represented by the complex of sheaves . This is weakly equivalent to 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 representing the counit is the image under 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 which models the (∞,1)-limit over in . Since ∞-stackification preserves finite -limits this models also the corresponding -limit in . Therefore the top left object is indeed a model for .
de Rham cohomology
The intrinsic de Rham cohomology of Smooth∞Grpd with coefficients in or 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 with .
Above in Flat U(1)-valued differential cohomology we found a fibrant representative of to be given by
Let be a good open cover. At Smooth∞Grpd is discussed that then the Cech nerve is a cofibrant resolution of in . Therefore we have
The right hand is the -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 , , etc. , such that this collection is annihilated by the total differentoal , where is the de Rham differential and 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 subordinate to the cover , i.e. and .
where we use that from 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 .
Such a cocycle being -closed says precisely that for a globally defined closed differential form. Moreover, coboundaries between two cocycles both of this form
are necessarily themselves of the form with for a globally defined differential -form and .
The intrinsic definition of the ∞-groupoid of cocycles for the intrinsic differential cohomology in with coefficients is the object in the (∞,1)-pullback
We show now that for this reproduces the Deligne cohomology of :
For 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 only one curvature form representative.
We give the proof below after some preliminary expositional discussion.
Circle bundles with connection
Before discussing the full theorem, it is instructive to start by looking at the special case in some detail, which is about ordinary -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 is modeled in 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 by an ordinary pullback in sSet we also want to resolve the curvature characteristic morphism by a fibration. We claim that this may be obtained by choosing the resolution given by
with the morphism given by