nLab
circle n-bundle with connection

This page is currently being reworked into the more comprehensive discussion at Smooth∞Grpd

Context

Bundles

Differential cohomology

Cohesive \infty-Toposes

cohesive topos

cohesive (∞,1)-topos

cohesive homotopy type theory

Backround

Definition

Presentation over a site

Structures in a cohesive (,1)(\infty,1)-topos

structures in a cohesive (∞,1)-topos

Structures with infinitesimal cohesion

infinitesimal cohesion?

Models

Contents

Idea

In every cohesive (∞,1)-topos H\mathbf{H} there is an intrinsic notion of differential cohomology with coefficients in an abelian group object AHA \in \mathbf{H} that classifies B n1A\mathbf{B}^{n-1}A-principal ∞-bundles with ∞-connection.

Here we discuss the specific realization for H=\mathbf{H} = Smooth∞Grpd the (∞,1)-topos of smooth ∞-groupoids and A=U(1)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 BG\mathbf{B}G of smooth ∞-groups GG. Differential cocycles on the latter support the ∞-Chern-Weil homomorphism that sends nonabelian ∞-connections to circle nn-bundles whose curvature form realizes a characteristic class in de Rham cohomology.

The ambient context

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

(ΠDiscΓcoDisc):SmoothGrpdcoDiscΓDiscΠGrpd (\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 intrinsic fundamental ∞-groupoid functor Π\Pi.

From this induced is the path ∞-groupoid adjunction

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

and the intrinsic de Rham cohomology adjunction

(Π dR dR):*/SmoothGrpd dRΠ dRSmoothGrpd. (\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 AA an abelian group object there for each integer nn is the universal curvature characteristic form, given by a cocycle-morphism

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

The cocycles for differential cohomology in degree nn with coefficients in AA are the points in the homotopy fiber H diff(,B nA)\mathbf{H}_{diff}(-, \mathbf{B}^n A) of the morphism on cohomology

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

induced by this. Every such cocycle H diff(X,B nA)\nabla \in \mathbf{H}_{diff}(X,\mathbf{B}^n A) we may think of as an ∞-connection on the B n1A\mathbf{B}^{n-1}A-principal ∞-bundle classified by the underlying cocycle in H(X,B nA)\mathbf{H}(X, \mathbf{B}^n A).

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

SmoothGrpd:=Sh (,1)(CartSp smooth) Smooth\infty Grpd := Sh_{(\infty,1)}(CartSp_{smooth})

on the site CartSp smooth{}_{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[CartSp^{op}, sSet]_{proj,loc}

SmoothGrpd([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 H=\mathbf{H} = Smooth∞Grpd a canonical choice for AA is the circle group

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

We show how the notion of smooth circle nn-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 H=\mathbf{H} = Smooth∞Grpd with values in B nU(1)\mathbf{B}^n U(1) is B nU(1)=LConstΓB nU(1)\mathbf{\flat} \mathbf{B}^n U(1) = LConst \Gamma \mathbf{B}^n U(1).

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

dRB nU(1) B nU(1) * B nU(1). \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 forms.

Proposition

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

B nU(1) chn:=Ξ[C (,U(1))d dRΩ 1()d dRd dRΩ cl n()], \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 dRB nU(1)\mathbf{\flat}_{dR} \mathbf{B}^n U(1) is

dRB nU(1) chn:=Ξ[0Ω 1()d dRd dRΩ cl n()]. \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 B nU(1)\mathbf{\flat}\mathbf{B}^n U(1) is that which defines flat Deligne cohomology, while that of dRB nU(1)\mathbf{\flat}_{dR} \mathbf{B}^n U(1) is essentially that which defines de Rham cohomology in degree n>1n \gt 1 (see below). Also notice that we denoted by d dRd_{dR} also the differential C (,U(1))d dRlogΩ 1()C^\infty(-,U(1)) \stackrel{d_{dR} log}{\to} \Omega^1(-); this is to stress that we are looking at U(1)U(1) as the quotient /\mathbb{R}/\mathbb{Z}.

Proof

Since the global section functor Γ\Gamma amounts to evaluation on the point 0\mathbb{R}^0 and since constant simplicial presheaves on CartSp satisfy descent (on objects in CartSpCartSp!), we have that B nU(1)\mathbf{\flat} \mathbf{B}^n U(1) is represented by the complex of sheaves Ξ[constU(1)00]\Xi[const U(1) \to 0 \to \cdots \to 0]. This is weakly equivalent to Ξ[C (,U(1))d dRΩ 1()d dRd dRΩ cl n()]\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

constU(1) 0 0 C (,U(1)) d dR Ω 1() Ω cl n() \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 quasi-isomorphism.

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

C (,U(1)) Ω 1() Ω cl n() = C (,U(1)) 0 0. \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

Ξ[0Ω 1()Ω cl n()] Ξ[C (,U(1))Ω 1()Ω cl n()] Ξ[000] Ξ[C (,U(1))00] \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[CartSp^{op}, sSet]_{proj} which models the (∞,1)-limit over *B nU(1)B nU(1)* \to \mathbf{B}^n U(1) \leftarrow \mathbf{\flat}\mathbf{B}^n U(1) in PSh (,1)(CartSp)PSh_{(\infty,1)}(CartSp). Since ∞-stackification preserves finite (,1)(\infty,1)-limits this models also the corresponding (,1)(\infty,1)-limit in H\mathbf{H}. Therefore the top left object is indeed a model for dRB nU(1)\mathbf{\flat}_{dR} \mathbf{B}^n U(1).

de Rham cohomology

The intrinsic de Rham cohomology of Smooth∞Grpd with coefficients in \mathbb{R} or U(1)=/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 nn \in \mathbb{N} with n2n \geq 2.

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

Ξ[Ω 1()d dRΩ 2()d dRΩ cl n()] \Xi[\Omega^1(-) \stackrel{d_{dR}}{\to} \Omega^2(-) \stackrel{d_{dR}}{\to} \cdots \to \Omega^n_{cl}(-)]

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

Proposition

For XSmoothGrpdX \in Smooth\infty Grpd a paracompact smooth manifold we have in for H=SmoothGrpd\mathbf{H} = Smooth \infty Grpd a natural isomorphism

H dR(X,B nU(1)):=π 0H(X, dRB nU(1))H dR n(X), 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 de Rham cohomology of a smooth manifold.

Proof

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

H(X, dRB nU(1))[CartSp op,sSet](C({U i}),Ξ[Ω 1()d dRΩ cl n()]). \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 Cech hypercohomology of the complex of sheaves of differential forms. A cocycle is given by a collection

(C i,B ij,A ijk,,Z i 0,,i n) (C_i, B_{i j}, A_{i j k}, \cdots , Z_{i_0, \cdots, i_n})

of differential forms, with C iΩ cl n(U i)C_i \in \Omega^n_{cl}(U_i), B ijΩ n1(U iU j)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±δD = d_{dR} \pm \delta, where d dRd_{dR} is the de Rham differential and δ\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

(F i,0,0,,0) (F_i, 0, 0, \cdots, 0)

by using a partition of unity (ρ iC (X))(\rho_i \in C^\infty(X)) subordinate to the cover {U iX}\{U_i \to X\}, i.e. xU iρ i(x)=0x \in U_i \Rightarrow \rho_i(x) = 0 and iρ i=1\sum_i \rho_i = 1.

For consider

(C i,B ij,A ijk,,Y i 1,,i n,Z i 1,,i n+1) + D(0,0,, i 0ρ i 0Z i 0,i 1,,i n,0) = (C i,B ij,A ijk,,Y i 1,,i n+d dR i 0ρ i 0Z i 0,i 1,,i n,0), \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 (δZ) i 1,,i n+2=0(\delta Z)_{i_1, \cdots, i_{n+2}} = 0 it follows that

(δρZ) i 1,,i n+1 = i 0ρ i 0 k=1 n+1(1) kZ i 0,i 1,i^ k,,i n+1 = i 0ρ i 0Z i 1,,i n+1 =Z i 1,,i n+1. \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,,0)(F_i, 0, \cdots, 0).

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

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

are necessarily themselves of the form (λ i,λ ij,)=(λ i,0,,0)(\lambda_i, \lambda_{i j}, \cdots) = (\lambda_i, 0 ,\cdots, 0) with λ i=λ| U i\lambda_i = \lambda|_{U_i} for λΩ n1(X)\lambda \in \Omega^{n-1}(X) a globally defined differential nn-form and F=F+d dRλF = F' + d_{dR} \lambda.

Differential cohomology

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

H diff(X,B nU(1)) H dR(X,B n+1U(1)) H(X,B nU(1)) curv H dR(X,B n+1U(1)) \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 n1n \geq 1 this reproduces the Deligne cohomology H(X,(n+1) D )H(X,\mathbb{Z}(n+1)_D^\infty) of XX:

Theorem

For XX a paracompact smooth manifold we have

H diff(X,B nU(1))(H(X,(n+1) D ))× Ω cl n+1(X)H dR n+1int(X). 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 XX only one curvature form representative.

We give the proof below after some preliminary expositional discussion.

Remark

The restriction to single representatives in each de Rham class is a reflection of the fact that in the above (,1)(\infty,1)-pullback diagram the morphism H dR(X,B n+1U(1))H dR(X,B n+1U(1))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 (,1)(\infty,1)-pullback the homotopy pullback

H diff(X,B nU(1)) Ω cl n+1(X) H(X,B nU(1)) curv H dR(X,B n+1U(1)) \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 H dR(X,B n+1U(1))\mathbf{H}_{dR}(X, \mathbf{B}^{n+1} U(1)). With this definition we get the isomorphism

H diff(X,B nU(1))H(X,(n+1) D ). 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 (,1)(\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.

Circle bundles with connection

Before discussing the full theorem, it is instructive to start by looking at the special case n=1n=1 in some detail, which is about ordinary U(1)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 BU(1)H\mathbf{B}U(1) \in \mathbf{H} is modeled in [CartSp op,sSet][CartSp^{op}, sSet] by

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

Accordingly we have for the double delooping the model

B 2U(1)=Ξ(C (,U(1))00) \mathbf{B}^2 U(1) = \Xi( \; C^\infty(-,U(1)) \to 0 \to 0 \;)

and for the universal principal 2-bundle

EBU(1)=Ξ(C (,U(1))IdC (,U(1))0). \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

B 2U(1)=Ξ(constU(1)00). \mathbf{\flat} \mathbf{B}^2 U(1) = \Xi( \; const U(1) \to 0 \to 0 \; ) \,.

Above we already found the model

dRB 2U(1)=Ξ(0Ω 1()d dRΩ cl 2()). \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 H diff(,BU(1))\mathbf{H}_{diff}(-,\mathbf{B}U(1)) by an ordinary pullback in sSet we also want to resolve the curvature characteristic morphism BU(1) dRB 2U(1)\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 BU(1)BU(1) diff,chn\mathbf{B}U(1) \stackrel{\simeq}{\leftarrow} \mathbf{B} U(1)_{diff,chn} given by

BU(1) diff:=Ξ(C (,U(1))Ω 1()d dRIdΩ 1()) \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:B diffU(1) dRB 2U(1)curv : \mathbf{B}_{diff}U(1) \to \mathbf{\flat}_{dR}\mathbf{B}^2 U(1) given by

C (,U(1))Ω 1() d dR+Id Ω 1() p 2 d dR Ω 1() d dR