Schreiber
differential cohomology in an (∞,1)-topos -- examples

Differential cohomology in an (,1)(\infty,1)-topos

  1. survey

  2. general structures

  3. paths

  4. Lie theory

  5. differential cohomology

  6. examples

  7. references


We discuss examples for and applications of the theory presented at differential cohomology in an (∞,1)-topos -- survey .

Contents

Ambient \infty-connected (,1)(\infty,1)-toposes

Every (∞,1)-topos H\mathbf{H} comes with its intrinsic notion of cohomology. At differential cohomology in an (∞,1)-topos we define an intrinsic notion of differential cohomology whenever H\mathbf{H} has the special property that it is a (locally) ∞-connected (∞,1)-topos so that it has an intrinsic notion of path ∞-groupoids.

Here we discuss examples of such \infty-connected (,1)(\infty,1)-toposes.

LieGrpd\infty LieGrpd

The (∞,1)-category of (∞,1)-sheaves on the site CartSp of Cartesian spaces and smooth functions between them is an \infty-connected (,1)(\infty,1)-topos. Its objects we may think of as ∞-Lie groupoids.

Also \infty-connected is the infinitesimally thickened (∞,1)-topos of (,1)(\infty,1)-sheaves on the site ThCartSp of infinitesimally thickened Cartesian spaces. This is the (,1)(\infty,1)-version of what is called the Cahiers topos. It contains in the manner of synthetic differential geometry also \infty-Lie groupoids with infinitesimally extended morphisms, which we may think of as ∞-Lie algebroids.

For a list of examples related to this (,1)(\infty,1)-topos see

Most examples of differential cohomology in an (,1)(\infty,1)-topos discussed below take place in LieGrpd\infty LieGrpd.

Grpd\infty Grpd

The archetypical (,1)(\infty,1)-topos is ∞Grpd itself. This is \infty-connected, for trivial reasons.

For interpreting the following statements, it is useful to think of the objects in Grpd\infty Grpd as precisely those objects in ∞LieGrpd with discrete smooth structure, under the embedding LConst:GrpdLieGrpdLConst : \infty Grpd \to \infty Lie Grpd.

In Grpd\infty Grpd the intrinsic path ∞-groupoid-functor is the identity – Π=Id\mathbf{\Pi} = Id – and hence every cocycle is uniquely a flat differential cocycle in the sense of flat differential cohomology in an (∞,1)-topos.

To get an impression for what this statement amounts to, notice that for instance the analog of a (not necessarily flat) U(1)U(1)-principal bundle in inftyLieGrpd\inftyLieGrpd over the 0-truncated group object U(1)LieGrpdU(1) \in \infty LieGrpd (the circle group) is, in this terminology, a flat B\mathbf{B}\mathbb{Z}-principal 2-bundle over the 1-truncated group object B\mathbf{B}\mathbb{Z} in Grpd\infty Grpd (the strict 2-group coming from the crossed module [0][\mathbb{Z} \to 0]). While often these concepts are conflated, their conceptual difference is important. For instance, while their classification happens to match over good enough base objects, already their cocycle groupoids differ: for XX a smooth manifold we have π 1LieGrpd(X,BU(1))=C (X,U(1))\pi_1 \infty LieGrpd(X,\mathbf{B}U(1)) = C^\infty(X,U(1)), while π 1Grpd(SingX,BB)*\pi_1 \infty Grpd(Sing X,\mathbf{B}\mathbf{B}\mathbb{Z}) \simeq *.

Accordingly, for groupal objects AGrpdA \in \infty Grpd the curvature characteristic class curv A:A dRBA*curv_A : A \to \mathbf{\flat}_{dR} \mathbf{B}A \simeq * is trivial in Grpd\infty Grpd and the notion of differential cohomology with groupal coefficients as the curv Acurv_A-twisted cohomology reduces to just the ordinary intrinsic cohomology in H=Grpd\mathbf{H} = \infty Grpd.

Nevertheless, the intrinsic Chern character in an (∞,1)-topos in mathfbH=Grpd\mathfb{H} = \infty Grpd is nontrivial: it coincides precisely with the rationalization map

ch A:AAR ch_A : A \mapsto A \otimes R

in rational homotopy theory.

Accordingly, differential cohomology in Grpd\infty Grpd in the sense of Chern-character twisted cohomology is essentially the (unstabilized) definition of traditional differential cohomology following Hopkins-Singer, Freed, Bunke-Schick and others, if for XX a smooth manifold in forming the homotopy fibers of

Grpd(SingX,A)ch AGrpd(SingX,A) \infty Grpd(Sing X, A) \stackrel{ch_A}{\to} \infty Grpd(Sing X, A \otimes \mathbb{R})

we do so explicitly over over singular cohomology-cocycles with values in \mathbb{R} that come from smooth differential forms on XX.

A potentially noteworthy point here is that from the intrinsic perspective of the (,1)(\infty,1)-topos Grpd\infty Grpd, neither the notion of smooth manifold nor that of differential forms exists, so that the pairing of homotopy theory and differential geometry in the standard definition of differential cohomology is somewhat ad hoc .1

However, using the richer context of the (,1)(\infty,1)-topos H=\mathbf{H} = ∞LieGrpd we may lift bare \infty-groupoids AGrpdA \in \infty Grpd to ∞-Lie groupoids A˜LieGrpd\tilde A \in \infty Lie Grpd, in that under geometric realization Π:LieGrpdGrpd\Pi : \infty Lie Grpd \to \infty Grpd we have Π(A˜)A\Pi(\tilde A) \simeq A. For instance BGrpd\mathbf{B}\mathbb{Z} \in \infty Grpd may be lifted to U(1)LieGrpdU(1) \in \infty Lie Grpd. But quite generally we also always have the lift A˜:=LConstA\tilde A := LConst A to be the \infty-Lie groupoid structure on AA with discrete smooth structure.

Then the intrinsic definition of differential cohomology with groupal coefficients applies and may give the expected answers (which is the case for the lift to U(1)U(1) as we show below in the section abelian n-gerbes with connection) or at least the homotopy fibers of the intrinsic Chern-character map are naturally computed over cocycles represented by smooth differential forms, since that concept does exists intrinsically in LieGrpd\infty Lie Grpd (as we discuss below in Ordinary de Rham cohomology).

In summary, this suggests that we think of the tradiional definition of differential cohomology as being the intrisic (,1)(\infty,1)-topos theoretic differential cohomology in H=LieGrpd\mathbf{H} = \infty LieGrpd restricted to coefficient objects that happen to be in the image of LConst:GrpdLieGrpdLConst : \infty Grpd \to \infty LieGrpd.

Stable differential cohomology

Here we discuss examples for Differential cohomology with groupal coefficients.

This applies when the coeficcient object AHA \in \mathbf{H} is an ∞-group-object in that it has a delooping BA\mathbf{B}A. In that case there is a canonical morphism

curv A:A dRBA curv_A : A \to \mathbf{\flat}_{dR} \mathbf{B}A

representing a class in the intrinsic de Rham cohomology of AA – the ∞-Maurer-Cartan form on AA – and we define differential cocycles to be the objects in the homotopy fiber of the induced morphism

H(,A)curv AH dR(,BA) \mathbf{H}(-,A) \stackrel{curv_A}{\to} \mathbf{H}_{dR}(-, \mathbf{B}A)

of cocycle ∞-groupoids.

Circle nn-bundles with connection

This section is at

Holonomy

The abstract topos-theoretic definition of holonomy of differential cocycles in an \infty-connected (,1)(\infty,1)-topos H\mathbf{H} on an object XX with coefficients in an ∞-group AA along a morphism ϕ:ΣX\phi : \Sigma \to X such that H dR(Σ,BA)*\mathbf{H}_{dR}(\Sigma,\mathbf{B}A) \simeq * identifies this notion with the canonical morphism

hol ϕ:H diff(X,A)H flat(Σ,A). hol_\phi : \mathbf{H}_{diff}(X,A) \to H_{flat}(\Sigma,A) \,.

We demonstrate now how for H=\mathbf{H} = ∞LieGrpd, for XX a closed smooth manifold and A=BB n1U(1)A = \mathbf{B} \mathbf{B}^{n-1}U(1) the delooping of the circle Lie n-group this abstract definition reproduces the ordinary definition of holonomy of U(1)U(1)-principal bundles and their higher analogs.

The first proposition asserts that in this case the set H flat(X,A)H_{flat}(X,A) is indeed canonically identified with the circle group U(1)U(1), which we expect the ordinary holonomy of circle nn-bundles to take values in. The proof that we present uses an insight pointed out in joint discussion by Domenico Fiorenza.

Proposition

Let knk \leq n \in \mathbb{N} and let X nkH=X_{n-k} \in \mathbf{H} = ∞LieGrpd be a closed paracompact smooth manifold of dimension nkn-k. Write τ k\tau_k for kk-truncation.

Then we have an equivalence

τ kH flat(X,B nU(1)) kU(1) \tau_k \mathbf{H}_{flat}(X,\mathbf{B}^n U(1)) \simeq \mathcal{B}^k U(1)

of ∞-group objects in ∞Grpd. In particular for X nX_n nn-dimensional, we have

H flat(X,B nU(1))U(1). H_{flat}(X,\mathbf{B}^n U(1)) \simeq U(1) \,.

Here, as usual, we write B nU(1)\mathbf{B}^n U(1) for the nnth delooping of the circle group U(1)U(1) regarded naturally as a group object in ∞LieGrpd, and nU(1)\mathcal{B}^n U(1) for the delooping in ∞Grpd, whose homotopy type is that of the Eilenberg-MacLane space K(n,U(1))K(n, U(1)).

Proof

By definition of flat differential cohomology and using the path ∞-groupoid adjunction we have

H flat(X,B nU(1))=LieGrpd(Π(X),B nU(1))Grpd(Π(X), nU(1)). \mathbf{H}_{flat}(X,\mathbf{B}^n U(1)) = \infty LieGrpd(\mathbf{\Pi}(X), \mathbf{B}^n U(1)) \simeq \infty Grpd(\Pi(X), \mathcal{B}^n U(1)) \,.

Since XX is assumed to be a paracompact manifold, we have by the discussion at geometric realization that Π(X)SingX\Pi(X) \simeq Sing X is equivalent to the singular simplicial complex of XX. Accordingly

Grpd(Π(X), nU(1))Top(X, nU(1)) \infty Grpd(\Pi(X), \mathcal{B}^n U(1)) \simeq Top(X, \mathcal{B}^n U(1))

is the cocycle \infty-groupoid of ordinary singular cohomology H (X,U(1))H^\bullet(X,U(1)) of XX. So far this is just a special case of the general statement about cohomology with constant coefficients in ∞LieGrpd.

Observe that from general considerations (recalled for instance at cohomology and at fiber sequence) we have for the homotopy groups of the cocycle \infty-groupoid that

π iGrpd(Π(X), nU(1))H ni(X,U(1)). \pi_i \infty Grpd(\Pi(X), \mathcal{B}^n U(1)) \simeq H^{n-i}(X, U(1)) \,.

We can evaluate this using that XX is an (nk)(n-k)-dimensional manifold in the universal coefficient theorem, which asserts that we have an exact sequence

0Ext 1(H ni1(X,))H ni(X,U(1))Ab(H ni(X,),U(1))0. 0 \to Ext^1(H_{n-i-1}(X,\mathbb{Z})) \to H^{n-i}(X,U(1)) \to Ab(H_{n-i}(X,\mathbb{Z}), U(1)) \to 0 \,.

Since the circle group U(1)U(1) is an injective \mathbb{Z}-module the left term vanishes, Ext 1(,U(1))=0Ext^1(-, U(1)) = 0.

Therefore we have an isomorphism

H n1(X,U(1))Ab(H ni(X,),U(1)) H^{n-1}(X, U(1)) \simeq Ab(H_{n-i}(X,\mathbb{Z}), U(1))

between the cohomology group in question and the abelian group of homomorphisms from the homology group H ni(X,)H_{n-i}(X,\mathbb{Z}) to U(1)U(1).

By assumption on XX we have for i<ki \lt k that H ni(X,)=0H_{n-i}(X,\mathbb{Z}) = 0 and for i=ki = k that H ni(X,)=H_{n-i}(X,\mathbb{Z}) = \mathbb{Z}. It follows that

π iLieGrpd(X,B nU(1))={0 fori<k U(1) fori=k. \pi_i \infty LieGrpd(X, \mathbf{B}^n U(1)) = \left\{ \array{ 0 & for\; i \lt k \\ U(1) & for i = k } \right. \,.

Nonabelian differential cohomology

Here we discuss realization of the theory of Differential cohomology in an (∞,1)-topos with general coefficients.

We had seen above that a connection on a B nU(1)\mathbf{B}^n U(1)-principal ∞-bundle is an intermediate step in a model for the curvature characteristic class morphism

curv B nU(1)H(,B nU(1))H dR(,B n+1U(1)), curv_{\mathbf{B}^n U(1)} \mathbf{H}(-,\mathbf{B}^n U(1)) \to \mathbf{H}_{dR}(-, \mathbf{B}^{n+1}U(1)) \,,

namely a cocycle with values in the simplicial presheaf B nU(1) diff\mathbf{B}^n U(1)_{diff} that appears as the correspondence space of a span

B nU(1) diff,chn dRB n+1U(1) chn B nU(1) \array{ \mathbf{B}^n U(1)_{diff,chn} &\to& \mathbf{\flat}_{dR} \mathbf{B}^{n+1} U(1)_{chn} \\ \downarrow^{\simeq} \\ \mathbf{B}^n U(1) }

(an \infty-anafunctor) of simplicial presheaves that represents curv B nU(1)curv_{\mathbf{B}^n U(1)} in the model [CartSp op,sSet] proj,cov[CartSp^{op}, sSet]_{proj,cov}. We had also seen above that an equivalent model for B nU(1) diff,simp\mathbf{B}^n U(1)_{diff,simp} is given by relative Lie integration of the morphism b n1inn(b n1)b^{n-1}\mathbb{R} \to inn(b^{n-1}\mathbb{R}) of ∞-Lie algebras.

That latter model has an evident generalization BG diff\mathbf{B}G_{diff} to arbitrary ∞-Lie algebras and the ∞-Lie groups GG integrating them. Cocycles with values in such BG diff,simp\mathbf{B}G_{diff,simp} we discuss below as

These generalize the ordinary notion of a connection on a bundle to a notion of connections of GG-principal ∞-bundles, including connections on nonabelian gerbes.

As before in the abelian case, such connections may be understood as an interpolating step in a concrete model for an intrinsic (∞,1)-topos-theoretic notion of curvature characteristic classes, namely for the intrinsic Chern character?

ch BG:BG(ΠBG) ch_{\mathbf{B}G} : \mathbf{B}G \to (\mathbf{\Pi}\mathbf{B}G) \otimes \mathbb{R}

in H=\mathbf{H} = ∞LieGrpd. However, for general non-abelian GG, the relation between the intrinsic notion of curvature and its model by connections is a bit more involved than in the abelian case: instead of using a single span, we need to model a collection of morphisms

  • into the stages of the Postnikov tower-decomposition of (ΠBG)(\mathbf{\Pi}\mathbf{B}G) \otimes \mathbb{R};

  • and out of the stages of a smooth refinement of the intrinsic Whitehead tower of BG\mathbf{B}G.

This we discuss below in

Each smoothly refined stage in this decomposition corresponds to a invariant polynomial on the ∞-Lie algebra 𝔤\mathfrak{g} of GG and yields a curvature characteristic form obtained by evaluting this polynomial on the curvature? of the \infty-Lie algebra valued connection. The resulting map from GG-principal ∞-bundles to curvature characteristic forms subsumes and generalizes the ordinary Chern-Weil homomorphism. This we discuss in

But more is true: the combination of the underlying characteristic class in degree n+1n+1 with its curvature differential form lifts to a differential character: a cocycle in the abelian differential cohomology H diff(,B nU(1))\mathbf{H}_{diff}(-, \mathbf{B}^n U(1)). These refined differential characteristic classes

H(,BG)H diff(,B nU(1)) \mathbf{H}(-,\mathbf{B}G) \to \mathbf{H}_{diff}(-,\mathbf{B}^n U(1))

provide a differential refinement of the twisted cohomology induced by the bare underlying characteristic classes. As examples for this we discuss

\infty-Chern-Weil theory

The content of this section is at

Tools

We summarize here for reference and for convenience a handful of standard computational tools that we use to extract concrete models from the abstract (,1)(\infty,1)-topos-theoretic definitions.

For abelian differential cohomology

The main tools for abelian differential cohomology are those of abelian sheaf cohomology.

  1. Dold-Kan correspondence for presenting \infty-stacks with values in strictly abelian strict \infty-groupoids in terms of chain complexes of sheaves;

  2. Cech cohomology for computing derived hom-spaces between \infty-stacks;

  3. Partitions of unity for finding coboundaries between Cech hypercohomology cocyces.

Dold-Kan correspondence

We recall a basic tool for the discussion of stable differential cohomology: the Dold-Kan correspondence. This serves to model ∞-stacks with values in strictly abelian strict ∞-groupoids equivalently as sheaves with values in chain complexes of abelian groups. For more discussion of this standard procedure in cohomology-theory see the discussion at abelian sheaf cohomology.

We write

Ξ:=Hom Ch (N Δ[],):Ch +sSet \Xi := Hom_{Ch_\bullet}(N_\bullet \Delta[\bullet], -): Ch_\bullet^+ \to sSet

for the Dold-Kan map from non-negatively graded chain complexes of abelian groups to sSet. Notice that this map factors through crossed complexes and then through Kan complexes, hence in particular lands in fibrant objects in the standard model structure on simplicial sets. More generally, a fundamental property of the Dold-Kan correspondence that we make use of repeatedly is that Ξ\Xi preserves projective fibrations and weak equivalences: a degreewise surjection of chain complexes is sent to a Kan fibration and a quasi-isomorphism of chain complexes is sent to a weak homotopy equivalence of Kan complexes.

By convenient abuse of notation, we use the same symbol Ξ\Xi also for the evident objectwise extension of this functor to presheaves on a small category CC:

Ξ:[C op,Ch +][C op,sSet]. \Xi : [C^{op}, Ch_\bullet^+] \to [C^{op}, sSet] \,.

With the above remark we have the following

Standard fact

The functor Ξ:[C op,Ch ] proj[C op,sSet Quillen] proj\Xi : [C^{op}, Ch_\bullet]_{proj} \to [C^{op}, sSet_{Quillen}]_{proj} is a right Quillen functor that preserves all weak equivalences:

  1. it is a right adjoint and hence preserves all limits;

  2. For f:A B f : A_\bullet \to B_\bullet a morphims in [C op,Ch ][C^{op}, Ch_\bullet] that is a surjection for each object UCU \in C and in each degree nn \in \mathbb{N} we have that Ξ(f)\Xi(f) is a fibration in [C op,sSet] proj[C^{op}, sSet]_{proj}.

  3. If ff is over each UCU \in C a quasi-isomorphism then Ξ(f)\Xi(f) is a weak equivalence in [C op,sSet] proj[C^{op}, sSet]_{proj}.

The left adjoint functor

N :sAbCh + N_\bullet : sAb \to Ch_\bullet^+

to Ξ\Xi is the normalized chain complex functor. Of the various isomorphic definitions of this for our purposes the following is the most useful one:

for a simplicial abelian group AA with face and degeneracy maps {δ j k},{σ j k}\{\delta^k_j\}, \{\sigma^k_j\}, respectively, we have that the corresponding normalized chain complex N (A)N_\bullet(A) is

  • in degree kk given by the group A kA_k modulo the images of the degeneracy maps

    N k(A)=A k/ jimσ j N_k(A) = A_k / \prod_j im \sigma_j
  • with differential given by the alternating sum of the face maps

    N k(A):N k(A) i(1) iδ iN k1(A). \partial_{N_k(A)} : N_k(A) \stackrel{\sum_i (-1)^i \delta_i}{\to} N_{k-1}(A) \,.

Again, by convenient abuse of notation we also write

N :[CartSp op,sAb][CartSp op,Ch +] N_\bullet : [CartSp^{op}, sAb] \to [CartSp^{op}, Ch_\bullet^+]

for the evident objectwise prolongation of N N_\bullet to presheaves of simplicial abelian groups.

Cech cohomology

For computing the intrinsic cohomology of an (∞,1)-topos H\mathbf{H} – and its cocycle ∞-groupoids, we may present it by a model structure on simplicial presheaves on some sSet-site CC (which in many of our examples is an ordinary 1-categorical site) as

H([C op,sSet] proj,cov) . \mathbf{H} \simeq ([C^{op}, sSet]_{proj,cov})^\circ \,.

This is reviewed at models for ∞-stack (∞,1)-toposes. In terms of this, the cocycle \infty-groupoids are derived hom-spaces whose computation is effectively that of (possibly nonabelian) Cech? hypercohomology.

Proposition

In the model structure on simplicial presheaves [CartSp op,sSet] proj,cov[CartSp^{op}, sSet]_{proj,cov} modelling the (,1)(\infty,1)-topos H=\mathbf{H} = ∞LieGrpd a cofibrant resolution of a paracompact smooth manifold XX is given by the Cech nerve C({U i})XC(\{U_i\}) \to X of any good open cover of XX.

Corollary

If A=Ξ𝒜 inftyLieGrpdA = \Xi \mathcal{A}_\bullet \in \inftyLieGrpd is represented by a fibrant object in [CartSp op,sSet] proj[CartSp^{op}, sSet]_{proj} in the image of the Dold-Kan map as discussed above, then the intrinsic cocycle \infty-groupoid on a paracompact manifodl XX with coefficients in AA coinicides with the abelian sheaf cohomology cocycle \infty-groupoid for 𝒜 \mathcal{A}_\bullet computed as the Cech hypercohomology with respect to a good open cover {U iX}\{U_i \to X\} of XX.

An nn-cocycle cc then is given by a sequence of elements

(C i,B ij,A ijk,,Z i 1,,i n) (C_i, B_{i j}, A_{i j k}, \cdots , Z_{i_1, \cdots , i_n})

with C i𝒜 0(U i)C_i \in \mathcal{A}_0(U_i), B ij𝒜 1(U iU j)B_{i j} \in \mathcal{A}_1(U_i \cap U_j), etc., which is closed under the double differential D=d 𝒜±δD = d_{\mathcal{A}} \pm \delta, where δ\delta is the alternating sum of the pullbacks along the face maps of the Cech nerve.

Partition of unity

(…) partition of unity (…)

The Cech nerve C({U i})XC(\{U_i\}) \to X of a manifold has itself trivial cohomology in positive degree, with respect to the differential on degreewise functions

δ= k(1) kδ k * \delta = \sum_k (-1)^k \delta_k^*

given by the alternating sum of the pullbaks along the face maps.

An explicit choice of coboundaries δλ=η\delta \lambda = \eta for a given δη=0\delta \eta = 0 is obtained from a choice of partition of unity

(ρ iC (U i,))(\rho_i \in C^\infty(U_i, \mathbb{R})) subordinate to the cover {U iX}\{U_i \to X\}.

λ i 1,,i n:= i 0η i 0,i 1,,i n. \lambda_{i_1, \cdots, i_n} := \sum_{i_0} \eta_{i_0, i_1, \cdots, i_n} \,.

(…)

For nonabelian differential cohomology

The familiar tools for embedding abelian sheaf cohomology into the intrinsic cohomolog of the (∞,1)-topos over the given site that we recalled above have well known – albeit maby not as widely known – generalizations into the regime of nonabelian cohomology. These are known as tools in nonabelian algebraic topology and nonabelian homological algebra. We briefly recall what we need below.

Nonabelian Dold-Kan

(…)

Let Ξ:CrsdCplxsSet\Xi : CrsdCplx \to sSet now denote the inclusion of crossed complexes into all ∞-groupoids.

Ξ:CrsdCplxKanCplx. \Xi : CrsdCplx \to KanCplx \,.

This generalizes the functor Ξ\Xi used in the abelian case above as we have a sequence of inclusions

ChainCplx CrsdCplx KanCplx StrAbStrGrpd StrGrpd Grpd. \array{ ChainCplx &\to& CrsdCplx &\to& KanCplx \\ \downarrow^{\mathrlap{\simeq}} && \downarrow^{\mathrlap{\simeq}} && \downarrow^{\mathrlap{\simeq}} \\ StrAb Str\infty Grpd &\to& Str \infty Grpd &\to& \infty Grpd } \,.

See also the cosmic cube of higher category theory.

Again, we use the same notation for the objectwise extension of this function to presheaves

Ξ:[C op,CrsdCplx][C op,KanCplx]. \Xi : [C^{op}, CrsdCplx] \to [C^{op}, KanCplx] \,.

(…)

Nonabelian Cech cohomology

(…)

see Cech cohomology

(…)


Context

  1. survey

  2. general structures

  3. paths

  4. Lie theory

  5. differential cohomology

  6. examples

  7. references


  1. At the Fourth Annual Meeting of the Dutch Quantum Theory and Geometry Cluster conference in Amsterdam in June 2010, Sir Michael Atiyah raised this point in the discussion session after a talk about differential K-theory: he argued that the standard definition is “ugly” in that it mixes concepts from “different worlds” and suggested to look for a modified definition built coherently in a single context.

Revised on August 18, 2010 15:17:43 by Urs Schreiber (131.211.36.96)