differential cohomology > old


The smooth (∞,1)-topos H\mathbf{H} equipped with the path ∞-groupoid functor

Π():HH \Pi(-) : \mathbf{H} \to \mathbf{H}

and its right adjoint

HH:() flat \mathbf{H} \leftarrow \mathbf{H} : {(-)_{flat}}

induces a notion of flat differential cohomology. Flat cohomology with coefficients in AA is the ordinary cohomology with coefficients in the flat differential refinement A flatA_{flat} of AA:

H flat(X,A):=H(Π(X),A)H(X,A flat). H_{flat}(X,A) := H(\Pi(X),A) \simeq H(X,A_{flat}) \,.

In as far as cohomology H(X,A)H(X,A) classifies principal ∞-bundles, flat differential cohomology classifies principal ∞-bundles with flat connection.

As discussed at path ∞-groupoid there is a natural morphism A flatAA_{flat} \to A. The corresponding morphism H flat(X,A)H(X,A)H_{flat}(X,A) \to H(X,A) sends a differential cocycle for an ∞-bundle with connection to the cocycle for the underlying bundle without connection.

Conversely, lifting an AA-cocycle through A flatAA_{flat} \to A means equipping a plain ∞-bundle with a flat connection.

This leads us to a first characterization of general differential cohomology

In an (∞,1)-topos H\mathbf{H} equipped with a path ∞-groupoid functor Π()\Pi(-) and the corresponding flat differential refinement functor () flat:HH(-)_{flat} : \mathbf{H} \to \mathbf{H} we conceive differential cohomology with coefficients in AHA \in \mathbf{H} as the obstruction theory of lifts of cocycles through the morphism

H(X,A flat)H(X,A) \mathbf{H}(X,A_{flat}) \to \mathbf{H}(X,A)

from AA-valued cohomology to flat differential AA-valued cohomology.

There is a good obstruction theory for lifts through a morphism f:AAf : A' \to A when ff is the homotopy fiber of a morphism h:ABh : A \to B. In that case the obstruction to lifing a cocycle g:XAg : X \to A to a cocycle g^:XA\hat g : X \to A' is the class of the BB-valued cocycle XABX \to A \to B.

We exhibit such a fibration sequence that realizes A flatAA_{flat} \to A as a homotoy fiber

A flatABA dR A_{flat} \to A \to \mathbf{B}A_{dR}

for the case that AA is group-like in that its delooping BA\mathbf{B}A exists. The object BA dR\mathbf{B}A_{dR} here we interpret as the coefficient object for AA-valued nonabelian deRham cohomology.

The induced morphism on cohomology

char:H(X,A)H(X,BA dR) char : \mathbf{H}(X,A) \to \mathbf{H}(X,\mathbf{B}A_{dR})

we interpret as a (possibly nonabelian refinement) of the assignment of curvature characteristic classes to the principal ∞-bundles classified by H(X,A)H(X,A).

Every fibration sequence induces a notion of twisted cohomology. We conceive general (non-flat, nonabelian) differential cohomology as thetwisted cohomology whose twist is given by the curvature characteristic classes charchar.

This gives a precise and useful meaning to the following very plausible-sounding statement

Differential cocycles \nabla with curvature characteristic class [char()][char(\nabla)] are cocycles in [char()][char(\nabla)]-twisted flat differential cohomology.

In order to construct the objects BA dR\mathbf{B}A_{dR} and the corresponding fibration sequence we use techniques from relative cohomology. In terms of that the nonabelian deRham object in question is, as smooth ∞-stacks

BA dR:U[U Π(U),* BA]. \mathbf{B}A_{dR} : U \mapsto \left[ \array{ U \\ \downarrow \\ \Pi(U) } \,, \array{ {*} \\ \downarrow \\ \mathbf{B}A } \right] \,.

This is by construction the classifying object for those flat differential BA\mathbf{B}A-cocycles whose underlying plain BA\mathbf{B}A-cocycles are trivial.

In terms of principal ∞-bundles with connecton this means that

  • flat AA-principal bundles with connection are the fundamental notion;

  • BA\mathbf{B}A-valued differential forms are identified with those flat AA-principal bundles with connection whose underlying AA-principal bundle is trivial.

So to summarize the situation for grouplike coefficients: following the general principle of twisted cohomology applied to the fibration sequence A flatABA dRA_{flat} \to A \to \mathbf{B}A_{dR} the differential cohomology of XX with coefficients in AA and with curvature characteristic class PP is the homotopy pullback

H [P](X,A) * *P H(X,A) char H(Π(X),BA). \array{ \mathbf{H}^{[P]}(X,A) &\to& {*} \\ \downarrow && \;\downarrow^{{*} \mapsto P} \\ \mathbf{H}(X,A) &\stackrel{char}{\to} & \mathbf{H}(\Pi(X), \mathbf{B}A) } \,.

One finds that the cocycles H [P](X,A)\nabla \in \mathbf{H}^{[P]}(X,A) are presented by diagrams in H\mathbf{H} of the form

X g A underlyingcocycle Π(X) EA connection Π(X) Pchar() BA characteristicforms. \array{ X &\stackrel{g}{\to}& A &&& underlying cocycle \\ \downarrow && \downarrow &&& \\ \Pi(X) &\stackrel{\nabla}{\to}& \mathbf{E}A &&& connection \\ \downarrow && \downarrow &&& \\ \Pi(X) &\stackrel{P \simeq char(\nabla)}{\to} & \mathbf{B}A &&& characteristic forms } \,.

It is in these diagrams that the notion of connection manifests itself: the connection is the choice of a morphism :Π(X)EA\nabla : \Pi(X) \to \mathbf{E}A that interpolates between the plain AA-valued cocycle g:XAg : X \to A (a principal ∞-bundle with connection) and its curvature characteristic nonabelian deRham forms P:Π(X)BAP : \Pi(X) \to \mathbf{B}A.

Every connection in this sense induces (nonabelian) deRham differential form data that yields a description of these connections in a way that generalizes the Ehresmann connection formulation of a connection on a principal bundle.

This leads to the description of differential cocycles in terms of

If AA is not grouplike, one can uses grouplike approximations to AA and proceed with these through the above constructions. One drastic but useful grouplike approximation to any AA is given by an integral Chern character morphism

ch:A iB n iR//Z. ch : A \to \prod_i \mathbf{B}^{n_i} R//Z \,.

From this one obtains along the above lines a coefficient object

BA ch:U[U Π(U),* B iB n iR]. \mathbf{B}A_{ch} : U \mapsto \left[ \array{ U \\ \downarrow \\ \Pi(U) } \,, \array{ {*} \\ \downarrow \\ \mathbf{B}\prod_i \mathbf{B}^{n_i} R } \right] \,.

equipped with a morphism ABA chA \to \mathbf{B}A_{ch} such that the induced morphism on cohomology

ch:H(,A)H(,BA ch) ch : \mathbf{H}(-,A) \to \mathbf{H}(-, \mathbf{B}A_{ch})

generalizes the Chern character morphism in abelian cohomology.

Proceeding with this morphism as a substitute for the non-existing ABA dRA \to \mathbf{B}A_{dR} produces a the notion of differential nonabelian cohomology that measures obstructions to flatness only up to some approximation, but that is direct generalization of the defintition of classical abelian differential cohomology in terms of homotopy fibers of the Chern character map.


Fix a site CC as described at path ∞-groupoid.

Write SPSh(C) locSPSh(C)^{loc} for the injective or projective local model structure on simplicial presheaves on CC. This is a combinatorial simplicial model category.

Fix a cosimplicial object Δ C:ΔC\Delta_C : \Delta \to C as described at path ∞-groupoid at let Π():SPSh(C)SPSh(C)\Pi(-) : SPSh(C) \to SPSh(C) be the corresponding path \infty-groupoid functor and () flat(-)_{flat} its right adjoint.

Recall from path ∞-groupoid the definition of flat differential cohomology.

Definition (flat differential cohomoloy)

For ASPSh(C)A \in SPSh(C) the flat differential cohomology with coefficients in AA is cohomology with coefficients in A flatA_{flat}.

H flat(X,A):=π 0H(X,A flat). H_{flat}(X,A) := \pi_0 \mathbf{H}(X,A_{flat}) \,.

By the differential Quillen adjunction? a cocycle in H flat(X,A)H_{flat}(X,A) is represented by a morphism Π(X)A\Pi(X) \to A.

Since we have the canonical natural inclusion XΠ(X)X \to \Pi(X) we may consider relative cohomology with respect to this.

At nonabelian deRham cohomology the following definition is discussed in detail

Definition (deRham differential refinement)

For ASPSh(C)A \in SPSh(C) a pointed object with point pt A:*Apt_A : {*} \to A define A dRSPSh(C)A_{dR} \in SPSh(C) by

A dR:U[I,SPSh(C)](U Π(U),* pt A A). A_{dR} : U \mapsto [I,SPSh(C)] \left( \array{ U \\ \downarrow \\ \Pi(U) } \,,\; \array{ {*} \\ \downarrow^{pt_A} \\ A } \right) \,.

This we call the deRham differential refinement of AA.

The cohomology with coefficients in A dRA_{dR}

H dR(X,A):=π 0H(X,A dR) H_{dR}(X,A) := \pi_0 \mathbf{H}(X,A_{dR})

we call AA-valued deRham cohomology

Here [I,SPSh(C)][I,SPSh(C)] is the arrow category of SPSh(C)SPSh(C) as described at relative cohomology.


We conceive differential cohomology with coefficients in AA as being the obstruction classes to lifts from bare AA-cohomology to flat differential AA-cohomology through H(X,A flat)H(X,A)\mathbf{H}(X,A_{flat}) \to \mathbf{H}(X,A).

There are two cases to be distinguished in such an obstruction problem:

  • in the case that A diffAA_{diff} \to A is the homotopy fiber of some morphism AQA \to Q, the obstruction classes live precisely in the A flatA_{flat}-twisted cohomology defined by that morphism;

    in this case the obstuction classes in QQ-cohomology are precise: they vanish if and only if the lift exists: this is just the restatement of the assumed homotopy pullback property of A diffA_{diff}

  • in the case that A flatAA_{flat} \to A is not the homotopy fiber of any morphism in that it does not fit into a square

    A flat * A Q \array{ A_{flat} &\to& {*} \\ \downarrow && \downarrow \\ A &\to& Q }

    that is a homotopy pullback square, we don’t have precise obstruction classes as above. But as a next best approximation we can still find necessary (but generally not sufficient) obstruction classes by finding any homotopy commutative square as above.

    For any such square with some QQ we get the statement: for a bare AA-cocycle g:XAg : X \to A to have a lift g^:XA flat\hat g : X \to A_{flat} to a flat differential AA-cocycle the QQ-cocycle XgAQX \stackrel{g}{\to} A \to Q necessarily has to be have trivial cohomology class. This is just what the homotopy commutativity of this square means.

    This is discussed at differential cohomology - with general coefficients .

Revised on May 5, 2010 22:23:59 by Urs Schreiber (