nLab
relative cohomology

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Cohomology

cohomology

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Contents

Idea

Where cohomology classifies cocycles on an object XX with coefficients in some object AA, relative cohomology for a map morphism YXY \to X classifies cocycles on XX that satisfy some condition when pulled back to YY, such as being trivializable there. Or rather: cocycles equipped with some structure on their pullback to YY, such as a trivialization.

Definition

We first give a general abstract definition and then reduce to certain special cases.

General definition

Recall the general abstract definition of cohomology, as discussed there:

for H\mathbf{H} an (∞,1)-topos, and X,AHX, A \in \mathbf{H} two objects, a cocycle on XX with coefficients in AA is a morphism XAX \to A, the ∞-groupoid of cocycles, coboundaries and higher coboundaries is the (∞,1)-categorical hom space H(X,A)\mathbf{H}(X,A) and the cohomology classes themselves are the connected components / homotopy classes in there

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

Let i:YXi : Y \to X and f:BAf : B \to A be two morphisms in H\mathbf{H}. Then the relative cohomology of XX with coefficients in AA relative to these morphisms is the connected components of the \infty-groupoid of relative cocycles

H Y B(X,A):=π 0H I(YiX,BfA), H_{Y}^B(X,A) := \pi_0 \mathbf{H}^I(Y \stackrel{i}{\to} X\;,\; B \stackrel{f}{\to} A) \,,

where H I\mathbf{H}^I the arrow (∞,1)-topos of H\mathbf{H}, hence the (∞,1)-category of (∞,1)-functors IHI \to \mathbf{H}, where II is the interval category.

Remark

Often relative cohomology is considered for the special case where AA is a pointed object, B=*B = * is the terminal object and B*AB \simeq * \to A is the point inclusion. In this case we may write just

H Y(X,A):=H Y *(X,A). H_Y(X,A) := H_Y^*(X,A) \,.
Remark

The \infty-groupoid of relative cocycles is the (∞,1)-pullback in

H I(YX,BA) H(Y,B) f * H(X,A) i * H(Y,A). \array{ \mathbf{H}^I(Y \to X, B \to A) &\to& \mathbf{H}(Y, B) \\ \downarrow && \downarrow^{\mathrlap{f_*}} \\ \mathbf{H}(X, A) &\stackrel{i^*}{\to}& \mathbf{H}(Y,A) } \,.

This makes manifest the interpretation of relative cocycles as AA-cocycles on XX whose restriction to YY is equipped with a coboundary to a BB-cocycle on YY.

If BB is the point then this means: AA cocycles on XX which are equipped with a trivialization on YY.

Remark

If we fix an AA-cocycle c\mathbf{c} on XX, then we may consider the relative cohomology just of this cocycle, hence the homotopy fiber of the \infty-groupoid of relative cocycles over c\mathbf{c}

H c I(i,f) H I(i,f) * c H(X,A). \array{ \mathbf{H}^I_{\mathbf{c}}(i,f) &\to& \mathbf{H}^I(i,f) \\ \downarrow && \downarrow \\ * &\stackrel{\mathbf{c}}{\to}& \mathbf{H}(X,A) } \,.

By the pasting law it follows that this fiber is equivalently given by the following (∞,1)-pullback

H c I(i,f) H(Y,B) * i *c H(Y,A). \array{ \mathbf{H}^I_{\mathbf{c}}(i,f) &\to& \mathbf{H}(Y,B) \\ \downarrow && \downarrow \\ * &\stackrel{i^* \mathbf{c}}{\to}& \mathbf{H}(Y,A) } \,.

Comparison shows that this identifies H c I(i,f)\mathbf{H}^I_{\mathbf{c}}(i,f) as the cocycle \infty-groupoid of the [i *c][i^* \mathbf{c}]-twisted cohomology of YY with coefficients in the homotopy fiber of ff.

See for instance the example of twisted bundles on D-branes below.

In chain complexes

A special case of the general definition of cohomology is abelian sheaf cohomology, obtained by taking the coefficient objects AA and BB to be in the image of chain complexes under the Dold-Kan correspondence.

If moreover we restrict attention to the case that B=*B = *, then by remark 1 the relative cohomology is given by the homotopy fiber of a morphism of cochain complexes C (X,A)C (Y,A)C^\bullet(X,A) \to C^\bullet(Y,A), presenting the morphism H(X,A)H(Y,A)\mathbf{H}(X, A) \to \mathbf{H}(Y,A). Such homotopy fibers are given for instance by dual mapping cone complexes. Accordingly, in the abelian case the AA-cohomology on XX relative to YY is the cochain cohomology of this mapping cone complex. This is the definition one finds in much traditional literature.

Examples

Twisted bundles on D-branes

We consider an example of relative cohomology for the general case where BAB \to A is not the point inclusion, but exhibits an additional twist, according to 2.

This example is motivated from the physics of D-branes in type II string theory, as well as from twisted K-theory.

Let the ambient (∞,1)-topos be H:=\mathbf{H} := Smooth∞Grpd. For any nn \in \mathbb{N} consider the sequence of Lie groups

U(1)U(n)PU(n), U(1) \to U(n) \to PU(n) \,,

exhibiting the unitary group as a central extension of groups of the projective unitary group. The fiber sequence on delooping smooth ∞-groupoids induced by this is

BU(1)BU(n)BPU(n)fB 2U(1). \mathbf{B}U(1) \to \mathbf{B}U(n) \to \mathbf{B} PU(n) \stackrel{f}{\to} \mathbf{B}^2 U(1) \,.

For YY a smooth manifold, the ff-twisted cohomology of XX is classifies U(n)U(n)-principal twisted bundles on YY, as discussed there. These appear in string theory/twisted K-theory, where the twist is a B-field/circle 2-bundle/bundle gerbe classified by YB 2U(1)Y \to \mathbf{B}^2 U(1).

But in these applications, this 2-bundle is the restriction of an ambient 2-bundle classified by c:XB 2U(1)\mathbf{c} : X \to \mathbf{B}^2 U(1) on a spacetime XX along an embedding YXY \hookrightarrow X of the D-brane worldvolume. Therefore the moduli stack of total field configurations consisting of the ambient B-field and the gauge field on the D-brane (see Freed-Witten anomaly for the details of this mechanism) is the \infty-groupoid of (YX)(Y \hookrightarrow X)-relative cocycles with coefficients in (BPUB 2U(1))(\mathbf{B}PU \to \mathbf{B}^2 U(1)).

Revised on January 18, 2013 03:52:11 by Urs Schreiber (203.116.137.162)