nLab
relative cohomology

Idea

Where cohomology classifies cocycles on an object $X$ with coefficients in some object $A$, relative cohomology for a map morphism $Y\to X$ classifies cocycles on $X$ that satisfy some condition when pulled back to $Y$, such as being trivializable there. Or rather: cocycles equipped with some structure on their pullback to $Y$, 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 $H$ an (∞,1)-topos, and $X,A\in H$ two objects, a cocycle on $X$ with coefficients in $A$ is a morphism $X\to A$, the ∞-groupoid of cocycles, coboundaries and higher coboundaries is the (∞,1)-categorical hom space $H(X,A)$ and the cohomology classes themselves are the connected components / homotopy classes in there

$$H(X,A):={\pi }_{0}H(X,A).$$H(X,A) := \pi_0 \mathbf{H}(X,A) \,.

Definition

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

$$H_Y^B(X,A):={\pi }_0{H}^I(Y\stackrel{i}{\to }X,B\stackrel{f}{\to }A),$$H_{Y}^B(X,A) := \pi_0 \mathbf{H}^I(Y \stackrel{i}{\to} X\;,\; B \stackrel{f}{\to} A) \,,

where $H^I$ the arrow (∞,1)-topos of $H$, hence the (∞,1)-category of (∞,1)-functors $I\to H$, where $I$ is the interval category.

Remark

Often relative cohomology is considered for the special case where $A$ is a pointed object, $B=*$ is the terminal object and $B\simeq *\to A$ 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 $\mathrm{\infty }$-groupoid of relative cocycles is the (∞,1)-pullback in

$$\begin{array}{ccc}{H}^I(Y\to X,B\to A)& \to & H(Y,B)\\ \downarrow & & {\downarrow }^{f_{*}}\\ H(X,A)& \stackrel{i^{*}}{\to }& H(Y,A)\end{array}.$$\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 $A$-cocycles on $X$ whose restriction to $Y$ is equipped with a coboundary to a $B$-cocycle on $Y$.

If $B$ is the point then this means: $A$ cocycles on $X$ which are equipped with a trivialization on $Y$.

Remark

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

$$\begin{array}{ccc}{H}_c^I(i,f)& \to & H^I(i,f)\\ \downarrow & & \downarrow \\ *& \stackrel{c}{\to }& H(X,A)\end{array}.$$\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

$$\begin{array}{ccc}{H}_c^I(i,f)& \to & H(Y,B)\\ \downarrow & & \downarrow \\ *& \stackrel{i^{*}c}{\to }& H(Y,A)\end{array}.$$\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 $H_c^I(i,f)$ as the cocycle $\mathrm{\infty }$-groupoid of the $[i^{*}c]$-twisted cohomology of $Y$ with coefficients in the homotopy fiber of $f$.

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 $A$ and $B$ to be in the image of chain complexes under the Dold-Kan correspondence.

If moreover we restrict attention to the case that $B=*$, then by remark 1 the relative cohomology is given by the homotopy fiber of a morphism of cochain complexes $C^{•}(X,A)\to C^{•}(Y,A)$, presenting the morphism $H(X,A)\to H(Y,A)$. Such homotopy fibers are given for instance by dual mapping cone complexes. Accordingly, in the abelian case the $A$-cohomology on $X$ relative to $Y$ 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 $B\to A$ 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 $H:=$ Smooth∞Grpd. For any $n\in \mathbb{N}$ consider the sequence of Lie groups

$$U(1)\to U(n)\to \mathrm{PU}(n),$$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

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

For $Y$ a smooth manifold, the $f$-twisted cohomology of $X$ is classifies $U(n)$-principal twisted bundles on $Y$, 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 $Y\to B^{2}U(1)$.

But in these applications, this 2-bundle is the restriction of an ambient 2-bundle classified by $c:X\to B^{2}U(1)$ on a spacetime $X$ along an embedding $Y\hookrightarrow X$ 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 $\mathrm{\infty }$-groupoid of $(Y\hookrightarrow X)$-relative cocycles with coefficients in $(B\mathrm{PU}\to B^{2}U(1))$.