algebraic topology – application of higher algebra and higher category theory to the study of (stable) homotopy theory
group cohomology, nonabelian group cohomology, Lie group cohomology
Hochschild cohomology, cyclic cohomology?
cohomology with constant coefficients / with a local system of coefficients
differential cohomology
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.
We first give a general abstract definition and then reduce to certain special cases.
Recall the general abstract definition of cohomology, as discussed there:
for $\mathbf{H}$ an (∞,1)-topos, and $X, A \in \mathbf{H}$ 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 $\mathbf{H}(X,A)$ and the cohomology classes themselves are the connected components / homotopy classes in there
Let $i : Y \to X$ and $f : B \to A$ be two morphisms in $\mathbf{H}$. Then the relative cohomology of $X$ with coefficients in $A$ relative to these morphisms is the connected components of the $\infty$-groupoid of relative cocycles
where $\mathbf{H}^I$ the arrow (∞,1)-topos of $\mathbf{H}$, hence the (∞,1)-category of (∞,1)-functors $I \to \mathbf{H}$, where $I$ is the interval category.
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
The $\infty$-groupoid of relative cocycles is the (∞,1)-pullback in
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 $Y$.
If we fix an $A$-cocycle $\mathbf{c}$ on $X$, then we may consider the relative cohomology just of this cocycle, hence the homotopy fiber of the $\infty$-groupoid of relative cocycles over $\mathbf{c}$
By the pasting law it follows that this fiber is equivalently given by the following (∞,1)-pullback
Comparison shows that this identifies $\mathbf{H}^I_{\mathbf{c}}(i,f)$ as the cocycle $\infty$-groupoid of the $[i^* \mathbf{c}]$-twisted cohomology of $Y$ with coefficients in the homotopy fiber of $f$.
See for instance the example of twisted bundles on D-branes below.
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 the relative cohomology is given by the homotopy fiber of a morphism of cochain complexes $C^\bullet(X,A) \to C^\bullet(Y,A)$, presenting the morphism $\mathbf{H}(X, A) \to \mathbf{H}(Y,A)$. Such homotopy fibers are given for instance by dual mapping cone complexes. Accordingly, in the abelian case the $A$-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.
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 .
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 $\mathbf{H} :=$ Smooth∞Grpd. For any $n \in \mathbb{N}$ consider the sequence of Lie groups
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
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 $Y \to \mathbf{B}^2 U(1)$.
But in these applications, this 2-bundle is the restriction of an ambient 2-bundle classified by $\mathbf{c} : X \to \mathbf{B}^2 U(1)$ 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 $\infty$-groupoid of $(Y \hookrightarrow X)$-relative cocycles with coefficients in $(\mathbf{B}PU \to \mathbf{B}^2 U(1))$.
Last revised on June 8, 2022 at 13:42:34. See the history of this page for a list of all contributions to it.