group cohomology, nonabelian group cohomology, Lie group cohomology
Hochschild cohomology, cyclic cohomology?
cohomology with constant coefficients / with a local system of coefficients
differential cohomology
A concordance between cocycles in cohomology is a relation similar to but different from a plain coboundary, it is a “coboundary after geometric realization”.
A concordance is a left homotopy in an (∞,1)-topos with respect to a topological interval object, not with respect to the categorical interval .
For instance for $S = Diff$ the site of smooth manifolds, there is
the “topological interval” $I \in \mathbf{H}_{diff}$ which is the smooth ∞-stack on $Diff$ represented by the manifold $I = [0,1]$;
the “categorical interval” $Ex^\infty \Delta^1 \in \mathbf{H}_{Diff}$ is the smooth ∞-stack that is constant on the free groupoid on a single morphism.
For $\mathbf{H}$ and (∞,1)-topos with a fixed notion of topological interval object $I$, for $A \in \mathbf{A}$ any coefficient object and $X \in \mathbf{H}$ any other object, a concordance between two objects
(two cocycles in $A$-cohomology on $X$)
is an object $\eta \in A(X \times I)$ such that
(concordant topological principal bundles are isomorphic)
For $X \,\in\, TopSp$ a topological space and $G \,\in\, Grp(TopSp)$ a topological group, consider a concordance between a pair of $G$-principal bundles over $X$,
If
or
(e.g. if $X$ admits the structure of a smooth manifold)
then there exists already an isomorphism of principal bundles
Recall that isomorphisms between principal bundles $P$ and $P'$ over $X$ are equivalently global sections of the fiber bundle $(P \times_X P')/G$. In particular, for every $P$ the identity morphism on it corresponds to a canonical section of $(P \times_X P)/G$.
In the given situation, this means that we have a canonical local section $\sigma_0$ making the following solid diagram commute, exhibiting that the restriction of the bundle $P_0 \times [0,1]$ to $\{0\} \subset [0,1]$ is isomorphic to $P_0$, by construction:
Now
assuming the first condition:
the right vertical map is a Serre fibration, as all locally trivial fiber bundles are Serre fibrations (by this Prop.);
the left vertical map is a Serre-Quillen-acyclic cofibration – since (see this Prop.) it is the product $id_X \times (D^0 \hookrightarrow D^0 \times [0,1] )$ of the cofibrant object $X$ with a generating acyclic cofibration (see this Def.) –, hence is a Serre-Quillen-acyclic cofibration and as such has the left lifting property against the right map;
or
assuming the second condition:
the right vertical map is a Hurewicz fibration, by this Prop.,
hence it has the right lifting property against the left map, by definition of Hurewicz fibrations.
In either case, this implies that a lift exists, as shown by the dashed arrow above.
The resulting commutativity of the bottom right triangle says that this lift is a global section which hence exhibits an isomorphism of principal bundles (over $X \times [0,1]$) of this form:
The restriction of this isomorphism to $\{1\} \subset [0,1]$ is hence an isomorphism of the form $P_1 \,\simeq_X\, P_0$, as required.
For topological vector bundles over paracompact Hausdorff spaces, concordance classes coincide with plain isomorphism classes:
(concordance of topological vector bundles)
Let $X$ be a paracompact Hausdorff space. If $E \to X \times [0,1]$ is a topological vector bundle over the product space of $X$ with the closed interval (hence a concordance of topological vector bundles on $X$), then the two endpoint-restrictions
are isomorphic topological vector bundles over $X$.
For proof see at topological vector bundle this Prop..
For $A = VectrBund(-)$ the difference between concordance of vectorial bundles and isomorphism of vectorial bundles plays a crucial rule in the construction of K-theory from this model.
The notions of coboundary and concordance exist in every cohesive (∞,1)-topos.
Discussion of concordance of topological principal bundles (in fact for simplicial principal bundles parameterized over some base space):
Discussion of concordance in terms of the shape modality in the cohesive (∞,1)-topos of smooth ∞-groupoids (see at shape via cohesive path ∞-groupoid for more):
Dmitri Pavlov, Structured Brown representability via concordance, 2014 (pdf, pdf)
Daniel Berwick-Evans, Pedro Boavida de Brito, Dmitri Pavlov, Classifying spaces of infinity-sheaves (arXiv:1912.10544)
Last revised on October 21, 2021 at 11:22:56. See the history of this page for a list of all contributions to it.