group cohomology, nonabelian group cohomology, Lie group 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
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 June 19, 2021 at 11:52:44. See the history of this page for a list of all contributions to it.