nLab
concordance

Idea

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.

Definition

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

$$c,d \in \mathbf{H}(X,A)$$

(two cocycles in $A$-cohomology on $X$)

is an object $\eta \in A(X \times I)$ such that

$$\begin{matrix} X&&\\ \downarrow&\searrow^{c}&\\ X \times I&\stackrel{\eta}{\to}& A\\ \uparrow& \nearrow_{d}&\\ X&& \end{matrix} \,.$$

Examples

For topological vector bundles

For topological vector bundles over paracompact Hausdorff spaces, concordance classes coincide with plain isomorphism classes:

Proposition

(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

$$E|_{X \times \{0\}} \phantom{AA} \text{and} \phantom{AA} E|_{X \times \{1\}}$$

are isomorphic topological vector bundles over $X$.

For proof see at topological vector bundle this Prop..

More examples

• 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.

References

Discussion of concordance of topological principal bundles (in fact for simplicial principal bundles parameterized over some base space):

• David Roberts, Danny Stevenson, Cor. 15 in: Simplicial principal bundles in parametrized spaces, New York Journal of Mathematics Volume 22 (2016) 405-440 (arXiv:1203.2460, nyjm:22-19)

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)