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

• concordance of topological vector bundles is discussed there

• 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.