nLab
concordance

Idea

A concordance between cocycles in cohomology is a relation similar to but different from a coboundary.

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 $S=\mathrm{Diff}$ the site of smooth manifolds, there is

• the “topological interval” $I\in H_{\mathrm{diff}}$ which is the smooth ∞-stack on $\mathrm{Diff}$ represented by the manifold $I=[0,1]$;

• the “categorical interval” ${\mathrm{Ex}}^{\mathrm{\infty }}{\Delta }^1\in H_{\mathrm{Diff}}$ is the smooth ∞-stack that is constant on the free groupoid on a single morphism.

Definition

For $H$ and (∞,1)-topos with a fixed notion of topological interval object $I$, for $A\in A$ any coefficient object and $X\in H$ any other object, a concordance between two objects

$$c,d\in H(X,A)$$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

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

Examples

• For $A=\mathrm{VectrBund}(-)$ 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.