Special and general types
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 the site of smooth manifolds, there is
the “topological interval” which is the smooth ∞-stack on represented by the manifold ;
the “categorical interval” is the smooth ∞-stack that is constant on the free groupoid on a single morphism.
For and (∞,1)-topos with a fixed notion of topological interval object , for any coefficient object and any other object, a concordance between two objects
(two cocycles in -cohomology on )
is an object such that
For 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.
Revised on October 27, 2013 22:34:49
by Urs Schreiber