Classifying spaces of infinity-sheaves

We prove that the set of concordance classes of sections of an 1-sheaf on a manifold is representable, extending a theorem of Madsen and Weiss for sheaves of sets.This is reminiscent of an h-principle in which the role of isotopy is played by concordance.As an application, we offer an answer to the question: what does the classifying space of a Segal space classify?18F20, 18N60, 55N30 1. Introduction 4891 2. The concordance resolution is concordance-invariant 4896 3. Weak Kan fibrations 4901 4. Weak Kan fibrancy of the concordance resolution 4908 5.The shape functor preserves the 1-sheaf property 4920 6.What does the classifying space of an 1-category classify?4927 Appendix.Technical lemmas on simplicial sets and spaces 4930