Higher Segal Spaces

摘要: This is the first paper in a series on new higher categorical structures called Segal spaces. For every d C 1, we introduce notion of d-Segal space which simplicial satisfying locality conditions related to triangulations cyclic polytopes dimension d. In case = recover Rezk’s theory The present focuses 2-Segal starting point observation that Hall algebras, as previously studied, are only shadow much richer structure governed by system coherences captured datum space. given Waldhausen’s S-construction, familiar algebraic K-theory. Other examples spaces arise naturally classical topics such Hecke bar constructions, configuration flags, solutions pentagon equation, and mapping class groups.