Higher Segal spaces I

TL;DR

This paper introduces higher Segal spaces, a new framework generalizing Segal spaces, with a focus on 2-Segal spaces that reveal richer algebraic structures in various mathematical contexts.

Contribution

It defines d-Segal spaces for all d, generalizing Segal spaces, and demonstrates the significance of 2-Segal spaces in understanding complex algebraic and geometric structures.

Findings

2-Segal spaces encompass Hall algebras and other classical structures.

Waldhausen's S-construction exemplifies a 2-Segal space.

Higher Segal spaces unify various mathematical topics.

Abstract

This is the first paper in a series on new higher categorical structures called higher Segal spaces. For every d > 0, we introduce the notion of a d-Segal space which is a simplicial space satisfying locality conditions related to triangulations of cyclic polytopes of dimension d. In the case d=1, we recover Rezk's theory of Segal spaces. The present paper focuses on 2-Segal spaces. The starting point of the theory is the observation that Hall algebras, as previously studied, are only the shadow of a much richer structure governed by a system of higher coherences captured in the datum of a 2-Segal space. This 2-Segal space is given by Waldhausen's S-construction, a simplicial space familiar in algebraic K-theory. Other examples of 2-Segal spaces arise naturally in classical topics such as Hecke algebras, cyclic bar constructions, configuration spaces of flags, solutions of the pentagon equation, and mapping class groups.