Relative $2$-Segal spaces

TL;DR

This paper introduces the concept of relative 2-Segal spaces, expanding the framework of 2-Segal spaces to include new examples and showing their role in representing Hall algebras categorically, with connections to higher categories.

Contribution

It defines relative 2-Segal spaces, provides examples, and demonstrates their use in categorical Hall algebra representations, extending the existing theory of 2-Segal spaces.

Findings

Examples include categorified cyclic nerve and real pseudoholomorphic polygons.

Relative 2-Segal spaces define categorical Hall algebra representations.

Decategorification recovers known Hall algebra constructions.

Abstract

We introduce a relative version of the $2$-Segal simplicial spaces defined by Dyckerhoff and Kapranov and G\'{a}lvez-Carrillo, Kock and Tonks. Examples of relative $2$-Segal spaces include the categorified unoriented cyclic nerve, real pseudoholomorphic polygons in almost complex manifolds and the $\mathcal{R}_{\bullet}$-construction from Grothendieck-Witt theory. We show that a relative $2$-Segal space defines a categorical representation of the Hall algebra associated to the base $2$-Segal space. In this way, after decategorification we recover a number of known constructions of Hall algebra representations. We also describe some higher categorical interpretations of relative $2$-Segal spaces.