Relative $2$–Segal spaces

Abstract

We introduce a relative version of the $2$–Segal simplicial spaces defined by Dyckerhoff and Kapranov, and Gálvez-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.