We introduce a relative version of the –Segal simplicial spaces defined by Dyckerhoff and Kapranov, and Gálvez-Carrillo, Kock and Tonks. Examples of relative –Segal spaces include the categorified unoriented cyclic nerve, real pseudoholomorphic polygons in almost complex manifolds and the –construction from Grothendieck–Witt theory. We show that a relative –Segal space defines a categorical representation of the Hall algebra associated to the base –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 –Segal spaces.
"Relative $2$–Segal spaces." Algebr. Geom. Topol. 18 (2) 975 - 1039, 2018. https://doi.org/10.2140/agt.2018.18.975