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.

45 pages. Final section split into two sections, with added details

Cited by in corpus (4)

• Incidence bicomodules, Möbius inversion, and a Rota formula for infinity adjunctions
• The incidence comodule bialgebra of the Baez-Dolan construction
• Comparison of Waldhausen constructions
• Quasi-2-Segal sets