#### Volume 18, issue 2 (2018)

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Relative $2\mskip-1mu$–Segal spaces

### Matthew B Young

Algebraic & Geometric Topology 18 (2018) 975–1039
##### 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 ${\mathsc{ℛ}}_{\bullet }$–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.

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##### Keywords
higher Segal spaces, categorified Hall algebra representations, categories with duality, Grothendieck-Witt theory
##### Mathematical Subject Classification 2010
Primary: 18G30
Secondary: 18G55, 16G20, 19G38