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

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 –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.

higher Segal spaces, categorified Hall algebra representations, categories with duality, Grothendieck-Witt theory
Mathematical Subject Classification 2010
Primary: 18G30
Secondary: 18G55, 16G20, 19G38
Received: 8 February 2017
Revised: 7 October 2017
Accepted: 30 October 2017
Published: 12 March 2018
Matthew B Young
The Institute of Mathematical Sciences and Department of Mathematics
The Chinese University of Hong Kong
Hong Kong