Volume 18, issue 2 (2018)

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ISSN (electronic): 1472-2739
ISSN (print): 1472-2747
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 –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.

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