Relative 2–Segal spaces

Young, Matthew

doi:10.2140/agt.2018.18.975

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Relative $2\mskip-1mu$–Segal spaces

Matthew B Young

Algebraic & Geometric Topology 18 (2018) 975–1039

DOI: 10.2140/agt.2018.18.975

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 $ℛ_{∙}$ –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

Volume 18, issue 2 (2018)