Algebraic & Geometric Topology

Relative $2$–Segal spaces

Matthew B Young

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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.

Article information

Algebr. Geom. Topol., Volume 18, Number 2 (2018), 975-1039.

Received: 8 February 2017
Revised: 7 October 2017
Accepted: 30 October 2017
First available in Project Euclid: 22 March 2018

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 18G30: Simplicial sets, simplicial objects (in a category) [See also 55U10]
Secondary: 18G55: Homotopical algebra 16G20: Representations of quivers and partially ordered sets 19G38: Hermitian $K$-theory, relations with $K$-theory of rings

higher Segal spaces categorified Hall algebra representations categories with duality Grothendieck-Witt theory


Young, Matthew B. Relative $2$–Segal spaces. Algebr. Geom. Topol. 18 (2018), no. 2, 975--1039. doi:10.2140/agt.2018.18.975.

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