## Algebraic & Geometric Topology

### Relative $2$–Segal spaces

Matthew B Young

#### 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.

#### Article information

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

Dates
Revised: 7 October 2017
Accepted: 30 October 2017
First available in Project Euclid: 22 March 2018

https://projecteuclid.org/euclid.agt/1521684027

Digital Object Identifier
doi:10.2140/agt.2018.18.975

Mathematical Reviews number (MathSciNet)
MR3773745

Zentralblatt MATH identifier
06859611

#### Citation

Young, Matthew B. Relative $2$–Segal spaces. Algebr. Geom. Topol. 18 (2018), no. 2, 975--1039. doi:10.2140/agt.2018.18.975. https://projecteuclid.org/euclid.agt/1521684027

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