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arXiv · 1611.09234

Relative $2$-Segal spaces

Matthew B. Young
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Abstract

We introduce a relative version of the $2$-Segal simplicial spaces defined by Dyckerhoff and Kapranov and G\'{a}lvez-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 $\mathcal{R}_{\bullet}$-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.

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BibTeXRIS

Matthew B. Young. 2016-11-28. Relative $2$-Segal spaces. https://doi.org/10.2140/agt.2018.18.975

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