We introduce a relative version of the
–Segal simplicial
spaces defined by Dyckerhoff and Kapranov, and Gálvez-Carrillo, Kock and Tonks. Examples of
relative
–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
–Segal
space defines a categorical representation of the Hall algebra associated to the base
–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
–Segal
spaces.
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