Higher Segal Spaces

By Tobias Dyckerhoff, M. Kapranov. Lecture notes in mathematics, 2012

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This is the first paper in a series on new higher categorical structures called higher Segal spaces. For every d C 1, we introduce the notion of a d-Segal space which is a simplicial space satisfying locality conditions related to triangulations of cyclic polytopes of dimension d. In the case d = 1, we recover Rezk’s theory of Segal spaces. The present paper focuses on 2-Segal spaces. The starting point of the theory is the observation that Hall algebras, as previously studied, are only the shadow of a much richer structure governed by a system of higher coherences captured in the datum of a 2-Segal space. This 2-Segal space is given by Waldhausen’s S-construction, a simplicial space familiar in algebraic K-theory. Other examples of 2-Segal spaces arise naturally in classical topics such as Hecke algebras, cyclic bar constructions, configuration spaces of flags, solutions of the pentagon equation, and mapping class groups.

Subfields: Algebra and Number Theory, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/73097052e4f9936dd4d2498fdf2bc62655ce6c11

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Citations in the corpus

Decomposition spaces, incidence algebras and Möbius inversion II: Completeness, length filtration, and finiteness (2015, background): “ and Corollary 5.10). This fact is both conceptually pleasing and facilitates eﬃcient arguments. Related work. The notion of decomposition space was discovered independently by Dyckerhoﬀ and Kapranov [7], who call them unital 2-Segal spaces. While some of the basic results in [12] were also proved i”
Decomposition spaces, incidence algebras and Möbius inversion I: Basic theory (2015, methodology): “lish that the S• construction of an abelian category or a stable ∞-category is a decomposition space. This result was ﬁrst proved by Dyckerhoﬀ and Kapranov and constitutes a cornerstone in their work [17], [18], [16], [15], to which we refer for the remarkable richness of this class of examples. Bri”