Higher Segal spaces I

Abstract

This is the first paper in a series on new higher categorical structures called higher Segal spaces. For every d > 0, 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.

221 pages

References in corpus (10)

• Stability structures, motivic Donaldson-Thomas invariants and cluster transformations
• On the Classification of Topological Field Theories
• Derived Algebraic Geometry III: Commutative Algebra
• Homotopy limits and colimits and enriched homotopy theory
• (Infinity,2)-Categories and the Goodwillie Calculus I
• The pentagon relation and incidence geometry
• On the Hall algebra of semigroup representations over F_1
• Derived Algebraic Geometry II: Noncommutative Algebra
• Lectures on Hall algebras
• The spherical Hall algebra of Spec(Z)

Cited by in corpus (51)

• Decomposition spaces, incidence algebras and Möbius inversion I: basic theory
• Triangulated surfaces in triangulated categories
• Two-dimensional categorified Hall algebras
• Decomposition spaces, incidence algebras and Möbius inversion II: completeness, length filtration, and finiteness
• Crossed simplicial groups and structured surfaces
• Decomposition spaces, incidence algebras and Möbius inversion III: the decomposition space of Möbius intervals
• Simplicial structures in higher Auslander-Reiten theory
• Decomposition spaces and restriction species
• Higher categorical aspects of Hall Algebras
• External triangulation of the homotopy category of exact quasi-category
• Quantization via Linear homotopy types
• On the Hall algebra of semigroup representations over F_1
• Cyclic structures and broken cycles
• The Fukaya category pairs with Lagrangian cobordisms
• Derived Algebraic Geometry
• 2-Segal spaces as invertible infinity-operads
• 2-Segal objects and the Waldhausen construction
• A bivariant Yoneda lemma and $(\infty ,2)$-categories of correspondences
• Legendrian skein algebras and Hall algebras
• Simplicial spaces, lax algebras and the 2-Segal condition
• Incidence bicomodules, Möbius inversion, and a Rota formula for infinity adjunctions
• Operadic categories and décalage
• Cyclic symmetries of A_n-quiver representations
• Enhanced twisted arrow categories
• The incidence comodule bialgebra of the Baez-Dolan construction
• Comparison of Waldhausen constructions
• Cospan construction of the graph category of Borisov and Manin
• Coefficient Quivers, ${\mathbb{F}}_1$-Representations, and Euler Characteristics of Quiver Grassmannians
• McKay correspondence, cohomological Hall algebras and categorification
• Frobenius objects in the category of spans
• Spherical adjunctions of stable $\infty$-categories and the relative S-construction
• A geometric approach to Hall algebras I: Higher Associativity
• Weak cartesian properties of simplicial sets
• The Duskin nerve of 2-categories in Joyal's cell category Theta_2
• Contingency Tables with Variable Margins (with an Appendix by Pavel Etingof)
• Quasi-2-Segal sets
• Free decomposition spaces
• Scalar extensions of quiver representations over ${\mathbb{F}}_1$
• The universal Hall bialgebra of a double 2-Segal space
• The Gálvez-Kock-Tonks conjecture for locally discrete decomposition spaces
• Gabriel-Zisman Cohomology and spectral sequences
• Perverse sheaves on Riemann surfaces as Milnor sheaves
• Parabolic induction and perverse sheaves on h/W
• On Infinite Matroids with Strong Maps: Proto-exactness and Finiteness Conditions
• Culf maps and edgewise subdivision
• Hall categories and KLR categorification
• Combinatorics and simplicial groupoids
• Higher Segal spaces and Lax ${\mathbb{A}}_{\infty }$-algebras
• Monoidal envelopes and Grothendieck construction for dendroidal Segal objects
• A universal characterization of noncommutative motives and secondary algebraic K-theory
• Generalizing quasi-categories via model structures on simplicial sets