Decomposition spaces, incidence algebras and Möbius inversion I: basic theory

Abstract

This is the first in a series of papers devoted to the theory of decomposition spaces, a general framework for incidence algebras and Möbius inversion, where algebraic identities are realised by taking homotopy cardinality of equivalences of $\infty$-groupoids. A decomposition space is a simplicial $\infty$-groupoid satisfying an exactness condition, weaker than the Segal condition, expressed in terms of active and inert maps in Delta. Just as the Segal condition expresses composition, the new condition expresses decomposition, and there is an abundance of examples in combinatorics. After establishing some basic properties of decomposition spaces, the main result of this first paper shows that to any decomposition space there is an associated incidence coalgebra, spanned by the space of 1-simplices, and with coefficients in $\infty$-groupoids. We take a functorial viewpoint throughout, emphasising conservative ULF functors; these induce coalgebra homomorphisms. Reduction procedures in the classical theory of incidence coalgebras are examples of this notion, and many are examples of decalage of decomposition spaces. An interesting class of examples of decomposition spaces beyond Segal spaces is provided by Hall algebras: the Waldhausen S-construction of an abelian (or stable infinity) category is shown to be a decomposition space. Note: The notion of decomposition space was arrived at independently by Dyckerhoff and Kapranov (arXiv:1212.3563) who call them unital 2-Segal spaces. Our theory is quite orthogonal to theirs: the definitions are different in spirit and appearance, and the theories differ in terms of motivation, examples and directions.

50 pages. This paper is one of six papers that formerly constituted the long manuscript arXiv:1404.3202. v4: minor expository improvements. Final version to appear in Adv. Math

References in corpus (6)

• Higher Topos Theory
• Decomposition spaces, incidence algebras and Möbius inversion II: completeness, length filtration, and finiteness
• Decomposition spaces, incidence algebras and Möbius inversion III: the decomposition space of Möbius intervals
• Homotopy linear algebra
• $\infty$-Operads as Analytic Monads
• Polynomial functors and combinatorial Dyson-Schwinger equations

Cited by in corpus (33)

• Decomposition spaces, incidence algebras and Möbius inversion II: completeness, length filtration, and finiteness
• Decomposition spaces, incidence algebras and Möbius inversion III: the decomposition space of Möbius intervals
• Higher Segal structures in algebraic $K$-theory
• Decomposition spaces and restriction species
• 2-Segal spaces as invertible infinity-operads
• Operads of (noncrossing) partitions, interacting bialgebras, and moment-cumulant relations
• Polynomial functors and combinatorial Dyson-Schwinger equations
• 2-Segal objects and the Waldhausen construction
• Simplicial spaces, lax algebras and the 2-Segal condition
• Whole-grain Petri nets and processes
• Faà di Bruno for operads and internal algebras
• Incidence bicomodules, Möbius inversion, and a Rota formula for infinity adjunctions
• Operadic categories and décalage
• The incidence comodule bialgebra of the Baez-Dolan construction
• Cospan construction of the graph category of Borisov and Manin
• Comparison of Waldhausen constructions
• Frobenius objects in the category of spans
• The universal Hall bialgebra of a double 2-Segal space
• Frobenius and commutative pseudomonoids in the bicategory of spans
• Free decomposition spaces
• Quasi-2-Segal sets
• Weak cartesian properties of simplicial sets
• A simplicial groupoid for plethysm
• Gabriel-Zisman Cohomology and spectral sequences
• The Gálvez-Kock-Tonks conjecture for locally discrete decomposition spaces
• The equifibered approach to $\infty$-properads
• Tracelet Hopf Algebras and Decomposition Spaces (Extended Abstract)
• Equivariant motivic Hall algebras
• Combinatorics and simplicial groupoids
• Antipodes of monoidal decomposition spaces
• Generalizing quasi-categories via model structures on simplicial sets
• Semisimple Field Theories Detect Stable Diffeomorphism
• Culf maps and edgewise subdivision