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

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Cited by in corpus (34)

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