Decomposition Spaces and Restriction Species

Gálvez-Carrillo, Imma; Kock, Joachim; Tonks, Andrew

doi:10.1093/imrn/rny089

Cited by 16 publications

(31 citation statements)

References 62 publications

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“…The aims of this paper are to establish a Möbius inversion principle in the framework of complete decomposition spaces, and also to introduce the necessary finiteness conditions on a complete decomposition space to ensure that incidence (co)algebras and Möbius inversion descend to classical vector-space-level coalgebras on taking the homotopy cardinality of the objects involved. Along the way we also establish some auxiliary results of a more technical nature which are needed in the applications in the sequel papers [12,13,14].…”

Section: Introductionmentioning

confidence: 99%

Decomposition spaces, incidence algebras and Möbius inversion II: Completeness, length filtration, and finiteness

Gálvez-Carrillo

Kock

Tonks

2018

Advances in Mathematics

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This is the second in a trilogy of papers introducing and studying the notion of decomposition space as a general framework for incidence algebras and Möbius inversion, with coefficients in ∞-groupoids. A decomposition space is a simplicial ∞-groupoid satisfying an exactness condition weaker than the Segal condition. Just as the Segal condition expresses composition, the new condition expresses decomposition.In this paper, we introduce various technical conditions on decomposition spaces. The first is a completeness condition (weaker than Rezk completeness), needed to control simplicial nondegeneracy. For complete decomposition spaces we establish a general Möbius inversion principle, expressed as an explicit equivalence of ∞-groupoids. Next we analyse two finiteness conditions on decomposition spaces. The first, that of locally finite length, guarantees the existence of the important length filtration for the associated incidence coalgebra. We show that a decomposition space of locally finite length is actually the left Kan extension of a semi-simplicial space. The second finiteness condition, local finiteness, ensures we can take homotopy cardinality to pass from the level of ∞-groupoids to the level of Q-vector spaces.These three conditions -completeness, locally finite length, and local finitenesstogether define our notion of Möbius decomposition space, which extends Leroux's notion of Möbius category (in turn a common generalisation of the locally finite posets of Rota et al. and of the finite decomposition monoids of Cartier-Foata), but which also covers many coalgebra constructions which do not arise from Möbius categories, such as the Faà di Bruno and Connes-Kreimer bialgebras.Note: The notion of decomposition space was arrived at independently by Dyckerhoff and Kapranov (arXiv:1212.3563) who call them unital 2-Segal spaces.

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Section: Introductionmentioning

confidence: 99%

Decomposition spaces, incidence algebras and Möbius inversion II: Completeness, length filtration, and finiteness

Gálvez-Carrillo

Kock

Tonks

2018

Advances in Mathematics

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“…The relationship with Dür's construction is this (cf. [25]): the 'raw' decomposition space N(C op ) is the decalage of H, in close analogy with Lemma 10.2:…”

Section: Examplementioning

confidence: 82%

“…Other examples of coalgebras that can be realised as incidence coalgebras of decomposition spaces but not of categories are Schmitt's Hopf algebra of graphs [66] and the Butcher-Connes-Kreimer Hopf algebra of rooted trees [11]. In a sequel paper [25], these examples are subsumed as examples of decomposition spaces induced from restriction species and directed restriction species.…”

Section: The Idea Of Decomposition Spacesmentioning

confidence: 99%

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

Gálvez-Carrillo

Kock

Tonks

2018

Advances in Mathematics

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183

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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 ∞-groupoids. A decomposition space is a simplicial ∞-groupoid satisfying an exactness condition, weaker than the Segal condition, expressed in terms of active and inert maps in ∆. Just as the Segal condition expresses composition, the new exactness 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 ∞-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.In the second paper in this series we impose further conditions on decomposition spaces, to obtain a general Möbius inversion principle, and to ensure that the various constructions and results admit a homotopy cardinality. In the third paper we show that the Lawvere-Menni Hopf algebra of Möbius intervals is the homotopy cardinality of a certain universal decomposition space. Two further sequel papers deal with numerous examples from combinatorics.Note: The notion of decomposition space was arrived at independently by Dyckerhoff and Kapranov [17] 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.

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“…Layered finite posets and layered finite sets We refer to [15] for the following material. An n-layering of a finite poset P is a monotone map l : P → n, where n = {1, .…”

Section: Incidence Coalgebras and Culf Functorsmentioning

confidence: 99%

Incidence bicomodules, Möbius inversion and a Rota formula for infinity adjunctions

Carlier

2020

Algebr. Geom. Topol.

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In the same way decomposition spaces, also known as unital 2-Segal spaces, have incidence (co)algebras, and certain relative decomposition spaces have incidence (co)modules, we identify the structures that have incidence bi(co)modules: they are certain augmented double Segal spaces subject to some exactness conditions. We establish a Möbius inversion principle for (co)modules, and a Rota formula for certain more involved structures called Möbius bicomodule configurations. The most important instance of the latter notion arises as mapping cylinders of infinity adjunctions, or more generally of adjunctions between Möbius decomposition spaces, in the spirit of Rota's original formula.

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Decomposition Spaces and Restriction Species

Cited by 16 publications

References 62 publications

Decomposition spaces, incidence algebras and Möbius inversion II: Completeness, length filtration, and finiteness

Decomposition spaces, incidence algebras and Möbius inversion II: Completeness, length filtration, and finiteness

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

Incidence bicomodules, Möbius inversion and a Rota formula for infinity adjunctions

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