Faà di Bruno for operads and internal algebras

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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“…These examples are subsumed in the general notion of incidence bialgebra of an operad, cf. [26] and [46]. 10.6. q-binomials: F q -vector spaces.…”

Section: Examplementioning

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 variation, which is subsumed in the class of decomposition spaces coming from operads [21,35], has some different features which have been exploited to good effect in various contexts [15,31,32,34]. In particular it is important that the cut locus expresses a type match between the roots of the crown forest and the leaves of the bottom tree, and that there is a grading [19] given by number of leaves minus number of roots.…”

Section: Examples: Various Flavours Of Trees (Actually Forests)mentioning

confidence: 99%

Decomposition Spaces and Restriction Species

Gálvez-Carrillo

Kock

Tonks

2018

International Mathematics Research Notices

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We show that Schmitt's restriction species (such as graphs, matroids, posets, etc.) naturally induce decomposition spaces (a.k.a. unital 2-Segal spaces), and that their associated coalgebras are an instance of the general construction of incidence coalgebras of decomposition spaces. We introduce the notion of directed restriction species that subsume Schmitt's restriction species and also induce decomposition spaces. Whereas ordinary restriction species are presheaves on the category of finite sets and injections, directed restriction species are presheaves on the category of finite posets and convex maps. We also introduce the notion of monoidal (directed) restriction species, which induce monoidal decomposition spaces and hence bialgebras, most often Hopf algebras. Examples of this notion include rooted forests, directed graphs, posets, double posets, and many related structures. A prominent instance of a resulting incidence bialgebra is the Butcher-Connes-Kreimer Hopf algebra of rooted trees. Both ordinary and directed restriction species are shown to be examples of a construction of decomposition spaces from certain cocartesian fibrations over the category of finite ordinals that are also cartesian over convex maps. The proofs rely on some beautiful simplicial combinatorics, where the notion of convexity plays a key role. The methods developed are of independent interest as techniques for constructing decomposition spaces. Proposition 7.9. Monoidal directed restriction species naturally induce monoidal decomposition spaces and hence bialgebras.

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“…The monoidal structure is disjoint union and the monoidal unit is the (identity of the) empty set. More generally, for any reduced operad, the so-called two-sided bar construction X is a monoidal (complete) decomposition space [12]. The groupoid X 0 is the free symmetric monoidal category on the set of objects of the operad.…”

Section: Connectednessmentioning

confidence: 99%

“…The groupoid X 1 is the free symmetric monoidal category on the action groupoid of the symmetric-group actions on the set of operations. The generalisation of the classical Faà di Bruno formula to any operad [3,12] (the classical case being that of the terminal reduced operad) crucially exploits the typing constraints expressed by the objects in X 0 (which are invisible in the connected quotient Hopf algebra).…”

Section: Connectednessmentioning

confidence: 99%

“…Recent developments have shown the utility of avoiding this reduction, which destroys useful information. For example, the Faà di Bruno formula for general operads [3], [12] crucially exploits the finer structure of the zeroth graded piece of the incidence bialgebra, and in the bialgebra version [10] of BPHZ renormalisation in perturbative quantum field theory, the zeroth graded piece of the bialgebra of Feynman graphs contains the terms of the Lagrangian (not visible in the quotient Hopf algebra usually employed).…”

Section: Introductionmentioning

confidence: 99%

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Antipodes of monoidal decomposition spaces

Carlier

Kock

2018

Commun. Contemp. Math.

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We introduce a notion of antipode for monoidal (complete) decomposition spaces, inducing a notion of weak antipode for their incidence bialgebras. In the connected case, this recovers the usual notion of antipode in Hopf algebras. In the non-connected case it expresses an inversion principle of more limited scope, but still sufficient to compute the Möbius function as µ = ζ • S, just as in Hopf algebras. At the level of decomposition spaces, the weak antipode takes the form of a formal difference of linear endofunctors S even − S odd , and it is a refinement of the general Möbius inversion construction of Gálvez-Kock-Tonks, but exploiting the monoidal structure.

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Faà di Bruno for operads and internal algebras

Cited by 15 publications

References 50 publications

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

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

Decomposition Spaces and Restriction Species

Antipodes of monoidal decomposition spaces

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