Aprenentatge actiu de conceptes en probabilitat i estadística per a l'enginyeria

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Decomposition spaces in combinatorics

A decomposition space (also called unital 2-Segal space) is a simplicial object satisfying an exactness condition weaker than the Segal condition: just as the Segal condition expresses (up to homotopy) composition, the new condition expresses decomposition. It is a general framework for incidence (co)algebras. In the present contribution, after establishing a formula for the section coefficients, we survey a large supply of examples, emphasising the notion's firm roots in classical combinatorics. The first batch of examples, similar to binomial posets, serves to illustrate two key points: (1) the incidence algebra in question is realised directly from a decomposition space, without a reduction step, and reductions are often given by CULF functors; (2) at the objective level, the convolution algebra is a monoidal structure of species. Specifically, we encounter the usual Cauchy product of species, the shuffle product of L-species, the Dirichlet product of arithmetic species, the Joyal-Street external product of q-species and the Morrison `Cauchy' product of q-species, and in each case a power series representation results from taking cardinality. The external product of q-species exemplifies the fact that Waldhausen's S-construction on an abelian category is a decomposition space, yielding Hall algebras. The next class of examples includes Schmitt's chromatic Hopf algebra, the Fa\`a di Bruno bialgebra, the Butcher-Connes-Kreimer Hopf algebra of trees and several variations from operad theory. Similar structures on posets and directed graphs exemplify a general construction of decomposition spaces from directed restriction species. We finish by computing the M\Preprin

Three Hopf algebras and their common simplicial and categorical background

We consider three a priori totally different setups for Hopf algebras from number theory, mathematical physics and algebraic topology. These are the Hopf algebras of Goncharov for multiple zeta values, that of Connes--Kreimer for renormalization, and a Hopf algebra constructed by Baues to study double loop spaces. We show that these examples can be successively unified by considering simplicial objects, cooperads with multiplication and Feynman categories at the ultimate level. These considerations open the door to new constructions and reinterpretation of known constructions in a large common frameworkPreprin

Three Hopf algebras from number theory, physics & topology, and their common background I: operadic & simplicial aspects

We consider three a priori totally different setups for Hopf algebras from number theory, mathematical physics and algebraic topology. These are the Hopf algebra of Goncharov for multiple zeta values, that of Connes-Kreimer for renormalization, and a Hopf algebra constructed by Baues to study double loop spaces. We show that these examples can be successively unified by considering simplicial objects, co-operads with multiplication and Feynman categories at the ultimate level. These considerations open the door to new constructions and reinterpretations of known constructions in a large common framework, which is presented step-by-step with examples throughout. In this first part of two papers, we concentrate on the simplicial and operadic aspectsPeer ReviewedPostprint (author's final draft

Periodic orbits of planar integrable birational maps

A birational planar map F possessing a rational first integral preserves a foliation of the plane given by algebraic curves which, if F is not globally periodic, is given by a foliation of curves that have generically genus 0 or 1. In the genus 1 case, the group structure of the foliation characterizes the dynamics of any birational map preserving it. We will see how to take advantage of this structure to find periodic orbits of such maps.Preprin

Decomposition spaces and restriction species

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 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 spacesPeer ReviewedPostprint (author's final draft

Decomposition spaces, incidence algebras and Möbius inversion

Preprin

Decomposition spaces, incidence algebras and Mobius inversion III: the decomposition space of Möbius intervals

Decomposition spaces are simplicial 8-groupoids subject to a certain exactness condition, needed to induce a coalgebra structure on the space of arrows. Conservative ULF functors (CULF) between decomposition spaces induce coalgebra homomorphisms. Suitable added finiteness conditions define the notion of Möbius decomposition space, a far-reaching generalisation of the notion of Möbius category of Leroux. In this paper, we show that the Lawvere–Menni Hopf algebra of Möbius intervals, which contains the universal Möbius function (but is not induced by a Möbius category), can be realised as the homotopy cardinality of a Möbius decomposition space U of all Möbius intervals, and that in a certain sense U is universal for Möbius decomposition spaces and CULF functors.Peer ReviewedPostprint (author's final draft

Gabriel–Zisman cohomology and spectral sequences

Extending constructions by Gabriel and Zisman, we develop a functorial framework for the cohomology and homology of simplicial sets with very general coefficient systems given by functors on simplex categories into abelian categories. Furthermore we construct Leray type spectral sequences for any map of simplicial sets. We also show that these constructions generalise and unify the various existing versions of cohomology and homology of small categories and as a bonus provide new insight into their functoriality.Peer ReviewedPostprint (author's final draft

Groupoids and Faà di Bruno Formulae for green functions in bialgebras of trees

We prove a Faa di Bruno formula for the Green function in the bialgebra of P-trees, for any polynomial endofunctor P. The formula appears as relative homotopy cardinality of an equivalence of groupoids. For suitable choices of P, the result implies also formulae for Green functions in bialgebras of graphsPreprin