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 dir...

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Detalles Bibliográficos
Autores: Gálvez Carrillo, Maria Immaculada|||0000-0002-8338-0437, Kock, Joachim, Tonks, Andrew
Tipo de recurso: artículo
Fecha de publicación:2019
País:España
Institución:Universitat Politècnica de Catalunya (UPC)
Repositorio:UPCommons. Portal del coneixement obert de la UPC
Idioma:inglés
OAI Identifier:oai:upcommons.upc.edu:2117/130730
Acceso en línea:https://hdl.handle.net/2117/130730
https://dx.doi.org/10.1093/imrn/rny089
Access Level:acceso abierto
Palabra clave:Algebraic topology
Topologia algebraica
Classificació AMS::18 Category theory
homological algebra::18G Homological algebra
Classificació AMS::06 Order, lattices, ordered algebraic structures::06A Ordered sets
Classificació AMS::55 Algebraic topology::55P Homotopy theory
Àrees temàtiques de la UPC::Matemàtiques i estadística::Topologia::Topologia algebraica
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spelling Decomposition spaces and restriction speciesGálvez Carrillo, Maria Immaculada|||0000-0002-8338-0437Kock, JoachimTonks, AndrewAlgebraic topologyTopologia algebraicaClassificació AMS::18 Category theoryhomological algebra::18G Homological algebraClassificació AMS::06 Order, lattices, ordered algebraic structures::06A Ordered setsClassificació AMS::18 Category theoryhomological algebra::18G Homological algebraClassificació AMS::55 Algebraic topology::55P Homotopy theoryÀrees temàtiques de la UPC::Matemàtiques i estadística::Topologia::Topologia algebraicaWe 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 ReviewedOxford University Press20202020-11-0120192019-03-21journal articlehttp://purl.org/coar/resource_type/c_6501AMhttp://purl.org/coar/version/c_ab4af688f83e57aainfo:eu-repo/semantics/articleapplication/pdfhttps://hdl.handle.net/2117/130730https://dx.doi.org/10.1093/imrn/rny089reponame:UPCommons. Portal del coneixement obert de la UPCinstname:Universitat Politècnica de Catalunya (UPC)InglésengMinisterio de Economía y Competitividad http://doi.org/10.13039/501100003329 MTM2012-38122-C03-01 GEOMATRIA ALGEBRAICA, SIMPLECTICA, ARITMETICA Y APLICACIONESMinisterio de Economía y Competitividad http://doi.org/10.13039/501100003329 MTM2015-69135-P GEOMETRIA Y TOPOLOGIA DE VARIEDADES, ALGEBRA Y APLICACIONESMinisterio de Economía y Competitividad http://doi.org/10.13039/501100003329 MTM2013-42178-P HOMOTOPIA ESTRUCTURADAopen accesshttp://purl.org/coar/access_right/c_abf2info:eu-repo/semantics/openAccessoai:upcommons.upc.edu:2117/1307302026-05-27T15:37:01Z
dc.title.none.fl_str_mv Decomposition spaces and restriction species
title Decomposition spaces and restriction species
spellingShingle Decomposition spaces and restriction species
Gálvez Carrillo, Maria Immaculada|||0000-0002-8338-0437
Algebraic topology
Topologia algebraica
Classificació AMS::18 Category theory
homological algebra::18G Homological algebra
Classificació AMS::06 Order, lattices, ordered algebraic structures::06A Ordered sets
Classificació AMS::18 Category theory
homological algebra::18G Homological algebra
Classificació AMS::55 Algebraic topology::55P Homotopy theory
Àrees temàtiques de la UPC::Matemàtiques i estadística::Topologia::Topologia algebraica
title_short Decomposition spaces and restriction species
title_full Decomposition spaces and restriction species
title_fullStr Decomposition spaces and restriction species
title_full_unstemmed Decomposition spaces and restriction species
title_sort Decomposition spaces and restriction species
dc.creator.none.fl_str_mv Gálvez Carrillo, Maria Immaculada|||0000-0002-8338-0437
Kock, Joachim
Tonks, Andrew
author Gálvez Carrillo, Maria Immaculada|||0000-0002-8338-0437
author_facet Gálvez Carrillo, Maria Immaculada|||0000-0002-8338-0437
Kock, Joachim
Tonks, Andrew
author_role author
author2 Kock, Joachim
Tonks, Andrew
author2_role author
author
dc.subject.none.fl_str_mv Algebraic topology
Topologia algebraica
Classificació AMS::18 Category theory
homological algebra::18G Homological algebra
Classificació AMS::06 Order, lattices, ordered algebraic structures::06A Ordered sets
Classificació AMS::18 Category theory
homological algebra::18G Homological algebra
Classificació AMS::55 Algebraic topology::55P Homotopy theory
Àrees temàtiques de la UPC::Matemàtiques i estadística::Topologia::Topologia algebraica
topic Algebraic topology
Topologia algebraica
Classificació AMS::18 Category theory
homological algebra::18G Homological algebra
Classificació AMS::06 Order, lattices, ordered algebraic structures::06A Ordered sets
Classificació AMS::18 Category theory
homological algebra::18G Homological algebra
Classificació AMS::55 Algebraic topology::55P Homotopy theory
Àrees temàtiques de la UPC::Matemàtiques i estadística::Topologia::Topologia algebraica
description 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 spaces
publishDate 2019
dc.date.none.fl_str_mv 2019
2019-03-21
2020
2020-11-01
dc.type.none.fl_str_mv journal article
http://purl.org/coar/resource_type/c_6501
AM
http://purl.org/coar/version/c_ab4af688f83e57aa
dc.type.openaire.fl_str_mv info:eu-repo/semantics/article
format article
dc.identifier.none.fl_str_mv https://hdl.handle.net/2117/130730
https://dx.doi.org/10.1093/imrn/rny089
url https://hdl.handle.net/2117/130730
https://dx.doi.org/10.1093/imrn/rny089
dc.language.none.fl_str_mv Inglés
eng
language_invalid_str_mv Inglés
language eng
dc.relation.none.fl_str_mv Ministerio de Economía y Competitividad http://doi.org/10.13039/501100003329 MTM2012-38122-C03-01 GEOMATRIA ALGEBRAICA, SIMPLECTICA, ARITMETICA Y APLICACIONES
Ministerio de Economía y Competitividad http://doi.org/10.13039/501100003329 MTM2015-69135-P GEOMETRIA Y TOPOLOGIA DE VARIEDADES, ALGEBRA Y APLICACIONES
Ministerio de Economía y Competitividad http://doi.org/10.13039/501100003329 MTM2013-42178-P HOMOTOPIA ESTRUCTURADA
dc.rights.none.fl_str_mv open access
http://purl.org/coar/access_right/c_abf2
dc.rights.openaire.fl_str_mv info:eu-repo/semantics/openAccess
rights_invalid_str_mv open access
http://purl.org/coar/access_right/c_abf2
eu_rights_str_mv openAccess
dc.format.none.fl_str_mv application/pdf
dc.publisher.none.fl_str_mv Oxford University Press
publisher.none.fl_str_mv Oxford University Press
dc.source.none.fl_str_mv reponame:UPCommons. Portal del coneixement obert de la UPC
instname:Universitat Politècnica de Catalunya (UPC)
instname_str Universitat Politècnica de Catalunya (UPC)
reponame_str UPCommons. Portal del coneixement obert de la UPC
collection UPCommons. Portal del coneixement obert de la UPC
repository.name.fl_str_mv
repository.mail.fl_str_mv
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score 15,300724

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