Show simple item record

dc.contributor.authorGálvez Carrillo, Maria Immaculada
dc.contributor.authorKock, Joachim
dc.contributor.authorTonks, Andrew
dc.contributor.otherUniversitat Politècnica de Catalunya. Departament de Matemàtiques
dc.identifier.citationGalvez, M.; Kock, J.; Tonks, A. Decomposition spaces and restriction species. "International mathematics research notices", 12 Setembre 2018, p. 1-44.
dc.description.abstractWe 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
dc.format.extent44 p.
dc.publisherOxford University Press
dc.subjectÀrees temàtiques de la UPC::Matemàtiques i estadística::Topologia::Topologia algebraica
dc.subject.lcshAlgebraic topology
dc.titleDecomposition spaces and restriction species
dc.subject.lemacTopologia algebraica
dc.contributor.groupUniversitat Politècnica de Catalunya. GEOMVAP - Geometria de Varietats i Aplicacions
dc.description.peerreviewedPeer Reviewed
dc.subject.amsClassificació AMS::18 Category theory; homological algebra::18G Homological algebra
dc.subject.amsClassificació AMS::06 Order, lattices, ordered algebraic structures::06A Ordered sets
dc.subject.amsClassificació AMS::18 Category theory; homological algebra::18G Homological algebra
dc.subject.amsClassificació AMS::55 Algebraic topology::55P Homotopy theory
dc.rights.accessOpen Access
dc.description.versionPostprint (author's final draft)
upcommons.citation.authorGalvez, M.; Kock, J.; Tonks, A.
upcommons.citation.publicationNameInternational mathematics research notices

Files in this item


This item appears in the following Collection(s)

Show simple item record

All rights reserved. This work is protected by the corresponding intellectual and industrial property rights. Without prejudice to any existing legal exemptions, reproduction, distribution, public communication or transformation of this work are prohibited without permission of the copyright holder