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dc.contributor.authorGálvez Carrillo, Maria Immaculada
dc.contributor.authorKock, Joachim
dc.contributor.authorTonks, Andrew
dc.contributor.otherUniversitat Politècnica de Catalunya. Departament de Matemàtiques
dc.date.accessioned2016-03-10T08:54:52Z
dc.date.available2016-03-10T08:54:52Z
dc.date.issued2015-12
dc.identifier.citationGalvez, M., Kock, J., Tonks, A. "Decomposition spaces, incidence algebras and Möbius inversion I: basic theory". 2015.
dc.identifier.urihttp://hdl.handle.net/2117/84102
dc.descriptionarXiv:1512.07573 [math.CT]
dc.description.abstractThis 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 8-groupoids. A decomposition space is a simplicial 8-groupoid satisfying an exactness condition, weaker than the Segal condition, expressed in terms of generic and free maps in ¿. Just as the Segal condition expresses up-to-homotopy composition, the new condition expresses decomposition, and there is an abundance of examples coming from 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 8-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. We treat a few examples of decomposition spaces beyond Segal spaces, the most interesting being that of Hall algebras: the Waldhausen S·-construction of an abelian (or stable infinity) category is shown to be a decomposition space.
dc.format.extent46 p.
dc.language.isoeng
dc.subjectÀrees temàtiques de la UPC::Matemàtiques i estadística::Topologia::Topologia algebraica
dc.subject.lcshAlgebraic topology
dc.subject.lcshCombinatorial topology
dc.subject.otherAlgebraic Topology
dc.subject.otherCombinatorics
dc.titleDecomposition spaces, incidence algebras and Möbius inversion I: basic theory
dc.typeExternal research report
dc.subject.lemacTopologia algebraica
dc.subject.lemacTopologia combinatòria
dc.contributor.groupUniversitat Politècnica de Catalunya. GEOMVAP - Geometria de Varietats i Aplicacions
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::55 Algebraic topology::55P Homotopy theory
dc.relation.publisherversionhttp://arxiv.org/abs/1512.07573
dc.rights.accessOpen Access
local.identifier.drac17543094
dc.description.versionPreprint
dc.relation.projectidinfo:eu-repo/grantAgreement/MINECO//MTM2012-38122-C03-01/ES/GEOMATRIA ALGEBRAICA, SIMPLECTICA, ARITMETICA Y APLICACIONES/
dc.relation.projectidinfo:eu-repo/grantAgreement/AGAUR/PRI2010-2013/2014SGR634
dc.relation.projectidinfo:eu-repo/grantAgreement/MINECO//MTM2015-69135-P/ES/GEOMETRIA Y TOPOLOGIA DE VARIEDADES, ALGEBRA Y APLICACIONES/
local.citation.authorGalvez, M.; Kock, J.; Tonks, A.


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