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Decomposition spaces, incidence algebras and Möbius inversion I: basic theory
dc.contributor.author | Gálvez Carrillo, Maria Immaculada |
dc.contributor.author | Kock, Joachim |
dc.contributor.author | Tonks, Andrew |
dc.contributor.other | Universitat Politècnica de Catalunya. Departament de Matemàtiques |
dc.date.accessioned | 2016-03-10T08:54:52Z |
dc.date.available | 2016-03-10T08:54:52Z |
dc.date.issued | 2015-12 |
dc.identifier.citation | Galvez, M., Kock, J., Tonks, A. "Decomposition spaces, incidence algebras and Möbius inversion I: basic theory". 2015. |
dc.identifier.uri | http://hdl.handle.net/2117/84102 |
dc.description | arXiv:1512.07573 [math.CT] |
dc.description.abstract | 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 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.extent | 46 p. |
dc.language.iso | eng |
dc.subject | Àrees temàtiques de la UPC::Matemàtiques i estadística::Topologia::Topologia algebraica |
dc.subject.lcsh | Algebraic topology |
dc.subject.lcsh | Combinatorial topology |
dc.subject.other | Algebraic Topology |
dc.subject.other | Combinatorics |
dc.title | Decomposition spaces, incidence algebras and Möbius inversion I: basic theory |
dc.type | External research report |
dc.subject.lemac | Topologia algebraica |
dc.subject.lemac | Topologia combinatòria |
dc.contributor.group | Universitat Politècnica de Catalunya. GEOMVAP - Geometria de Varietats i Aplicacions |
dc.subject.ams | Classificació AMS::18 Category theory; homological algebra::18G Homological algebra |
dc.subject.ams | Classificació AMS::06 Order, lattices, ordered algebraic structures::06A Ordered sets |
dc.subject.ams | Classificació AMS::55 Algebraic topology::55P Homotopy theory |
dc.relation.publisherversion | http://arxiv.org/abs/1512.07573 |
dc.rights.access | Open Access |
local.identifier.drac | 17543094 |
dc.description.version | Preprint |
dc.relation.projectid | info:eu-repo/grantAgreement/MINECO//MTM2012-38122-C03-01/ES/GEOMATRIA ALGEBRAICA, SIMPLECTICA, ARITMETICA Y APLICACIONES/ |
dc.relation.projectid | info:eu-repo/grantAgreement/AGAUR/PRI2010-2013/2014SGR634 |
dc.relation.projectid | info:eu-repo/grantAgreement/MINECO//MTM2015-69135-P/ES/GEOMETRIA Y TOPOLOGIA DE VARIEDADES, ALGEBRA Y APLICACIONES/ |
local.citation.author | Galvez, M.; Kock, J.; Tonks, A. |
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