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Decomposition spaces in combinatorics
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 | 2017-03-09T13:10:06Z |
dc.date.available | 2017-03-09T13:10:06Z |
dc.date.issued | 2016-12 |
dc.identifier.citation | Galvez, M., Kock, J., Tonks, A. "Decomposition spaces in combinatorics". 2016. |
dc.identifier.uri | http://hdl.handle.net/2117/102202 |
dc.description.abstract | 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\ |
dc.format.extent | 67 p. |
dc.language.iso | eng |
dc.subject | Àrees temàtiques de la UPC::Matemàtiques i estadística::Topologia::Topologia algebraica |
dc.subject.lcsh | Combinatorial topology |
dc.subject.lcsh | Categories (Mathematics) |
dc.subject.lcsh | Combinatorial analysis |
dc.subject.other | Combinatorics |
dc.subject.other | Category Theory |
dc.title | Decomposition spaces in combinatorics |
dc.type | External research report |
dc.subject.lemac | Topologia combinatòria |
dc.subject.lemac | Categories (Matemàtica) |
dc.subject.lemac | Anàlisi combinatòria |
dc.contributor.group | Universitat Politècnica de Catalunya. GEOMVAP - Geometria de Varietats i Aplicacions |
dc.subject.ams | Classificació AMS::05 Combinatorics::05A Enumerative combinatorics |
dc.subject.ams | Classificació AMS::16 Associative rings and algebras |
dc.subject.ams | Classificació AMS::06 Order, lattices, ordered algebraic structures::06A Ordered sets |
dc.subject.ams | Classificació AMS::18 Category theory; homological algebra |
dc.subject.ams | Classificació AMS::55 Algebraic topology::55P Homotopy theory |
dc.relation.publisherversion | https://arxiv.org/abs/1612.09225 |
dc.rights.access | Open Access |
local.identifier.drac | 19740792 |
dc.description.version | Preprint |
local.citation.author | Galvez, M.; Kock, J.; Tonks, A. |
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