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| Formato: | Preprint |
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2016
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| Acceso en línea: | https://arxiv.org/abs/1612.09225 |
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| _version_ | 1866917806625259520 |
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| author | Gálvez-Carrillo, Imma Kock, Joachim Tonks, Andrew |
| author_facet | Gálvez-Carrillo, Imma Kock, Joachim Tonks, Andrew |
| contents | A decomposition space (also called 2-Segal space) is a simplicial object satisfying an exactness condition weaker than the Segal condition: just as the Segal condition expresses composition, the new condition expresses decomposition. It is a general framework for incidence (co)algebras. In this 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 2 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. 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. 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 Faà di Bruno bialgebra, the Butcher-Connes-Kreimer Hopf algebra of trees and variations from operad theory. Similar structures on posets and directed graphs exemplify a general construction of decomposition spaces from directed restriction species. An appetiser on decomposition spaces of symmetric functions is included. We finish by computing the Möbius function in a few cases, and commenting on certain cancellations that occur in the process of taking cardinality, substantiating that these cancellations are not possible at the objective level. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_1612_09225 |
| institution | arXiv |
| publishDate | 2016 |
| record_format | arxiv |
| spellingShingle | Decomposition spaces in Combinatorics Gálvez-Carrillo, Imma Kock, Joachim Tonks, Andrew Combinatorics Category Theory 05A19, 16T10, 06A07, 18G30, 18B40, 18-XX, 55Pxx A decomposition space (also called 2-Segal space) is a simplicial object satisfying an exactness condition weaker than the Segal condition: just as the Segal condition expresses composition, the new condition expresses decomposition. It is a general framework for incidence (co)algebras. In this 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 2 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. 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. 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 Faà di Bruno bialgebra, the Butcher-Connes-Kreimer Hopf algebra of trees and variations from operad theory. Similar structures on posets and directed graphs exemplify a general construction of decomposition spaces from directed restriction species. An appetiser on decomposition spaces of symmetric functions is included. We finish by computing the Möbius function in a few cases, and commenting on certain cancellations that occur in the process of taking cardinality, substantiating that these cancellations are not possible at the objective level. |
| title | Decomposition spaces in Combinatorics |
| topic | Combinatorics Category Theory 05A19, 16T10, 06A07, 18G30, 18B40, 18-XX, 55Pxx |
| url | https://arxiv.org/abs/1612.09225 |