Decomposition spaces in combinatorics
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 (c...
| Autores: | , , |
|---|---|
| Formato: | informe técnico |
| Fecha de publicación: | 2016 |
| País: | España |
| Recursos: | Universitat Politècnica de Catalunya (UPC) |
| Repositorio: | UPCommons. Portal del coneixement obert de la UPC |
| Idioma: | inglés |
| OAI Identifier: | oai:upcommons.upc.edu:2117/102202 |
| Acesso em linha: | https://hdl.handle.net/2117/102202 |
| Access Level: | acceso abierto |
| Palavra-chave: | Combinatorial topology Categories (Mathematics) Combinatorial analysis Combinatorics Category Theory Topologia combinatòria Categories (Matemàtica) Anàlisi combinatòria Classificació AMS::05 Combinatorics::05A Enumerative combinatorics Classificació AMS::16 Associative rings and algebras Classificació AMS::06 Order, lattices, ordered algebraic structures::06A Ordered sets Classificació AMS::18 Category theory homological algebra Classificació AMS::55 Algebraic topology::55P Homotopy theory Àrees temàtiques de la UPC::Matemàtiques i estadística::Topologia::Topologia algebraica |
| Resumo: | 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\ |
|---|