Decomposition spaces, incidence algebras and Mobius inversion II: Completeness, length filtration, and finiteness
This is the second in a trilogy of papers introducing and studying the notion of decomposition space as a general framework for incidence algebras and Möbius inversion, with coefficients in 8-groupoids. A decomposition space is a simplicial 8-groupoid satisfying an exactness condition weaker than th...
| Autores: | , , |
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| Tipo de recurso: | artículo |
| Fecha de publicación: | 2018 |
| País: | España |
| Institución: | Universitat Politècnica de Catalunya (UPC) |
| Repositorio: | UPCommons. Portal del coneixement obert de la UPC |
| Idioma: | inglés |
| OAI Identifier: | oai:upcommons.upc.edu:2117/125234 |
| Acceso en línea: | https://hdl.handle.net/2117/125234 https://dx.doi.org/10.1016/j.aim.2018.03.017 |
| Access Level: | acceso abierto |
| Palabra clave: | Algebraic topology Combinatorial topology decomposition space 2-Segal space incidence algebra Mobius inversion homotopy cardinality length filtration Topologia algebraica Topologia combinatòria Classificació AMS::18 Category theory homological algebra::18G Homological algebra Classificació AMS::06 Order, lattices, ordered algebraic structures::06A Ordered sets Classificació AMS::55 Algebraic topology::55P Homotopy theory Àrees temàtiques de la UPC::Matemàtiques i estadística::Topologia::Topologia algebraica |
| Sumario: | This is the second in a trilogy of papers introducing and studying the notion of decomposition space as a general framework for incidence algebras and Möbius inversion, with coefficients in 8-groupoids. A decomposition space is a simplicial 8-groupoid satisfying an exactness condition weaker than the Segal condition. Just as the Segal condition expresses composition, the new condition expresses decomposition. In this paper, we introduce various technical conditions on decomposition spaces. The first is a completeness condition (weaker than Rezk completeness), needed to control simplicial nondegeneracy. For complete decomposition spaces we establish a general Möbius inversion principle, expressed as an explicit equivalence of 8-groupoids. Next we analyse two finiteness conditions on decomposition spaces. The first, that of locally finite length, guarantees the existence of the important length filtration for the associated incidence coalgebra. We show that a decomposition space of locally finite length is actually the left Kan extension of a semi-simplicial space. The second finiteness condition, local finiteness, ensures we can take homotopy cardinality to pass from the level of 8-groupoids to the level of -vector spaces. These three conditions — completeness, locally finite length, and local finiteness — together define our notion of Möbius decomposition space, which extends Leroux's notion of Möbius category (in turn a common generalisation of the locally finite posets of Rota et al. and of the finite decomposition monoids of Cartier–Foata), but which also covers many coalgebra constructions which do not arise from Möbius categories, such as the Faà di Bruno and Connes–Kreimer bialgebras. |
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