Decomposition spaces, incidence algebras and Möbius inversion II: Completeness, length filtration, and finiteness
Decomposition spaces, incidence algebras and Möbius inversion II: Completeness, length filtration, and finiteness
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分解空间、关联代数和莫比乌斯反演 II：完备性、长度过滤和有限性
DOI：
10.1016/j.aim.2018.03.017
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发表时间：
2015
影响因子：
1.7
通讯作者：
A. Tonks
中科院分区：
文献类型：
--
作者：
Imma G'alvez;Joachim Kock;A. Tonks
这是第二个三部曲的文件介绍和研究的概念，分解空间作为一个一般框架的发病代数和莫比乌斯反演，系数在∞-广群。分解空间是满足比Segal条件弱的正合性条件的单纯∞-广群。正如西格尔条件表示合成，新条件表示分解。本文介绍了分解空间的各种技术条件。第一个是完备性条件（比Rezk完备性弱），需要控制单纯非退化。对于完全分解空间，我们建立了一个一般的莫比乌斯反演原理，表示为∞-广群胚的显式等价。其次，我们分析了分解空间的两个有限性条件。第一个是局部有限长的，保证了关联余代数的重要长度过滤的存在性。我们证明了局部有限长的分解空间实际上是半单纯空间的左Kan扩张。第二个有限性条件，局部有限性，确保我们可以把同伦基数从∞-群胚的水平传递到Q-向量空间的水平。这三个条件--完备性、局部有限长和局部有限性--共同定义了我们的莫比乌斯分解空间的概念，它推广了勒鲁的莫比乌斯范畴的概念（反过来，Rota等人的局部有限偏序集和Cartier-Foata的有限分解幺半群的共同推广），但它也涵盖了许多不产生于莫比乌斯范畴的余代数结构，例如Faà di Bruno和Connes-Kreimer双代数。注：分解空间的概念是由Dyckerhoff和Kapranov [6]独立提出的，他们称之为酉2-Segal空间。
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∞-groupoids. A decomposition space is a simplicial∞-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∞-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∞-groupoids to the level of Q-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. Note: The notion of decomposition space was arrived at independently by Dyckerhoff and Kapranov [6] who call them unital 2-Segal spaces.