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Decomposition spaces, incidence algebras and Möbius inversion. II: Completeness, length filtration, and finiteness. (English) Zbl 1403.18016
Adv. Math. 333, 1242-1292 (2018); corrigendum ibid. 371, Article ID 107267, 5 p. (2020).
In the first part of this trilogy [Adv. Math. 331, 952–1015 (2018; Zbl 1403.00023)] the authors have introduced the notion of decomposition space as a generalization for incidence coalgebras, which is equivalent to the notion of unital $$2$$-Segal space of [T. Dyckerhoff and M. Kapranov, Higher Segal spaces. I, Lect. Notes Math. 2244. Cham: Springer (2019; Zbl 1459.18001), see also arXiv:1212.3563]. The principal objective in this second part is to establish a Möbius inversion principle within the framework of complete decomposition spaces, and to analyze their associated finiteness conditions to ensure incidence coalgebras and Möbius inversion descent to classical level of $$\mathbb{Q}$$-vector spaces on taking the homotopy cardinality of the objects involved, which results in Möbius decomposition spaces as a generalization of Möbius categories in [P. Leroux, Cah. Topologie Géom. Différ. Catégoriques 16, 280–282 (1976; Zbl 0364.18001)]. Möbius decomposition spaces cover more coalgebra constructions than Möbius categories, comprehending the Faà di Bruno and Connes-Kremer bialgebras.

MSC:
 18N50 Simplicial sets, simplicial objects 16T10 Bialgebras 06A11 Algebraic aspects of posets 05A19 Combinatorial identities, bijective combinatorics 55U35 Abstract and axiomatic homotopy theory in algebraic topology
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