Decomposition spaces, incidence algebras and Möbius inversion III: the decomposition space of Möbius intervals

Abstract

Decomposition spaces are simplicial $\infty$-groupoids subject to a certain exactness condition, needed to induce a coalgebra structure on the space of arrows. Conservative ULF functors (CULF) between decomposition spaces induce coalgebra homomorphisms. Suitable added finiteness conditions define the notion of Möbius decomposition space, a far-reaching generalisation of the notion of Möbius category of Leroux. In this paper, we show that the Lawvere-Menni Hopf algebra of Möbius intervals, which contains the universal Möbius function (but is not induced by a Möbius category), can be realised as the homotopy cardinality of a Möbius decomposition space $U$ of all Möbius intervals, and that in a certain sense $U$ is universal for Möbius decomposition spaces and CULF functors.

35 pages. This paper is one of six papers that formerly constituted the long manuscript arXiv:1404.3202. v3: minor expository improvements. Final version to appear in Adv. Math

References in corpus (4)

• Higher Topos Theory
• Decomposition spaces, incidence algebras and Möbius inversion I: basic theory
• Decomposition spaces, incidence algebras and Möbius inversion II: completeness, length filtration, and finiteness
• Homotopy linear algebra

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