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arXiv · 1512.07580

Decomposition spaces, incidence algebras and M\"obius inversion III: the decomposition space of M\"obius intervals

Imma Gálvez-Carrillo·Joachim Kock·Andrew Tonks
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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 M\"obius decomposition space, a far-reaching generalisation of the notion of M\"obius category of Leroux. In this paper, we show that the Lawvere-Menni Hopf algebra of M\"obius intervals, which contains the universal M\"obius function (but is not induced by a M\"obius category), can be realised as the homotopy cardinality of a M\"obius decomposition space $U$ of all M\"obius intervals, and that in a certain sense $U$ is universal for M\"obius decomposition spaces and CULF functors.

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Imma Gálvez-Carrillo, Joachim Kock, Andrew Tonks. 2015-12-23. Decomposition spaces, incidence algebras and M\"obius inversion III: the decomposition space of M\"obius intervals. https://doi.org/10.1016/j.aim.2018.03.018

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