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

TL;DR

This paper constructs a universal M"obius decomposition space of intervals, generalizing classical M"obius categories, and relates it to the Lawvere-Menni Hopf algebra through homotopy cardinality.

Contribution

It introduces a universal M"obius decomposition space of intervals and connects it to the Lawvere-Menni Hopf algebra, extending the framework of M"obius inversion.

Findings

Realizes the Lawvere-Menni Hopf algebra as the homotopy cardinality of a universal decomposition space.

Shows that this space is universal for M"obius decomposition spaces and CULF functors.

Generalizes the notion of M"obius categories to decomposition spaces.

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.