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

Decomposition spaces are simplicial $\mathrm{\infty }$-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.