Decomposition spaces are simplicial infinity-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 Mobius decomposition space, a far-reaching generalisation of the notion of Mobius category of Leroux. In this paper, we show that the Lawvere-Menni Hopf algebra of Mobius intervals, which contains the universal Mobius function (but is not induced by a Mobius category), can be realised as the homotopy cardinality of a Mobius decomposition space U of all Mobius intervals, and that in a certain sense U is universal for Mobius decomposition spaces and CULF functors. (C) 2018 Elsevier Inc. All rights reserved.