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

• Published in: Advances in mathematics (New York. 1965) Vol. 334; pp. 544 - 584
• Main Authors: Gálvez-Carrillo, Imma, Kock, Joachim, Tonks, Andrew
• Format: Journal Article Publication
• Language: English
• Published: Elsevier Inc 20.08.2018
• Subjects: 06 Order, lattices, ordered algebraic structures; 06A Ordered sets; 18 Category theory; homological algebra; 18G Homological algebra; 2-Segal space; 55 Algebraic topology; 55P Homotopy theory; Algebraic topology; Classificació AMS; Combinatorial topology; CULF functor; decomposition space; Matemàtiques i estadística; Möbius interval; Möbius inversion; Topologia; Topologia algebraica; Topologia combinatòria; Àrees temàtiques de la UPC; secondary; Möbius interval; decomposition space; CULF functor; 2-Segal space; Möbius inversion; primary
• ISSN: 0001-8708; 1090-2082
• DOI: 10.1016/j.aim.2018.03.018

Decomposition spaces are simplicial ∞-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.