The Gálvez-Carrillo–Kock–Tonks conjecture for locally discrete decomposition spaces

Abstract

Gálvez-Carrillo et al. [Decomposition spaces, incidence algebras and Möbius inversion III: The decomposition space of Möbius intervals, Adv. Math. 334 (2018) 544–584] constructed a decomposition space [Formula: see text] of all Möbius intervals, as a recipient of Lawvere’s interval construction for Möbius categories, and conjectured that [Formula: see text] enjoys a certain universal property: for every Möbius decomposition space [Formula: see text], the space of culf functors from [Formula: see text] to [Formula: see text] is contractible. In this paper, we work at the level of homotopy 1-types to prove the first case of the conjecture, namely for locally discrete decomposition spaces. This provides also the first substantial evidence for the general conjecture. This case is general enough to cover all locally finite posets, Cartier–Foata monoids, Möbius categories and strict (directed) restriction species. The proof is 2-categorical. First, we construct a local strict model of [Formula: see text], which is then used to show by hand that the Lawvere interval construction, considered as a natural transformation, does not admit other self-modifications than the identity.