The G\'alvez-Kock-Tonks conjecture for locally discrete decomposition spaces

Abstract

G\'alvez-Carrillo, Kock, and Tonks constructed a decomposition space $U$ of all M\"obius intervals, as a recipient of Lawvere's interval construction for M\"obius categories, and conjectured that $U$ enjoys a certain universal property: for every M\"obius decomposition space $X$, the space of culf functors from $X$ to $U$ 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, M\"obius categories and strict (directed) restriction species. The proof is 2-categorical. First, we construct a local strict model of $U$, 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.