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

TL;DR

This paper proves a special case of the Gálvez-Kock-Tonks conjecture for locally discrete decomposition spaces, providing evidence for the general conjecture and covering various mathematical structures.

Contribution

It establishes the first case of the conjecture at the homotopy 1-type level for locally discrete decomposition spaces, using a 2-categorical proof approach.

Findings

First case of the conjecture proved for locally discrete decomposition spaces.

Constructed a local strict model of the universal decomposition space.

Lawvere interval construction admits no self-modifications besides the identity.

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.