Sur l'existence d'une cat\'egorie ayant une matrice strictement positive donn\'ee

TL;DR

This paper investigates which strictly positive integer matrices can be realized as the Leinster matrix of a finite category, providing a characterization for such matrices.

Contribution

It offers a complete answer to the question of which positive matrices correspond to finite categories when all entries are strictly positive.

Findings

Characterization of strictly positive matrices realizable as Leinster matrices

Conditions under which a positive matrix corresponds to a finite category

Clarification of the relationship between matrix entries and morphism counts

Abstract

The Leinster matrix corresponding to a finite category has entries counting the number of morphisms between objects. A first question is to know which positive integer matrices come from at least one finite category. Here, that question will be answered when the entries are strictly positive.