Tilting theory on continuous quivers (I) maximal rigid and tilting subcategories

Abstract

Let $\mathrm{rep}(\mathbb I)$ be the category of finitely generated persistence modules on the interval $\mathbb I\subseteq\mathbb R$. We study the maximal rigid subcategories of $\mathrm{rep}(\mathbb I)$. Maximal rigid interval consistent subcategories are classified via tilting modules over the corresponding discrete type $A$ quiver. We compare several related notions, (weak) tilting and maximal rigidity, in both the context of hereditary abelian categories with enough projective objects and $\mathrm{rep}(\mathbb I)$ and prove that every maximal rigid interval consistent subcategory is weak tilting and every maximal rigid consistent below subcategory is tilting. We also establish a continuous analogue of Ingalls-Thomas's bijection by showing a bijection between integrally supported maximal rigid consistent below subcategories and integrally generated torsion classes.