Cotorsion torsion triples and the representation theory of filtered hierarchical clustering

Abstract

We give a full classification of representation types of the subcategories of representations of an $m\times n$ rectangular grid with monomorphisms (dually, epimorphisms) in one or both directions, which appear naturally in the context of clustering as two-parameter persistent homology in degree zero. We show that these subcategories are equivalent to the category of all representations of a smaller grid, modulo a finite number of indecomposables. This equivalence is constructed from a certain cotorsion torsion triple, which is obtained from a tilting subcategory generated by said indecomposables.

arXiv comment39 pages; corrected the lists appearing in Cor. 1.6 and minor changes throughout

References in corpus (2)

• Cotorsion pairs in categories of quiver representations
• Tilting via torsion pairs and almost hereditary noetherian rings

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