# Cotorsion torsion triples and the representation theory of filtered hierarchical clustering

## TL;DR

This paper classifies the representation types of subcategories of grid representations relevant to clustering and persistent homology, using cotorsion torsion triples and tilting theory to establish categorical equivalences.

## Contribution

It provides a complete classification of these subcategories' representation types and constructs categorical equivalences via cotorsion torsion triples derived from tilting subcategories.

## Key findings

- Subcategories are equivalent to representations of smaller grids.
- Classification of representation types for grid subcategories.
- Use of cotorsion torsion triples to establish categorical equivalences.

## 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.