  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.

Representation Theory;Algebraic Topology