The Lattice of Subobject Closed Subcategories and Colocal Type

We consider abelian length categories, a generalization of module categories over Artin algebras. Let $\mathcal{A}$ be an abelian length category of colocal type. We show that the lattice $\mathsf{S}(\mathcal{A})$ of full additive subobject closed subcategories of $\mathcal{A}$ is distributive. Furthermore, we give a characterization of abelian length categories of colocal type. If $A$ is an algebra of colocal type over an algebraically closed field, then this characterization is especially simple and we can describe the lattice $\mathsf{S}(\mathsf{m}\mathsf{o}\mathsf{d}A)$ up to isomorphism.