Metric completions of triangulated categories from finite dimensional algebras

In this paper, we study metric completions of triangulated categories in a representation-theoretic context. We provide a concrete description of completions of bounded derived categories of hereditary finite dimensional algebras of finite representation type. In order to investigate completions of bounded derived categories of algebras of finite global dimension, we define image and preimage metrics under a triangulated functor and use them to induce a triangulated equivalence between two completions. Furthermore, for a given metric on a triangulated category we construct a new, closely related good metric called the improvement and compare the respective completions.