Comparing metric completions of triangulated categories with Verdier quotients

Abstract

Neeman's metric on a triangulated category assigns lengths to morphisms and allows us to construct triangulated metric completions of the category. We prove that the metric completion of a triangulated category with respect to a constant metric is always a subcategory of the Verdier quotient by the thick subcategory generated by cones of zero-length morphisms, and we provide a sufficient criterion for equality. To that end, we define continuous functors between completions which induce triangulated functors between completions. Moreover, we give two applications. First, we show that the additive equivalence discovered by Cummings and Gratz between the Paquette-Yildirim completion and a specific metric completion of an Igusa-Todorov discrete cluster category is, in fact, a triangulated equivalence. Second, we fully classify metric completions of bounded derived categories of coherent sheaves over weighted projective lines of domestic or tubular type.