Metric completions of discrete cluster categories

TL;DR

This paper explores how completing discrete cluster categories with specific metrics results in new triangulated categories, providing insights into their topological and algebraic structures.

Contribution

It introduces a method to compute completions of discrete cluster categories using internal t-structure induced metrics, revealing new triangulated categories and their properties.

Findings

Completion with coaisle metric yields a new triangulated category.

Completion with internal aisle metric results in a thick subcategory.

Provides a topological interpretation of combinatorial models.

Abstract

Neeman shows that the completion of a triangulated category with respect to a good metric yields a triangulated category. We compute completions of discrete cluster categories with respect to metrics induced by internal t-structures. In particular, for a coaisle metric this yields a new triangulated category which can be interpreted as a topological completion of the associated combinatorial model. Moreover, we show that the completion of any triangulated category with respect to an internal aisle metric is a thick subcategory of the triangulated category itself.