Metrics on triangulated categories and their enhancements

Abstract

In this paper we investigate the uniqueness of enhancements of the natural subcategories of weakly approximable triangulated categories. The main idea is to enhance at the level of $\infty$-categories the recently developed theory of excellent metrics. The applications of our results include a vast generalization of the known results about the (strong) uniqueness of enhancements in the linear and nonlinear setting, providing positive answers to some open questions. In addition we prove that, under some natural assumptions, the equivalences between such subcategories can be lifted through their natural inclusions. This completes the picture started in our previous paper arXiv:2505.10374 and extends the known results about Margolis Uniqueness Conjecture for the homotopy category of spectra.