# Metrics on triangulated categories

## TL;DR

This survey explores metrics on triangulated categories, showing that their completions contain interesting triangulated subcategories, with applications to derived categories of coherent sheaves and perfect complexes.

## Contribution

It specializes general metric constructions to triangulated categories and proves that completions contain triangulated subcategories, revealing new structural insights.

## Key findings

- Completion of a triangulated category with a good metric contains a triangulated subcategory.
- Special cases include derived categories of coherent sheaves and perfect complexes.
- Provides new examples of triangulated categories arising from metric completions.

## Abstract

In this survey we explain the results of the recent article arXiv:1806.06471. Following a 1973 article by Lawvere one can define metrics on categories, and following Kelly's 1982 book one can complete a category with respect to its metric. We specialize these general constructions to triangulated categories, and restrict our attention to "good metrics". And the remarkable new theorem is that, when we start with a triangulated category $\mathcal S$ with a good metric, its completion $\mathfrak{L}(\mathcal{S})$ contains an interesting subcategory $\mathfrak{S}(\mathcal{S})$ which is always triangulated. As special cases we obtain $\mathcal{H}^0(\mathrm{Perf}(X))$ and $D^b_{\mathrm{coh}}(X)$ from each other. We also give a couple of other examples.