  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.
