The categories ${\mathcal T}^c$ and ${\mathcal T}^b_c$ determine each other

Given an essentially small triangulated category it is possible to give a metric on it, to complete it with respect to the metric, and to look at the subcategory of objects in the completion which are compactly supported with respect to the metric. The main theorem says that this procedure produces a new triangulated category. And then we give examples: for example we learn that it is possible, for suitable choices of metrics, to produce the categories $D^b(R-\text{mod})$ and $K^b(R-\text{proj})$ out of each other.