Metrics on triangulated categories

Abstract

In a 1973 article Lawvere defined (among many other things) metrics on categories—the article has been enormously influential over the years, spawning a huge literature. In recent work, which is surveyed in the current note, we pursue a largely-unexplored angle: we complete categories with respect to their Lawvere metrics.This turns out to be particularly interesting when the category is triangulated and the Lawvere metric is good; a metric is good if it is translation invariant and the balls of radius ε>0 shrink rapidly enough as ε decreases. The definitions are all made precise at the beginning of the note. And the main theorem is that a certain natural subcategory S(S), of the completion of S with respect to a good metric, is triangulated.There is also a theorem which, under restrictive conditions, gives a procedure for computing S(S). As examples we discuss the special cases (1) where S is the homotopy category of finite spectra, and (2) where S=Db(R–mod), the derived category of bounded complexes of finitely generated R–modules over a noetherian ring R.