A metric on a category assigns lengths to morphisms, with the triangle inequality holding. This notion goes back to a 1974 article by Lawvere. We'll begin with a quick review of some basic constructions, like forming the Cauchy completion of a category with respect to a metric.

We then discuss a string of surprising new results. It turns out that, in a triangulated category with a metric, there is a reasonable notion of Taylor series, and an approximable triangulated category can be thought of as a category where many objects are the limits of their Taylor expansions. Then come two types of theorems: (1) theorems providing examples, meaning showing that some category you might naturally want to look at is approximable, and (2) general structure theorems about approximable triangulated categories.

What makes it all interesting is (3) applications. These turn out to include the proof of a conjecture by Bondal and Van den Bergh, a vast generalization of a theorem of Rouquier's, and an amazingly short proof of Serre's GAGA theorem.

It should be noted that the results I will be presenting are the highlights of a series of five papers. Of these only the first has been published (May 2021). The second article in the series has been submitted for publication, with a 3rd submission to follow.

To join this seminar via Zoom please click here.

If you would like to join the seminar online and are not currently affiliated with ANU, please contact Martin Helmer at martin.helmer@anu.edu.au.