Motivated by the work of Cohn and Schofield on Sylvester rank functions on rings, Chuang and Lazarev have recently introduced the notion of a rank function on a triangulated category. A rank function is a nonnegative real-valued, additive, translation-invariant function on the objects of a triangulated category for which the triangle inequality on distinguished triangles holds. It turns out that a rank function on a triangulated category C can be recast as translation-invariant additive function on its abelianisation mod-C. This allows us to relate integral-valued rank functions on C with endofinite cohomological functors on C and, when C is the subcategory of compact objects of a compactly generated triangulated category T, with endofinite objects in T and certain closed sets of the Ziegler spectrum of T.

This talk is based on joint work with Mikhail Gorsky, Frederik Marks and Alexandra Zvonareva.