A functorial approach to rank functions on triangulated categories Gomes Cipriano Nabais Conde, Teresa Gomes Cipriano Nabais Conde Teresa Gorsky, Mikhail Gorsky Mikhail Marks, Frederik Marks Frederik Zvonareva, Alexandra Zvonareva Alexandra We study rank functions on a triangulated category C via its abelianisation mod C. We prove that every rank function on C can be interpreted as an additive function on mod C. As a consequence, every integral rank function has a unique decomposition into irreducible ones. Furthermore, we relate integral rank functions to a number of important concepts in the functor category Mod C. We study the connection between rank functions and functors from C to locally finite triangulated categories, generalising results by Chuang and Lazarev. In the special case C = T-c for a compactly generated triangulated category T, this connection becomes particularly nice, providing a link between rank functions on C and smashing localisations of T. In this context, any integral rank function can be described using the composition length with respect to certain endofinite objects in T. Finally, if C = per (A) for a differential graded algebra A, we classify homological epimorphisms A -> B with per (B) locally finite via special rank functions which we call idempotent. De Gruyter 2024