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Abstract We study rank functions on a triangulated category π via its abelianisation mod β‘ C modC. We prove that every rank function on π can be interpreted as an additive function on mod β‘ C modC. 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 β‘ C ModC. We study the connection between rank functions and functors from π to locally finite triangulated categories, generalising results by Chuang and Lazarev. In the special case C = T c C=T^c for a compactly generated triangulated category π―, this connection becomes particularly nice, providing a link between rank functions on π and smashing localisations of π―. In this context, any integral rank function can be described using the composition length with respect to certain endofinite objects in π―. Finally, if C = per β‘ (A) C=per (A) for a differential graded algebra π΄, we classify homological epimorphisms A β B A B with per β‘ (B) per (B) locally finite via special rank functions which we call idempotent.
Conde et al. (Fri,) studied this question.