Rank functions on $$(d+2)$$-angulated categories: a functorial approach

We introduce the notion of a rank function on a $$(d+2)$$ -angulated category $$\mathcal {C}$$ which generalises the notion of a rank function on a triangulated category. Inspired by work of Chuang and Lazarev, for $$d$$ an odd positive integer, we prove that there is a bijective correspondence between rank functions defined on objects in $$\mathcal {C}$$ and rank functions defined on morphisms in $$\mathcal {C}$$ . Inspired by work of Conde, Gorsky, Marks and Zvonareva, for $$d$$ an odd positive integer, we show there is a bijective correspondence between rank functions on $${\textsf{proj}}A$$ and certain additive functions on $${\textsf{mod}}({\textsf{proj}}A)$$ , where $$A$$ is a twisted $$(d+2)$$ -periodic algebra and $${\textsf{proj}}A$$ is endowed with the Amiot-Lin $$(d+2)$$ -angulated category structure. This allows us to show that every integral rank function on $${\textsf{proj}}A$$ can be decomposed into a (locally finite) sum of irreducible rank functions.