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

TL;DR

This paper develops a functorial framework for rank functions in $(d+2)$-angulated categories, establishing bijections with morphism-based rank functions and decomposing integral rank functions into irreducibles.

Contribution

It introduces a new notion of rank functions on $(d+2)$-angulated categories and proves bijections with morphism-based rank functions and additive functions, extending prior work.

Findings

Bijective correspondence between object and morphism rank functions for odd $d$.

Decomposition of integral rank functions into irreducible components.

Extension of rank function concepts to $(d+2)$-angulated categories.

Abstract

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 $\operatorname{\mathsf{Proj}}A$ and additive functions on $\operatorname{\mathsf{mod}}(\operatorname{\mathsf{Proj}}A)$, where $\operatorname{\mathsf{Proj}}A$ is endowed with the Amiot-Lin $(d+2)$-angulated category structure. This allows us to show that every integral rank function on $\operatorname{\mathsf{Proj}}A$ can be decomposed into irreducible rank functions.