The index with respect to a contravariantly finite subcategory

Abstract

Cluster algebras are categorified by cluster categories, and $g$-vectors are categorified by the classic index with respect to cluster tilting subcategories. However, the recently introduced completed discrete cluster categories of Dynkin type $\mathbb{A}$ have a very limited supply of cluster tilting subcategories, so we define the index with respect to additive, contravariantly finite subcategories of which there are many more. This permits us to extend several strong results from the classic theory to completed discrete cluster categories of Dynkin type $\mathbb{A}$. Notably, the index with respect to the subcategory generated by a fan triangulation distinguishes between rigid objects. We also prove that our index is additive on triangles up to an error term. This extends the key property which permits the classic index to be used in the categorification of cluster algebras.