Relative cluster tilting theory and $\tau$-tilting theory

Abstract

Let $\mathcal C$ be a Krull-Schmidt triangulated category with shift functor $[1]$ and $\mathcal R$ be a rigid subcategory of $\mathcal C$. We are concerned with the mutation of two-term weak $\mathcal R[1]$-cluster tilting subcategories. We show that any almost complete two-term weak $\mathcal R[1]$-cluster tilting subcategory has exactly two completions. Then we apply the results on relative cluster tilting subcategories to the domain of $\tau$-tilting theory in functor categories and abelian categories.