Stability conditions on morphisms in a category

Abstract

Let $\mathbf D$ be the homotopy category of a stable infinity category. Then the category $\mathbf D^{\Delta^1}$ of morphisms in $\mathbf{D}$ is also triangulated. Hence the space $\mathsf{Stab}\,{ \mathbf D^{\Delta^1}}$ of stability conditions on $\mathbf D^{\Delta^1}$ is well-defined though the non-emptiness of $\mathsf{Stab}\,{ \mathbf D^{\Delta^1}}$ is not obvious. We discuss a relation between $\mathsf{Stab}\,{ \mathbf D^{\Delta^1}}$ and $\mathsf{Stab}\,{ \mathbf D}$ by proposing some problems.