Stability conditions on morphisms in a category

Abstract

Let $\mathrm{h}(\mathcal{C})$ be the homotopy category of a stable infinity category $\mathcal{C}$. Then the homotopy category $\mathrm{h}({\mathcal{C}}^{{\mathrm{\Delta }}^1})$ of morphisms in the stable infinity category $\mathcal{C}$ is also triangulated. Hence, the space $\mathsf{Stab}\mathrm{h}({\mathcal{C}}^{{\mathrm{\Delta }}^1})$ of stability conditions on $\mathrm{h}({\mathcal{C}}^{{\mathrm{\Delta }}^1})$ is well-defined though the nonemptiness of $\mathsf{Stab}\mathrm{h}({\mathcal{C}}^{{\mathrm{\Delta }}^1})$ is not obvious. Our basic motivation is a comparison of the homotopy type of $\mathsf{Stab}\mathrm{h}(\mathcal{C})$ and that of $\mathsf{Stab}\mathrm{h}({\mathcal{C}}^{{\mathrm{\Delta }}^1})$. Under the motivation, we show that functors $d_0$ and $d_1:{\mathcal{C}}^{{\mathrm{\Delta }}^1}\rightrightarrows \mathcal{C}$ induce continuous maps from $\mathsf{Stab}\mathrm{h}(\mathcal{C})$ to $\mathsf{Stab}\mathrm{h}({\mathcal{C}}^{{\mathrm{\Delta }}^1})$ contravariantly where $d_0$ (resp., $d_1$) takes a morphism to the target (resp., source) of the morphism. As a consequence, if $\mathsf{Stab}\mathrm{h}(\mathcal{C})$ is nonempty, then so is $\mathsf{Stab}\mathrm{h}({\mathcal{C}}^{{\mathrm{\Delta }}^1})$. Assuming $\mathcal{C}$ is the derived infinity category of the projective line over a field, we further study basic properties of $d_0^{\ast }$ and $d_1^{\ast }$. In addition, we give an example of a derived category which does not have any stability condition.