Stability conditions on morphisms in a category

Abstract: Let $\mathrm{h}\mathscr{C}$ be the homotopy category of a stable infinity category $\mathscr{C}$. Then the homotopy category $\mathrm{h}\mathscr{C}^{\Delta^{1}}$ of morphisms in the stable infinity category $\mathscr{C}$ is also triangulated. Hence the space $\mathsf{Stab}\,{ \mathrm{h}\mathscr{C}^{\Delta^{1}}}$ of stability conditions on $\mathrm{h}\mathscr{C}^{\Delta^{1}}$ is well-defined though the non-emptiness of $\mathsf{Stab}\,{ \mathrm{h}\mathscr{C}^{\Delta^{1}}}$ is not obvious. Our basic motivation is a comparison of the homotopy type of $\mathsf{Stab}{\mathrm{h}\mathscr{C}}$ and that of $\mathsf{Stab}{\mathrm{h}\mathscr{C}^{\Delta^{1}}}$. Under the motivation we show that functors $d_{0}$ and $d_{1} \colon \mathscr{C}^{\Delta^{1}} \rightrightarrows \mathscr{C}$ induce continuous maps from $\mathsf{Stab} {\mathrm{h}\mathscr{C}}$ to $\mathsf{Stab}{\mathrm{h}\mathscr{C}^{\Delta^{1}}}$ contravariantly where $d_{0}$ (resp. $d_{1}$) takes a morphism to the target (resp. source) of the morphism. As a consequence, if $\mathsf{Stab}{\mathrm{h}\mathscr{C}}$ is nonempty then so is $\mathsf{Stab}{\mathrm{h}\mathscr{C}^{\Delta^{1}}}$. Assuming $\mathscr{C}$ is the derived infinity category of the projective line over a field, we further study basic properties of $d_{0}^{*} $ and $d_{1}^{*}$. In addition, we give an example of a derived category which does not have any stability condition.