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 $\mathrm{h}{\mathcal{C}}^{{\Delta }^1}$ of morphisms in the stable infinity category $\mathcal{C}$ is also triangulated. Hence the space $\mathsf{Stab}\mathrm{h}{\mathcal{C}}^{{\Delta }^1}$ of stability conditions on $\mathrm{h}{\mathcal{C}}^{{\Delta }^1}$ is well-defined though the non-emptiness of $\mathsf{Stab}\mathrm{h}{\mathcal{C}}^{{\Delta }^1}$ is not obvious. Our basic motivation is a comparison of the homotopy type of $\mathsf{Stab}\mathrm{h}\mathcal{C}$ and that of $\mathsf{Stab}\mathrm{h}{\mathcal{C}}^{{\Delta }^1}$. Under the motivation we show that functors $d_0$ and $d_1:{\mathcal{C}}^{{\Delta }^1}\rightrightarrows \mathcal{C}$ induce continuous maps from $\mathsf{Stab}\mathrm{h}\mathcal{C}$ to $\mathsf{Stab}\mathrm{h}{\mathcal{C}}^{{\Delta }^1}$ contravariantly where $d_0$ (resp. $d_1$) takes a morphism to the target (resp. source) of the morphism. As a consequence, if $\mathsf{Stab}\mathrm{h}\mathcal{C}$ is nonempty then so is $\mathsf{Stab}\mathrm{h}{\mathcal{C}}^{{\Delta }^1}$. 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.

26 pages, comments are welcome. For v2, added subsection 2.2 which gives a description of a Serre functor of the category of morphisms in $\mathbf{D}$. For v3, the proof of Proposition 3.3 has been updated. For v5, Section 6 was added. For v6, modified the proof of Proposition 6.1. For v7, minor revision for Proposition 6.1, final version

References in corpus (1)

• The space of stability conditions on the local projective plane