Let ${\mathbin{\cal T}}$ be a triangulated category. A (Bridgeland) stability condition $(Z,{\mathbin{\cal P}})$ on ${\mathbin{\cal T}}$ consists of a group homomorphism $Z:K_{0}({\mathbin{\cal T}})\rightarrow{\mathbin{\mathbb{C}}}$ called the central charge, and full additive subcategories ${\mathbin{\cal P}}(\phi)\subset{\mathbin{\cal T}}$ for each $\phi\in{\mathbin{\mathbb{R}}}$, satisfying the following properties:

• (i)

  If $A\in{\mathbin{\cal P}}(\phi)$ then $Z([A])=m(A)e^{i\pi\phi}$ for some $m(A)>0$.

• (ii)

  For all $\phi\in{\mathbin{\mathbb{R}}}$, ${\mathbin{\cal P}}(\phi+1)={\mathbin{\cal P}}(\phi)[1]$.

• (iii)

  If $\phi_{1}>\phi_{2}$ and $A_{j}\in{\mathbin{\cal P}}(\phi_{j})$ then $\mathop{\rm Hom}\nolimits_{\mathbin{\cal T}}(A_{1},A_{2})=0$.

• (iv)

  For each nonzero object $F\in{\mathbin{\cal T}}$ there is a finite sequence of real numbers $\phi_{1}>\phi_{2}>\cdots>\phi_{n}$ and a diagram in ${\mathbin{\cal T}}$

  where the triangles are distinguished and $A_{j}\in{\mathbin{\cal P}}(\phi_{j})$ for $j=1,\ldots,n$.

Objects in ${\mathbin{\cal P}}(\phi)$ for some $\phi\in{\mathbin{\mathbb{R}}}$ are called semistable.