  We develop a framework relating semiorthogonal decompositions of a
triangulated category $\mathcal{C}$ to paths in its space of stability
conditions. We prove that when $\mathcal{C}$ is the homotopy category of a
smooth and proper idempotent complete pre-triangulated dg-category, every
semiorthogonal decomposition whose factors admit a Bridgeland stability
condition can be obtained from our framework.
