A sphere of spherical objects

Given a Bridgeland stability condition on a 2-Calabi–Yau category, we define a simplicial complex that encodes the Harder–Narasimhan filtrations of spherical objects. For 2-Calabi–Yau categories of type A, we relate this complex to the complex of pointed pseudo-triangulations on configurations of points on the plane. Using this connection, we prove that the complex undergoes piecewise-linear wall-crossings as we vary the stability condition, and is piecewise-linearly homeomorphic to a sphere. Additionally, we prove that for a generic stability condition on a 2-Calabi–Yau category, a spherical object is determined by the ordered list of its Harder–Narasimhan factors.