Title: Spherelike objects and quiver constructions Abstract: The most simple kind of objects in a triangulated category are the exceptional objects. Next in complexity are objects with 2-dimensional graded endomorphism algebra; we call these "spherelike". In the first part of the talk, I will motivate why spherelike objects are interesting, and show examples occurring in derived categories of algebras. In the second part, I will explain how spherelike objects can be collected in a new invariant of triangulated categories, the "spherelike poset". I will mention some concrete operations on quivers with relations for which we can track the growth of this poset. (Joint with Andreas Hochenegger, Milano, and Martin Kalk, Edinburgh.)