This project is part of a larger goal to introduce and develop a fundamental topic from algebra (multiplicity) into the modern setting of triangulated categories admitting a central ring action. Multiplicity is a basic topic that has been extensively studied in many forms in algebra, geometry and topology, and is a topic that has found applications throughout other areas as well. In addition to introducing this topic into the realm of triangulated categories, this proposed project also provides new invariants attached to pairs of objects in triangulated categories. In this project, we will primarily investigate the notion of multiplicity as pertaining to intersections, an aspect of multiplicity theory with strong geometric foundations. Our goal is to develop this rich theory in the setting of triangulated categories, while also providing a path forward to understand other types of multiplicity in this abstract setting, which should be of interest to other algebraists working on various types of multiplicity and to category theorists working in abstract triangulated categories.