Abstract: Joyce and others have used shifted symplectic geometry to define Donaldson-Thomas Invariants. This kind of geometry naturally appears on derived moduli stacks of perfect complexes on Calabi-Yau varieties. One wonderful feature of shifted symplectic geometry (developed by Pantev, Toën, Vaquié and Vezzosi) is that fibre products (i.e. intersections) of Lagrangians automatically carry Lagrangian structures. Using a strange property of triple intersections from arXiv:1309.0596, this extra structure can be organized into a 2-category. We discuss a partial linearization using Joyce's perverse sheaves. One can also speculate on the relationship of this 2-category with TQFTs, algebraic versions of the Fukaya categories, infinity categories of Lagrangians and the gluing of stacks of matrix factorizations. This is joint work with Lino Amorim.