We can give a direct argument for why 2-category theory should be relevant for quantum theory. Suppose we have a quantum system with a finite-dimensional Hilbert space, seen as an object in the category Hilb of finite-dimensional Hilbert spaces. Then if a projector-valued measurement takes place, future dynamics can be different depending on which of the n possible measurement results occurred. We now effectively have n independent copies of our state space, conveniently described as an object in Hilbⁿ, the n-fold Cartesian product of the category of finite-dimensional Hilbert spaces. A categorical setting in which to study the measurement process must therefore include both Hilb and Hilbⁿ, since these give our mathematical context before and after the measurement takes place; and since they are themselves categories, the correct setting in which to study them will be a 2-category. The 2-category 2Hilb is then a natural choice, since up to equivalence, its objects are precisely the categories of the form Hilbⁿ.
James Vicary, Higher Quantum Theory
