  We aim to characterize the category of injective *-homomorphisms between
commutative C*-subalgebras of a given C*-algebra A. We reduce this problem to
finding a weakly terminal commutative subalgebra of A, and solve the latter for
various C*-algebras, including all commutative ones and all type I von Neumann
algebras. This addresses a natural generalization of the Mackey-Piron
programme: which lattices are those of closed subspaces of Hilbert space? We
also discuss the way this categorified generalization differs from the original
question.
