  We develop the viewpoint that the opposite of the category of W*-algebras and
unital normal *-homomorphisms is analogous to the category of sets and
functions. For each pair of W*-algebras, we construct their free exponential,
which in the context of this analogy corresponds to the collection of functions
from one of these W*-algebras to the other. We also show that every unital
normal completely positive map between W*-algebras arises naturally from a
normal state on their free exponential.
