Abstract
We develop the viewpoint that [Formula: see text], the opposite of the category of [Formula: see text]-algebras and unital normal ∗-homomorphisms, is analogous to the category of sets and functions. For each pair of [Formula: see text]-algebras [Formula: see text] and [Formula: see text], we construct their free exponential [Formula: see text], which in the context of this analogy corresponds to the collection of functions from [Formula: see text] to [Formula: see text]. We also show that every unital normal completely positive map [Formula: see text] arises naturally from a normal state on [Formula: see text].
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