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The inner product of vectors in a Hilbert space has many properties in common with a hom functor. I know that one can make a projectivized Hilbert space into a metric space with the Fubini-Study metric and then follow Lawvere in considering it a $[0,\infty)$-enriched category; it's particularly nice because the distance between two vectors is a function of the inner product.

Is there a way of defining a monoidal structure on $\mathbb{C}$ such that the inner product is the hom so that we can think of a (perhaps projectivized) Hilbert space as $\mathbb{C}$-enriched?

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Related question:… – Loop Space Nov 29 '10 at 8:18
You might also be interested in John Baez's paper on the subject: – Qiaochu Yuan Nov 29 '10 at 9:23

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