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The definition states that closedness is equivalent to the hom-sets being objects of the monoidal category itself. This is only intuitively true, the correct statement is that the generalized 1-elements (1 being the unit object) of the internal hom are isomorphic to the hom-set.
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In Hilb, f^\dagger is given by the adjoint of f, in Rel it is given by the opposite relation, and in the category of finitely generated projective modules it is... what, the transpose of the matrix representation of a linear map?
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What, exactly, is the deal with finite vs. infinite-dim hilbert space? Can we explicitly demonstrate why this doesn't work for the infinite case (yes, compactness. but spell it out in detail...)
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Quantum mechanics technically lives on complex projective hilbert space, not hilbert space in general. What would need to be added to make a category be projective? Is there a
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Stuff like the basis, eigenvectors, etc. should be worked out in detail for the category of relations, viz to have a clearer view of what it looks like in a different cat.
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Hilbert space is defined for any field; including finite fields; again, QM works only for the field being the complex numbers. Can anything intelligent be said about this?
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It may help to make the article more useful if somebody (more proficient at this than I) adds what is f^\dagger in each of the examples.
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I'm not an expert in category theory, but the part about eigenstates confuses me: shouldn't they be morphisms rather than objects?
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So, for example, normally, unitarity arises from the projective nature .. here, we might expect unitarity to be a
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Frobenius part has something to do with the movement of classical information (viz not
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The bit about defining a basis here makes use of relations from
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All the dagger does is reverse the direction of arrows. So, if
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