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I came across the claim that the category of finite dimensional Hilbert spaces, FHilb, where objects are Hilbert spaces and morphisms are bounded linear operators is a dagger category, i.e. it has a contravariant involutive functor. Then, it was suggested that this is a good reason of why we should use the framework of category theory because the category of sets, Set, does not have a dagger structure so it's better to think of Hilbert spaces from a categorical perspective rather than a set-theoretic one. Can someone explain why we cannot turn Set into a dagger structure?

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    $\begingroup$ This seems like a very dubious reason to think of Hilbert spaces from a categorical perspective rather than a set-theoretic one. $\endgroup$ – Eric Wofsey Mar 29 at 21:14
  • $\begingroup$ How would you define a functor on the function $\emptyset \to \{*\}?$ $\endgroup$ – QuantumSpace Mar 29 at 21:14
  • $\begingroup$ @EricWofsey the context here was the mathematical structure of quantum mechanics and its relations to the category of n-dimensional cobordisms. $\endgroup$ – mathripper Mar 29 at 21:19
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    $\begingroup$ The category of sets and maps doesn't support a dagger structure, but the category of sets and relations does. $\endgroup$ – Zhen Lin Mar 29 at 22:21
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Suppose $\dagger$ is a dagger structure on $\mathsf{Set}$. Let $X$ be any non-empty set, and let $f\colon \emptyset\to X$ be the empty function. Then $f^\dagger$ is a function $X\to \emptyset$. This is a contradiction, since are no functions from a non-empty set to the empty set.

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