# Can the category of sets become a daggery category?

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?

• This seems like a very dubious reason to think of Hilbert spaces from a categorical perspective rather than a set-theoretic one. – Eric Wofsey Mar 29 at 21:14
• How would you define a functor on the function $\emptyset \to \{*\}?$ – QuantumSpace Mar 29 at 21:14
• @EricWofsey the context here was the mathematical structure of quantum mechanics and its relations to the category of n-dimensional cobordisms. – mathripper Mar 29 at 21:19
• The category of sets and maps doesn't support a dagger structure, but the category of sets and relations does. – Zhen Lin Mar 29 at 22:21

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.