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?
Can the category of sets become a daggery category
category-theoryhilbert-spaces
Best Answer
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.