nLab
Hilb

$\mathrm{Hilb}$ is the category whose object are Hilbert spaces and whose morphisms are short linear maps (linear maps of norm at most $1$) between these.

We can also take $\mathrm{Hilb}$ to be the dagger category with the same object but whose morphisms are now all bounded linear maps between them. (The dagger operation is the usual adjoint operation on such maps.) The full subcategory $\mathrm{Fin}\mathrm{Hilb}$ of finite-dimensional Hilbert spaces becomes a dagger compact category.

Note that either way, the core (of isomorphisms in the first case, or of unitary isomorphisms in the other case) is the same groupoid, whose morphisms are all invertible linear maps of norm exactly $1$.

In any case, the forgetful functor from $\mathrm{Hilb}$ to Vect is faithful, confirming the intuition that a Hilbert space is a vector space equipped with extra structure. $\mathrm{Hilb}$ is also a full subcategory of Ban, the category of Banach spaces.

References

A pedagogical description of the monoidal structure on $\mathrm{Hilb}$ with an emphasis on their role in quantum mechanics and their relation to nCob is in

• John Baez, The monoidal category of Hilbert spaces (web)

category: category