Hilb

**algebraic quantum field theory** (perturbative, on curved spacetimes, homotopical)

**quantum mechanical system**, **quantum probability**

**interacting field quantization**

A category whose objects are Hilbert spaces is typically denoted $Hilb$ or similar.

There are different choices of morphisms in use, such as all linear maps or just the short linear maps (linear maps of norm at most $1$).

One may regard $Hilb$ as a dagger category with morphisms the bounded linear maps between them and the dagger operation assigning adjoint operators. The full subcategory $Fin Hilb$ 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 $Hilb$ to Vect is faithful, confirming the intuition that a Hilbert space is a vector space equipped with extra structure. $Hilb$ is also a full subcategory of Ban, the category of Banach spaces.

A pedagogical description of the monoidal category structure on $Hilb$ with an emphasis on their role in quantum mechanics and their relation to nCob:

An axiomatic characterization of the dagger-category of Hilbert spaces, with linear maps between them:

- Chris Heunen, Andre Kornell,
*Axioms for the category of Hilbert spaces*(arXiv:2109.07418)

category: category