Coproduct in the FdHilb category

Question

Consider the FdHilb category i.e. the category of finite dimensional Hilbert spaces that is important especially in the quantum mechanics.

I have some trouble with identification of coproduct in the category. What is it? Any help, hints are welcome.

Edit: The original question incorrectly identified the tensor product as the categorial product. It is very important in the quantum mechanics because it is used to describe compound systems. It will be also fine to get a categorial construction for the tensor product if it is possible

Answer 1

One of the features of FdHilb (the cateogry of finite dimensional Hilbert spaces) is that it possesses finite biproducts. A biproduct of a finite collection of objects is both a product and a coproduct.

In FdHilb the construction that is both a product and a coproduct is direct sum of Hilbert spaces, not tensor product. According to the definition on Wikipedia, tensor product is not a categorical product: there are not natural maps from a tensor product of two spaces to the two spaces separately, etc.

Answer 2

Be careful! It is not at all clear what the morphisms in $\text{FdHilb}$ should be. If you want them to be all linear maps, then the category you end up with is not really a category of Hilbert spaces, and in fact is equivalent to the category of finite-dimensional complex vector spaces; in other words this choice of morphism effectively forgets the inner product.

You could ask for the morphisms to be unitary maps or isometries, which does take the inner product into account, but this choice of morphism is very restrictive and hardly any categorical constructions exist in this category.

The "correct" thing to do appears to be to regard $\text{FdHilb}$ as something more complicated than a category: it is really a dagger category where the dagger is given by taking adjoints and this is somehow crucial to understanding its categorical nature. I do not really understand how "dagger category theory" is supposed to work, and in particular how to adjust the notion of limits and colimits to take the dagger into account.