Hilb as a Colimit in the Category of Scott Complete Categories (foundations)

Here is a paper I found by Adamek that generalizes Domain theory into categories of categories called Scott Complete Categories. The category of Scott Complete categories is denoted SCC. For years, I had a theory that the category describing quantum theory, Hilb, was a compact element in some domain of categories because Hilb has finitely many axioms. Actually, I am not sure if we even have a categorical model of Hilb that has exactly all the axioms we expect from quantum theory. If you have worked with Hilb, you know that you can see Hilbert spaces as the familiar topological spaces, metric spaces, modules, sets, Banach spaces etc. Right now, I want to say the following: Hilb is a colimit in SCC, and the diagram consists of categories {$A$, $B$...etc}. Is it the case that Hilb can be seen as a colimit in SCC where the diagram of the cone is all the familiar categories I have listed?

Alternatively, Heunen has done work on "compactly accessible categories". In order to get Hilb to be a colimit, do we need to modify Adamek's paper to use the category of compactly accessible categories? Are familiar categories, like those I listed, compactly accessible?

• This question confuses me. 1. Your question seems to be whether Hilb is a colimit of certain categories. I don't see the relevance of the idea that Hilb should be "compact". After all, I wouldn't expect "compact" objects to be closed under colimits. 2. Does Hilb mean Hilbert spaces and bounded maps? If so, it's a well-defined category, so what would a "categorical model" of Hilb be? 3. You seem to want to exhibit Hilb as a colimit of Scott-Complete categories. Have you checked whether Hilb itself is Scott-Complete? 4. It's not clear what diagram this colimit is supposed to be taken over. – Tim Campion Feb 22 '15 at 19:45
• According to Heunen's paper, Hilb is not cocomplete, which I think means it is not Scott complete. This is why I brought up his paper. I am wondering about modifying Adamek's work to regard compactly accessible categories as defined by Heunen. In that case, we could have a category of compactly accessible categories, wherein we find Hilb, and perhaps it is a colimit there. This question is about foundations in that we want to start seeing certain theories as idealizations, ie colimits, over diagrams of finite approximations (compact categories). – Ben Sprott Feb 23 '15 at 20:03