It's in everybody's interest here not to use too much of JB's time, so I am anxious to get help from someone who is not JB-----selfAdjoint, Hurkyl, arivero, Kea and probably several others, anyone who could help me understand twoHilberts. I am stuck with a detail of the categorified version of vector addition. Has to do with not understanding the idea of "canonical isomorphism". I understand the idea of an object being "universal" for a certain property. We are defining COPRODUCTS. (direct sum of vector spaces AND union of sets are BOTH examples of coproducts----coproducts is a good idea) in a category with coproducts, you have objects A and B and their coproduct which I will lazily write (A+B, i, j) is a triple consisting of this other object A+B and morphisms A--i-->A+B and B--j-->A+B which is UNIVERSAL for that double-stoker situation in the sense that for any other such triple (Z, i', j') there exists a unique morphism f that factors i' and j' so that A+B can come first. A+B can always act as a way-station in any other double-stoker. So there is a unique f with i' = if and j' = jf A --i--> A+B --f--> Z B --j--> A+B --f--> Z the thing I sense is true but dont see how to show is the UNIQUENESS of the coproduct A+B "up to canonical isomorphism" AAAHHH! maybe this is how you get the uniqueness. You suppose there is another coproduct (A&B, i', j') then because each is universal exist unique f and g with, so THESE morphisms constitute the "canonical isomorphism" they were talking about A --i--> A+B --f--> A&B B --j--> A+B --f--> A&B A --i'--> A&B --g--> A+B B --j'--> A&B --g--> A+B and we need to check fg = 1A+B but that is because we can factor the A+B double stoke THRU ITSELF and the definition says there must be a UNIQUE factor morphism that works and both fg works and also 1A+B works, so they have to be equal. I dont really need to write it out but copy/paste makes it easy so why not A --i--> A+B --f--> A&B --g--> A+B B --j--> A+B --f--> A&B --g--> A+B that shows that fg works and also 1A+B works (is that obvious?) so they are the same. Likewise for gf = 1A&B So up to "canonical isomorphism" which mean unavoidable preordained that you bark your shins on isomorphism there ISN'T ANY A&B. There is only the one coproduct A+B. OK. That went all right. Let us see what else there is what we have to do is to see how to REALIZE EVERY CONDITION IN THE DEFINITION OF A HILBERT SPACE in terms of category protocols Like a usual Hilbert has a ZERO, so now we are going to have a category where the objects are Hilberts and there is actually a ZERO HILBERT Fantastic, there is actually a Hilbert space that ACTS LIKE A ZERO VECTOR amongst other Hilbert spaces. Since on a usual Hilbert space we have the "inner product" of two vectors, we have to figure out what is an "inner product" of two Hilberts. ================ this is apt to be amusing. Please check in to see how I am doing, in case I get stuck or overlook something. ================ the motivation is something JB said about spin foams Just as Feynman diagrams and spin networks are diagrammatic ways of thinking about hookups between ordinary Hilberts, JB said that spinfoams are diagrammatic ways of thinking about hookups between TWOHILBERTS. what I am trying to understand is the definition of a twoHilbert. ================= being a highly skeptical sober and restrained person, i naturally doubt that such an abstract gizmo could be useful and have steadfastly resisted up til now understanding anything about them. But ORDINARY Hilberts have proven very useful----I picture an ordinary Hilbertspace as a handy device like a PalmPilot, or a ProgrammableRemote, or a PersonalDigitalAssistant that one constantly uses to keep track of the information one has about something. you can punch in stuff, or you can just let it run on autopilot (it has a clock in it and can unitarily evolve stuff when you arent watching). So it is one of these things about the size of a candybar which they make in Malaysia, with buttons, and a nice liquidcrystal display. That is what a Hilbert is. And these things are everywhere and indispensible. now a guy drives into town with a truckload of new gadgets called TWOHILBERTS. Are you going to buy one? Your neighbors ALREADY HAVE. Hard to resist, right? Mozart C-minor is playing in the kitchen. Almost time for lunch.