I should maybe clarify: the way I defined over at the entry it is manifestly a *small category* . But of course what I define is equivalent to various other definitions that one would want to write down. Instead of talking about n x n -matrices, we could be tallking about linear endomorphisms of arbitrary n-dimensional complex vector spacess. That would give, as David say, a definition of QChan that is at least *essentially small* .

There are various other generalizations that one can consider. However, I feel the discussion would profit from first focusing attention entirely to the simple setup of as it is currently defined in that entry. Once we all agree -- and Ian agrees -- about what is going on there, I am willing to consider more sophisticated versions and discuss them.

]]>I have replied at the entry

it would seem that the subcategories of QChan that I mentioned are "small."

I may not understand which subcategories are meant, but clearly Qchan itself is small, so *all* its subcategories are necessarily small.

Replied at quantum channel.

Reading both questions together (here and on the lab) I think that your question about commutative squares is asking whether you can take the arrow category and consider subcategories of this defined using subcategories of . Am I near the mark?

]]>findimVect is not a small category, but is equivalent to one. This may help for starters. ]]>

On the page on quantum channels, Urs defined the category QChan. Presumably we can make a subcategory (actually several of varying dimension, I would think) of QChan with the objects of the subcategory being the set of linear operators on the vector space of QChan. In Awodey's book on category theory he notes that a category is small if both the collection of objects on the category as well as the collection of arrows on the category are sets. Given that definition, it would seem that the subcategories of QChan that I mentioned are "small." Does that make sense?

Also, would it make sense to assume that, since the tensor product is a functor, we could make a commutative square using a couple of these subcategories (if that's the right wording) and thus have an arrow category of QChan?

Thank you! ]]>