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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?
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.
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.
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