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Atrium > Mathematics, Physics & Philosophy: Two questions on the category QChan

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    • CommentRowNumber1.
    • CommentAuthorIan_Durham
    • CommentTimeMar 1st 2010
    • (edited Mar 1st 2010)
    I'm still pretty new to nLab and I want to make sure I get the right etiquette, so if this type of question isn't appropriate for n-Forum, please let me know.

    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!
    • CommentRowNumber2.
    • CommentAuthorDavidRoberts
    • CommentTimeMar 1st 2010
    I think the general idea is that one asks a question at the appropriate point on the page (in a query box) and then puts a comment here that you asked the question, with a link.

    findimVect is not a small category, but is equivalent to one. This may help for starters.
    • CommentRowNumber3.
    • CommentAuthorIan_Durham
    • CommentTimeMar 1st 2010
    OK, thanks. I did put a query on the page about my first question. I'll put the second question in there and folks can go there to answer it. Thanks for the help!
    • CommentRowNumber4.
    • CommentAuthorDavidRoberts
    • CommentTimeMar 1st 2010
    • (edited Mar 1st 2010)

    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 QChan^\mathbf{2} and consider subcategories of this defined using subcategories of QChan. Am I near the mark?

    • CommentRowNumber5.
    • CommentAuthorUrs
    • CommentTimeMar 1st 2010

    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.

    • CommentRowNumber6.
    • CommentAuthorUrs
    • CommentTimeMar 1st 2010
    • (edited Mar 1st 2010)

    I should maybe clarify: the way I defined QChan 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 QChan 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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