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Atrium > Mathematics, Physics & Philosophy: created limit

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    • CommentRowNumber1.
    • CommentAuthorUrs
    • CommentTimeAug 22nd 2017
    • (edited Aug 22nd 2017)

    I am on vacation, with no real time, but I just received the following message by email, which I am quickly forwarding hereby.

    Somebody writes:

    The definition of “functor creates limits” in nCatLab https://ncatlab.org/nlab/show/created+limit

    seems to differ from the one used in Awodey’s Category Theory and from the one used in MacLane’s Categories for the Working Mathematician.

    The definition in nCatLab seems to imply that the creating functor is thus also preserving limits. I think that it is misleading, because such implication does not stem from definitions given by Awodey or MacLane.

    But this page by Kissinger, which is extremely useful besides that, seems to follow nCatLab’s definition : http://permalink.gmane.org/gmane.science.mathematics.categories/6644

    Thus I am quite confused.

    Todd Trimble seems to be involved in the nCatLab definition too, so later I may also post a question on MathOverflow so that he could give his view.

    • CommentRowNumber2.
    • CommentAuthorTodd_Trimble
    • CommentTimeAug 22nd 2017
    • (edited Aug 22nd 2017)

    I am surprised to see my name invoked, since I seem not to have been an author listed in the revision history. I don’t think I myself use the term to include limit preservation, but then again I tend to avoid the term and use the words “reflected” and “preserved” instead. “Created” to me refers to the Mac Lane definition. Maybe a semantic shift had taken place that I was unaware of.

    • CommentRowNumber3.
    • CommentAuthorsamo
    • CommentTimeAug 22nd 2017
    Hi Todd, I am the author of the original email. I am sorry to have mixed you with this ^^; Actually I wrote this line while looking at the history of "lifted limit"...
    Samuel Amo
    • CommentRowNumber4.
    • CommentAuthorsamo
    • CommentTimeAug 22nd 2017
    And thank you Todd to have given your view on the case! :)
    People actually involved in the history of "created limit" were, besides Urs, Mike Shulman and Sridhar Ramesh.
    • CommentRowNumber5.
    • CommentAuthorMike Shulman
    • CommentTimeAug 23rd 2017

    I believe Mac Lane’s definition does imply that the limits are preserved, at least as long as the limit exists in the base category, since then there is some limiting cone that lifts it and hence is preserved, and any other limiting cone upstairs is isomorphic to that one. (Mac Lane’s definition also differs from the nLab one in violating the principle of equivalence.) I suppose one might quibble about whether we should say that a functor “creates” a limit that exists upstairs but not downstairs; Mac Lane says yes, we say no. But in practice one only uses the notion of creation of limits for limits that exist downstairs.

    By the way, last year I said that I don’t think Kissinger’s definition is correct, but didn’t attract any replies.

    • CommentRowNumber6.
    • CommentAuthorsamo
    • CommentTimeAug 24th 2017
    • (edited Aug 25th 2017)
    Hi Mike, thank you so much for your input!
    Actually my mental image of limiting cone was plainly wrong, and now I see why the limits are preserved with nCatLab's definition but not with MacLane's.
    Maybe the nCatLab page should have a word on the preservation being a consequence of a weaker definition?

    Edited: I had mixed definitions.
    • CommentRowNumber7.
    • CommentAuthorMike Shulman
    • CommentTimeAug 24th 2017

    Yes, that would be good. In the non-strict case I guess the weaker definition would say that for each limiting cone downstairs there exists a limiting cone upstairs whose image downstairs is isomorphic to it.

    Anyone have thoughts about whether we should also modify our definition to align with Mac Lane’s convention about whether nonexisting limits can be created?

    • CommentRowNumber8.
    • CommentAuthorMike Shulman
    • CommentTimeAug 28th 2017

    I added some terminological remarks to hopefully clarify.

    • CommentRowNumber9.
    • CommentAuthorUrs
    • CommentTimeAug 29th 2017

    Thanks, Mike!

    • CommentRowNumber10.
    • CommentAuthorsamo
    • CommentTimeAug 29th 2017
    Ok, thank you very much!
    I'm not an expert, but the added details seem to fill perfectly the gap, in my opinion :)

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