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nLab: Clarification in adjoint functor theorem between locally presentable categories

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  1.  
    • CommentRowNumber1.
    • CommentAuthorHurkyl
    • CommentTimeApr 3rd 2018
    In the adjoint functor theorem page, Theorem 2.2 uses language that very strongly suggests the conclusions are meant to be "if and only if" statements, but could also be read as emphatic language for a mere "if" statement.

    In particular, if F is a functor between locally presentable categories and F has a left adjoint, it's unclear if the nLab page is asserting that F is accessible.

    While the following text does say that the condition of being accessible is "necessary", the example given at the top of section 3 strongly suggests that this really is sloppy language -- that the "necessity" is shown merely by giving a counterexample to theorem 2.2 if you remove the hypothesis that F is accessible.

    This section could use some rewording to make it more clear what the correct statement is.
    • CommentRowNumber2.
    • CommentAuthorTodd_Trimble
    • CommentTimeApr 3rd 2018

    The relevant statement in Adamek and Rosicky is: a functor G:C→DG: C \to D between locally presentable categories is a right adjoint if and only if it preserves limits and is accessible (meaning it preserves λ\lambda-directed colimits for some regular cardinal λ\lambda.

    I can make an adjustment to remove doubts.

    • CommentRowNumber3.
    • CommentAuthorTodd_Trimble
    • CommentTimeApr 3rd 2018

    Okay, I removed the ambiguity in language, to make clear we were indeed talking about logical necessity. Please have a look. (And thanks for pointing out how it read.)

    • CommentRowNumber4.
    • CommentAuthorHurkyl
    • CommentTimeApr 3rd 2018
    It's crystal clear now. Thank you!
    • CommentRowNumber5.
    • CommentAuthorUrs
    • CommentTimeApr 3rd 2018

    I usually write “precisely if” instead of “if and only if”, as it seems to be more decent prose to me. I wasn’t aware that it comes across as ambiguous.

    • CommentRowNumber6.
    • CommentAuthorTodd_Trimble
    • CommentTimeApr 3rd 2018

    Normally I don’t think it would be taken as ambiguous. But questions were raised about it after Hurkyl was trying to put it all together with other stuff on that page. So I was trying to remove every last vestige of doubt by using language whose meaning cannot be misread.

    • CommentRowNumber7.
    • CommentAuthorMike Shulman
    • CommentTimeApr 3rd 2018

    I would probably consider “if and only if” to be more precise than “precisely if”, and it doesn’t seem at all indecent to me. (-:

1 to 7 of 7

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