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nLab > Latest Changes: left adjoint

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  1.  
    • CommentRowNumber1.
    • CommentAuthorAdam Chalcraft
    • CommentTimeFeb 27th 2021
    (This is my first foray into nLab, so sorry if I'm making elementary errors.)
    In the definition of left adjoint of a functor U:C→ D, the claim is that it's a functor F:D → C s.t. ∃ natural transformations
    ι:id_C → F;U
    ϵ:U;F → id_D
    But F;U is a morphism in D and U;F is a morphism in C.
    Is something wrong here, have I misunderstood the notation F;U, is there a more general version of a natural transformation being used here, or what?
    Thank you.
    • CommentRowNumber2.
    • CommentAuthorHurkyl
    • CommentTimeFeb 27th 2021
    • (edited Feb 27th 2021)

    Some of the 2-category theory articles at nlab, I think, use the semicolon for horizontal composition: i.e. the functor hom(Y,Z)×hom(X,Y)→hom(X,Z)\hom(Y,Z) \times \hom(X,Y) \to \hom(X,Z).

    So, F;UF;U just means the ordinary composition of the two functors.

    Except… now that I’m looking at the article it looks like it’s the reversed convention? I.e. hom(X,Y)×hom(Y,Z)→hom(X,Z)\hom(X,Y) \times \hom(Y,Z) \to \hom(X,Z), so F;UF;U means the composite UFUF?

  3. Gave the page more structure, corrected a few typos and notational confusions, and generally tried to improve it.

    diff, v17, current

  4. Also deleted a vague sentence in the introduction which struck me as more likely to be confusing/misleading than helpful.

    diff, v17, current

    • CommentRowNumber5.
    • CommentAuthorRichard Williamson
    • CommentTimeFeb 27th 2021
    • (edited Feb 27th 2021)

    Hi Adam, thanks very much for raising this! The page was rather a mess before, with all sorts of notational confusion and typos; I have now, as described in #3 and #4, edited the page. Hopefully things are now clear, but if not just let us know (and feel free to edit this or any other page if you see something you wish to correct or add!).

  6. added text from HoTT book

    Anonymous

    diff, v20, current

    • CommentRowNumber7.
    • CommentAuthorUrs
    • CommentTimeJun 7th 2022

    Please, what you just added is the standard classical definition. This has nothing to do with HoTT. The edit should be reverted.

    • CommentRowNumber8.
    • CommentAuthorUrs
    • CommentTimeJun 8th 2022
    • (edited Jun 8th 2022)

    Hi Ali,

    I was just reverting your previous edit, but now I see that there is a second part that I had missed on first glance. The second part is contentful, so I deleted only the redundant first part.

    Since you have now signed a query box with your name, please also sign your edits with your name – or make up an individual pseudonym that you stick to. There are too many users currently all signing with the same default pseudonym. It should be in your own interest that you don’t get confused with the others.

    But please note that you are still producing broken typesetting, now of the form

    Lemma 3.1. Lemma 9.3.2

    If you really can’t think of a better name of your lemma than the number it has in some book, then at least put some parenthesis so that it does not collide with the new number it gets assigned here. I have shown you last time how it works, but as a reminder, here is one option:

    \begin{lemma}\label{Lemma932}
**(Lemma 9.3.2 in the HoTT book)**
\linebreak
...
\end{lemma}

    But better would be not to conflate the name of a lemma with the reference for its proof and instead go for something like this:

    \begin{lemma}\label{LeftAdjointnessIsMereProposition}
**(being left adjoint is a mere proposition)**
\linebreak
...
\end{lemma}

This is Lemma 9.3.2 in the HoTT book.

    Okay?

    diff, v21, current

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