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    • CommentRowNumber1.
    • CommentAuthorJohn Baez
    • CommentTimeJul 24th 2013
    • (edited Jul 24th 2013)

    I fixed a trivial typo in adjoint functor theorem but left wondering about this:

    … the limit

    Lc:=lim cRdd L c := \lim_{c\to R d} d

    over the comma category c/Rc/R (whose objects are pairs (d,f:cRd)(d,f:c\to R d) and whose morphisms are arrows ddd\to d' in DD making the obvious triangle commute in CC) of the projection functor

    Lc=lim (c/RD). L c = \lim_{\leftarrow} (c/R \to D ) \,.

    I don’t really understand this (and while I could figure it out, it’s probably not good to make readers do so). At first it sounds like someone is saying “the limit LcL c over the comma category of the projection functor LcL c”, which would be circular. But it must be that both formulas are intended as synonymous definitions of LcL c. At that point one is left wondering why one has a backwards arrow under it and the other does not. I guess old-fashioned people prefer writing limits with backwards arrows under them, so someone is trying to cater to all tastes? I think it’s better in this website to use limlim and colimcolim for limit and colimit.

    I could probably guess how to fix this, but I won’t since I might screw something up.

    • CommentRowNumber2.
    • CommentAuthorTodd_Trimble
    • CommentTimeJul 24th 2013

    I went ahead and made some changes per your comment. See if that looks better. (I think I’d try a different explanation if I were writing this – or writing this today in case I was the one who wrote that then! – but never mind.)

    • CommentRowNumber3.
    • CommentAuthorJohn Baez
    • CommentTimeAug 2nd 2013


    • CommentRowNumber4.
    • CommentAuthorTodd_Trimble
    • CommentTimeApr 3rd 2018
    • (edited Apr 3rd 2018)

    Clarified some language in the statements that characterize adjoints between locally presentable categories, in response to a comment made by user Hurkyl in another thread (here).

    diff, v46, current

    • CommentRowNumber5.
    • CommentAuthorTodd_Trimble
    • CommentTimeApr 3rd 2018

    Clarified the language in another relevant spot (where a counterexample was given).

    diff, v46, current

    • CommentRowNumber6.
    • CommentAuthorMike Shulman
    • CommentTimeJun 27th 2019

    Change notation in the statement of the theorem to match its proof (the functor is R:CDR:C\to D instead of G:DCG:D\to C).

    diff, v48, current

    • CommentRowNumber7.
    • CommentAuthorDmitri Pavlov
    • CommentTimeNov 16th 2019

    Added an adjoint functor theorem for cocomplete categories.

    diff, v49, current

  1. Explained the non-standard notation for the limit.

    Bartosz Milewski

    diff, v50, current

  2. Further explanation of syntax

    diff, v51, current

  3. Changed notation for presheaves

    Bartosz Milewski

    diff, v53, current

    • CommentRowNumber11.
    • CommentAuthorUrs
    • CommentTimeJun 26th 2020

    [ since Bartosz emailed me about this: ]

    The above edits concern the section Examples – In presheaf categories.

    Bartosz wanted to make notationally explicit the Yoneda embedding in the various formulas shown there. I have now touched the section myself, added the remaining instances of the Yoneda embedding; and also made some further cosmetic changes to the typesetting, such as height-aligned parenthesis etc.

  4. When reading the presheaf example, I was curious if one could make an argument that representables form a solution set, and this justifies the restriction of the coend to representables only.
    • CommentRowNumber13.
    • CommentAuthorThomas Holder
    • CommentTimeJul 1st 2020
    • (edited Jul 1st 2020)

    Added a reference to

    • Duško Pavlović, On completeness and cocompleteness in and around small categories , APAL 74 (1995) pp.121-152.

    diff, v57, current

    • CommentRowNumber14.
    • CommentAuthorTodd_Trimble
    • CommentTimeMar 13th 2021

    Strengthened the first of the two statements for adjoint functors in the locally presentable case.

    diff, v60, current

    • CommentRowNumber15.
    • CommentAuthorvarkor
    • CommentTimeApr 4th 2023

    Mention a generalisation of the AFT.

    diff, v64, current

    • CommentRowNumber16.
    • CommentAuthorvarkor
    • CommentTimeApr 20th 2023

    Add early reference for enriched adjoint functor theorem.

    diff, v66, current

    • CommentRowNumber17.
    • CommentAuthorUrs
    • CommentTimeJun 1st 2023

    none of the occurrences of “continuous functor” or “cocontinuous functor” here were hyperlinked. have changed that

    diff, v71, current

    • CommentRowNumber18.
    • CommentAuthorvarkor
    • CommentTimeOct 31st 2023

    Added a reference to Porst’s recent survey paper.

    diff, v74, current

    • CommentRowNumber19.
    • CommentAuthorUrs
    • CommentTimeOct 31st 2023

    have copied the reference also to Hans Porst

    diff, v75, current

    • CommentRowNumber20.
    • CommentAuthorDmitri Pavlov
    • CommentTimeDec 2nd 2023


    A stronger version for finitary functors between locally finitely presentable categories whose domain is ranked, requiring only the preservation of countable limits for the existence of a left adjoint, is discussed in

    diff, v78, current