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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

    Thanks.

    • CommentRowNumber4.
    • CommentAuthorTodd_Trimble
    • CommentTimeApr 2nd 2018
    • (edited Apr 2nd 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 2nd 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

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