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    • CommentRowNumber1.
    • CommentAuthorFinnLawler
    • CommentTimeJun 11th 2010

    Created doctrinal adjunction. The page could probably use some examples and/or fleshing out.

    • CommentRowNumber2.
    • CommentAuthorMike Shulman
    • CommentTimeJun 11th 2010
    • (edited Jun 11th 2010)

    Thanks! I added some remarks about the way I prefer to think of doctrinal adjunction in terms of double categories. This motivated me to finally create companion pair and conjunction.

    • CommentRowNumber3.
    • CommentAuthorUrs
    • CommentTimeJun 21st 2017

    Is the statement of prop. 2.2 at doctrinal adjunction as intended:

    For the unit and counit of the adjunction fuf \dashv u to be TT-transformations, and hence for the adjunction to live in TT-Alg lAlg_l, it is necessary and sufficient that f˜\tilde f have an inverse f¯\bar f that makes (f,f¯)(f,\bar f) into a lax TT-morphism, and hence (f,f¯)(f,\bar f) into a strong TT-morphism.


    The last line repeats the symbls (f,f¯)(f,\bar f). If this is what is really meant, it would be more clear to write “into a lax TT-morphism, which is then necessarily a strong TT-morphism”.

    And in the lines before the proposition, symbols “f¯\bar f” refer to 2-morphisms, while here they refer to (inverses of) 1-morphisms. I am not sure if I am parsing this correctly.

    • CommentRowNumber4.
    • CommentAuthorMike Shulman
    • CommentTimeJun 21st 2017

    Actually, it would be enough to stop at “have an inverse.” The fact that this makes (f,f¯)(f,\bar{f}) into a lax and a strong TT-morphism then follows automatically. But f¯\bar{f} is a 2-morphism here too, since it is an inverse of the 2-morphism f˜\tilde{f}.

    • CommentRowNumber5.
    • CommentAuthorUrs
    • CommentTimeJun 21st 2017
    • (edited Jun 21st 2017)

    Thanks, I see now where I made a mistake in parsing.

    Maybe I’ll find some time to re-arrange the notation of the entry a little. On my system at least, it is somewhat hard to decypher, with different kinds of nn-morphisms all in the same font and with the difference between small twiddles and small bars hard to make out.

    Generally, for better reading experience, I found that



     \bar{ }

    and it is good to type


    even for single symbols to be decorated. But in the entry at hand, maybe one should find altogether different decoration, for readability.

    • CommentRowNumber6.
    • CommentAuthorDavid_Corfield
    • CommentTimeMay 4th 2018

    I added a sentence to explain the ’doctrine’ part of ’doctrinal adjunction’.

    diff, v16, current

    • CommentRowNumber7.
    • CommentAuthormattecapu
    • CommentTimeApr 27th 2024

    added dual case

    diff, v17, current