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    • CommentRowNumber1.
    • CommentAuthorUrs
    • CommentTimeMar 16th 2012
    • (edited Mar 16th 2012)

    at 2-adjunction I would like to list a bunch of 2-category theoretic analogs of standard facts about ordinary adjunctions. Such as: a right adjoint is a full and faithful 2-functor precisely if the counit of the 2-adjunction is an equivalence, etc.

    But I haven’t really thought deeply about 2-adjunctions myself yet. Is there some reference where we could take such a list of properties from?

    • CommentRowNumber2.
    • CommentAuthorDavid_Corfield
    • CommentTimeMar 16th 2012

    Is there a nice pictorial representation of the equations for 2-adjunctions, just as in the case of ordinary adjunctions (week 174)?

    • CommentRowNumber3.
    • CommentAuthorTodd_Trimble
    • CommentTimeMar 16th 2012

    David, the triangulators satisfy a swallowtail coherence condition, for which there’s a nice picture…

    • CommentRowNumber4.
    • CommentAuthorzskoda
    • CommentTimeMar 16th 2012
    • (edited Mar 16th 2012)

    1: Some of these “obvious” categorifications are not worked out, many are known. For example there are few papers of Marmolejo and of Vitale which work out quite much including Beck-type monadicity/descent theorems. Some relevant stuff should also be in Pronk’s papers on bicategorical localization and some in older papers of Austrakian school, e.g. Kelly.

    • CommentRowNumber5.
    • CommentAuthorDavid_Corfield
    • CommentTimeMar 16th 2012

    That reminds me of the beginning of a long conversation at the Cafe, which ended up looking at generalizing the tangle hypothesis.

    I remember another interesting conversation we all had once about how from thinking about one dimensional things in the plane, one could be led to important category theoretic structures, and how this might happen with surfaces. Ah, here it is.