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nLab > Latest Changes: 2-category of adjunctions

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  1.  
    • CommentRowNumber1.
    • CommentAuthorDavid_Corfield
    • CommentTimeJun 16th 2018

    Started this page. No doubt it could be more elegant.

    v1, current

    • CommentRowNumber2.
    • CommentAuthorDavid_Corfield
    • CommentTimeJun 17th 2018

    Added something on the map from adjoints in KK to monafs in KK.

    diff, v2, current

    • CommentRowNumber3.
    • CommentAuthorMike Shulman
    • CommentTimeJun 18th 2018

    I suppose we ought also to have a page on the double category of adjunctions that figures in the mate correspondence.

    • CommentRowNumber4.
    • CommentAuthorDavid_Corfield
    • CommentTimeJun 18th 2018

    I see at mate it speaks about the double category, Adj(K)Adj(K). But this is the notation I just used for the 2-category of adjunctions in KK.

    • CommentRowNumber5.
    • CommentAuthorMike Shulman
    • CommentTimeJun 18th 2018

    Yes, well, in a latex paper I would use two different fonts for the “AdjAdj”.

    • CommentRowNumber6.
    • CommentAuthorDavid_Corfield
    • CommentTimeJun 18th 2018

    Made corrections.

    diff, v3, current

    • CommentRowNumber7.
    • CommentAuthorDavid_Corfield
    • CommentTimeJun 18th 2018

    Re #5, can’t we have two fonts here? Which should they be?

    • CommentRowNumber8.
    • CommentAuthorMike Shulman
    • CommentTimeJun 18th 2018

    I would probably write 𝒜dj(K)\mathcal{A}\mathit{dj}(K) for the 2-category and 𝔸dj(K)\mathbb{A}\mathsf{dj}(K) for the double category. That doesn’t work in a page title, though.

    • CommentRowNumber9.
    • CommentAuthorJohn Baez
    • CommentTimeJun 21st 2018
    • (edited Jun 21st 2018)

    [I wish I could have deleted this comment.]

    • CommentRowNumber10.
    • CommentAuthorMike Shulman
    • CommentTimeSep 27th 2018

    The inclusion of MndMnd, the free monad, in AdjAdj induces a 2-functor from the 2-category of adjunctions in KK to the 2-category of monads in KK.

    This doesn’t make sense to me. Adj(K)Adj(K) is not the functor 2-category [Adj,K][Adj,K] – as it says earlier on the page, the objects of Adj(K)Adj(K) are the objects of KK while its morphisms are the functors Adj→KAdj\to K – so I don’t see any “precomposition” functor going on.

    • CommentRowNumber11.
    • CommentAuthorDavid_Corfield
    • CommentTimeSep 27th 2018

    Have I garbled the end of The free adjunction?

    • CommentRowNumber12.
    • CommentAuthorMike Shulman
    • CommentTimeSep 27th 2018

    Yes, their “2-category of adjunctions” is by definition [Adj,K][Adj,K], not the 2-category we’re calling Adj(K)Adj(K) on this page. Perhaps this page should mention both, since this confusion seems likely to be common.

    • CommentRowNumber13.
    • CommentAuthorMike Shulman
    • CommentTimeSep 27th 2018

    Disambiguate between the two meanings of “2-category of adjunctions”

    diff, v4, current

    • CommentRowNumber14.
    • CommentAuthorvarkor
    • CommentTimeJul 7th 2021

    Add reference to The free adjunction.

    diff, v7, current

  15. The inclusion of MndMnd, the free monad, in AdjAdj induces a 2-functor from the 2-category of adjunctions in KK to the 2-category of monads in KK.

    Presumably this “free monad” should be referring to a page better termed the “walking monad” to chime with AdjAdj the walking adjunction, the topic of these notes.

    Or do people prefer the expression “free-standing monad”, since we have also “free-standing adjunction” at Adj?

    • CommentRowNumber16.
    • CommentAuthorvarkor
    • CommentTime2 days ago

    Personally, I prefer “free-standing monad”, because I feel this is a self-explanatory description, as opposed to “walking monad”, which I feel is completely unclear unless explained (and even then invokes a rather uncommon English construction).

    • CommentRowNumber17.
    • CommentAuthorUrs
    • CommentTime2 days ago

    Yes. A yet better term might be “archetype”. “MndMnd is the archetypical monad.”

  18. So I just called the page Mnd to resemble Adj, and added details as there.

    • CommentRowNumber19.
    • CommentAuthorUrs
    • CommentTime2 days ago

    Another thought re #17: Still better than “archetypical” may be “quintessential”.

  20. Both good suggestions. I just took a brief look to see how broad the issue is. There are 5 pages with ’walking’ in the title, one explaining what the term means in general. Maybe not enough to fix things systematically, although walking isomorphism seems a little embarrassed about it.

    • CommentRowNumber21.
    • CommentAuthorUrs
    • CommentTime2 days ago
    • (edited 2 days ago)

    although walking isomorphism seems a little embarrassed about it.

    Yeah, I had put in the clause “if you insist” (revision 10), since it seems a step in the wrong direction.

    If “quintessential” sounds good to you, I can make the change throughout.

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