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    • CommentRowNumber1.
    • CommentAuthorFinnLawler
    • CommentTimeMay 18th 2010

    I’ve created mate, but I can’t get my nice fancy diagrams to display – I just get the source. Help! What have I done wrong?

    • CommentRowNumber2.
    • CommentAuthorUrs
    • CommentTimeMay 18th 2010

    I have fixed it.

    I think the following things you had caused problems (though there is a chance that not all these did, and that I changed some superfluously)

    • I don’t think arrays inside other arrays are supported, that was a problem I think,

    • some of your \underset{–}{–} commands were lacking their second argument. I think that causes the parser to choke.

    Then the following at least sometimes cause an error:

    • sometimes (but not always) a diagram doesn’t display if it doesn’t have a blank line before and after its double dollars

    • similarly blank lines inside the diagram might not work

    Here is a general hint for diagrams: if you enclose labels on arrows inside \mathrlap{…} or \mathllap{..} they don’t off-set the position of the arrow to which they are attached, but instead stick out to the left or the right, as desired.

    • CommentRowNumber3.
    • CommentAuthorFinnLawler
    • CommentTimeMay 18th 2010

    Great, thanks! I actually didn’t have nested arrays in the first version, but the diagrams still wouldn’t display, probably because of the \underset typo (I cut and pasted the first diagram to make the second, which ended up making more work instead of less. D’oh!).

    • CommentRowNumber4.
    • CommentAuthorzskoda
    • CommentTimeMay 18th 2010

    I added Leinster’s book, section 6.1 as a reference for mates and lax maps of monads.

    • CommentRowNumber5.
    • CommentAuthorMike Shulman
    • CommentTimeMay 18th 2010

    Awesome, we’ve needed this for a while. I added a redirect for “mates” (hint, hint).

    • CommentRowNumber6.
    • CommentAuthorFinnLawler
    • CommentTimeJun 8th 2010

    Added a section to mate on the naturality of of the correspondence, using the double category of adjoints.

    • CommentRowNumber7.
    • CommentAuthorzskoda
    • CommentTimeJun 8th 2010

    There is this business about the convention what is a (lax or colax) morphism of monads. Steet has opposite direction of the arrow to the one coming from generalizing the strict morphisms of monads in a fixed category, as monoids, what is unpleasant having in mind that special case. I am now writing a paper in which i touch this at one place and I am a bit undecided, with inclinaton not to follow Street here.

    • CommentRowNumber8.
    • CommentAuthorMike Shulman
    • CommentTimeJun 8th 2010

    I feel strongly that Street’s convention is the correct one, because it is a special case of the general notion of lax and colax morphisms of algebras for 2-monads. Specifically, there is a 2-monad on Cat whose algebras are “categories equipped with a monad” and for which the lax resp. oplax morphisms are the lax resp. oplax monad functors (or “monad functors” and “monad opfunctors”) with Street’s convention. I think it just creates confusion when people start using “lax” for things that are properly called “oplax;” why not just call them oplax? It’s just two more letters and one more syllable.

    A natural transformation STS\to T between two monads on a fixed category which is a morphism of monoids in the endofunctor category can be considered as equipping the identity functor either with the structure of a lax monad functor from TT to SS, or a colax monad functor from SS to TT. If it bothers you that the lax structure goes in the other direction, why not just think instead about the colax structure going in the same direction? (In fact, these two structures make the identity adjunction IdIdId \dashv Id into a “doctrinal” or “colax/lax” adjunction, which would also get extra confusing to say if people started switching the meanings of lax and oplax.)

    • CommentRowNumber9.
    • CommentAuthorzskoda
    • CommentTimeJun 8th 2010

    Thank you for your kind discussion. I am not sure if I can easily get used to such thinking (I arrived at definitions myself through steps of generalizations as they were needed in my applied research, so it is difficult to straighten in a different way, but maybe I get gradually persuaded, your arguments are sensitive enough).

    • CommentRowNumber10.
    • CommentAuthorDavid_Corfield
    • CommentTimeJun 10th 2010

    Any more good examples of mates?

    • CommentRowNumber11.
    • CommentAuthorFinnLawler
    • CommentTimeJun 10th 2010
    • (edited Jun 10th 2010)

    As Mike intimated, they come into doctrinal adjunction (not sure if that page exists yet (edit: it doesn’t – another mini-project for me!)). If KK is a 2-category and TT a 2-monad on it, then you can ask whether adjoint morphisms fuf \dashv u in KK actually live in TT-Alg. Kelly’s result is that uu is a lax TT-morphism iff ff is a colax one, and then the 2-cells making them so are mates under TfTuT f \dashv T u and fuf \dashv u. Also, the entire adjunction lives in TT-Alg iff the mate of uu’s 2-cell is invertible, and then ff is always a pseudo TT-morphism.

    I think this was first observed and is (probably) best known when TT-Alg = monoidal categories.

    • CommentRowNumber12.
    • CommentAuthorFinnLawler
    • CommentTimeJun 11th 2010

    A couple more properties at mate from Kelly–Street.

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