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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?
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.
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!).
I added Leinster’s book, section 6.1 as a reference for mates and lax maps of monads.
Awesome, we’ve needed this for a while. I added a redirect for “mates” (hint, hint).
Added a section to mate on the naturality of of the correspondence, using the double category of adjoints.
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.
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 $S\to T$ 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 $T$ to $S$, or a colax monad functor from $S$ to $T$. 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 $Id \dashv Id$ 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.)
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).
Any more good examples of mates?
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 $K$ is a 2-category and $T$ a 2-monad on it, then you can ask whether adjoint morphisms $f \dashv u$ in $K$ actually live in $T$-Alg. Kelly’s result is that $u$ is a lax $T$-morphism iff $f$ is a colax one, and then the 2-cells making them so are mates under $T f \dashv T u$ and $f \dashv u$. Also, the entire adjunction lives in $T$-Alg iff the mate of $u$’s 2-cell is invertible, and then $f$ is always a pseudo $T$-morphism.
I think this was first observed and is (probably) best known when $T$-Alg = monoidal categories.
A couple more properties at mate from Kelly–Street.
An obvious question: mates are preserved by 2-functors. But what about pseudofunctors? Has this appeared in print before?
I’m thinking in the generality of Johnson and Yau’s book 2-Dimensional Categories, where they define mates in bicategories, and are careful about showing pseudofunctors send adjunctions to adjunctions.
added pointer to:
added pointer to
highlighting that the notion of mates is already there (though not under this name)
In the first example (here), where there was vague reference to the “usual notion” of isomorphism of adjoints, I have added explicit mentioning of conjugate transformation of adjoints.
in the main definition (here) I have added more informative referencing
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