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I suppose we ought also to have a page on the double category of adjunctions that figures in the mate correspondence.
I see at mate it speaks about the double category, Adj(K). But this is the notation I just used for the 2-category of adjunctions in K.
Yes, well, in a latex paper I would use two different fonts for the “Adj”.
Re #5, can’t we have two fonts here? Which should they be?
I would probably write 𝒜dj(K) for the 2-category and 𝔸dj(K) for the double category. That doesn’t work in a page title, though.
[I wish I could have deleted this comment.]
The inclusion of Mnd, the free monad, in Adj induces a 2-functor from the 2-category of adjunctions in K to the 2-category of monads in K.
This doesn’t make sense to me. Adj(K) is not the functor 2-category [Adj,K] – as it says earlier on the page, the objects of Adj(K) are the objects of K while its morphisms are the functors Adj→K – so I don’t see any “precomposition” functor going on.
Have I garbled the end of The free adjunction?
Yes, their “2-category of adjunctions” is by definition [Adj,K], not the 2-category we’re calling Adj(K) on this page. Perhaps this page should mention both, since this confusion seems likely to be common.
The inclusion of Mnd, the free monad, in Adj induces a 2-functor from the 2-category of adjunctions in K to the 2-category of monads in K.
Presumably this “free monad” should be referring to a page better termed the “walking monad” to chime with Adj 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?
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).
Yes. A yet better term might be “archetype”. “Mnd is the archetypical monad.”
Another thought re #17: Still better than “archetypical” may be “quintessential”.
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.
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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