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New entry adjoint monad with redirect adjoint comonad. Last night I had to struggle with proving the lemma at the bottom so I claimed the credit in brackets.
Added a query box in adjoint monad stating that the the construction in item 2 is what Mac Lane calls conjugate natural transformations (it’s in a query box because I’m not entirely sure if this is useful information for this entry - feel free to delete this).
Also, I was a little confused by “there is a bijection of sets between natural transformations $\phi$ and natural transformations $\psi$ … such that …” (in item 2), because the condition involves both $\phi$ and $\psi$. Wouldn’t it be better to say somethig like “given $\phi$ there is a unique $\psi$ such that…” and vice versa?
Eilenberg and Moore write symbol $\dashv$ “adjoint”, i.e. $\phi\dashv \psi$: this conjugation, or duality depends on two adjunctions to start with. Thanks for confirming other terminology.
In my opinion both formulations are equivalent. One can start with any $\phi$ and find $\psi$ and other way around. Now what is such refering to. If it is refering to $\psi$ then it is precisely what you said. If it is misunderstood that such refers to the bijection, does not matter, as all $\phi$-s and all $\psi$-s are involved and the shape of bijection is determined by the condition. We can make different formulation, though.
OK, I incorporated your query into text as
Eilenberg and Moore would write $\phi\dashv\psi$ and talk about “adjointness for morphisms” (of functors), which is of course relative to the given adjunctions among functors. MacLane calls the correspondence conjugation (Categories for Working Mathematician, 99-102).
Thanks!
By the way, to make the various square diagrams display with correct alignement, one can use the
\mathllap
and
\mathrlap
command to make arow labels stick out to the left or the right of the arrow alignement, respectively. For instance
$$
\array{
A (S,-) &\to& B(-,T)
\\
{}^{\mathllap{A(\phi,-)}}\downarrow &&\downarrow {}^{\mathrlap{B(-,\psi)}}
\\
A(S',-)&\to & B(-,T')
}
$$
gives
$\array{ A (S,-) &\to& B(-,T) \\ {}^{\mathllap{A(\phi,-)}}\downarrow &&\downarrow {}^{\mathrlap{B(-,\psi)}} \\ A(S',-)&\to & B(-,T') }$Yes, great, you told me once, but I could not recall the command and had no time to search through old images.
As I said at the other thread, I think this really is exactly a special case of what the Australians call mates; why do you say it isn’t? And isn’t the lemma at the end just a special case of the fact that mates are natural/functorial? There’s nothing special about the fact that the three adjunctions involved are all composites of the same given adjunction.
the category End(A) of endofunctors… is monoidal with respect to the composition… Hence we can consider operads and algebras over operads
I don’t know how to define an operad in a monoidal category that isn’t at least braided.
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