Jason (jcreed) wrote,
Jason
jcreed

Aha! The notion of relative monads in "Monads Need Not Be Endofunctors" (which, by the way, is a fantastic title: provocative without being goofy) generalize Hughes's notion of arrows without making me memorize a huge bunch of fiddly unmotivated axioms. (Don't even get me started on Freyd categories...) Instead, arrows are relative monads on the yoneda embedding! This only requires me to remember a (slight variant on) a small bunch of fiddly axioms that I already memorized, namely the monad axioms.

The proof that it is a great title: I only heard about this by skimming a tiny workshop announcement I got in my email this morning and seeing the title, and figured I just had to read it.

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The thing that surprises me just a little is the blatant asymmetry of definition 2: a relative adjunction between the functor J : JC and the category D is given by two functors L : JD and R : DC such that there is a natural isomorphism C(JX, RY) = D(LX, Y).

What happens if we instead have a "doubly-relative" adjunction that has
J : JC
L : JD
R : KC
M : KD
such that there is a natural isomorphism
C(JX, RY) = D(LX, MY)
?
Surely we ought to get some relative comonads out of this, too, at the very least.

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Oh, oops, Arrows are actually strong relative monads on the yoneda embedding. That explains all the fiddly axioms about products and projections and stuff I guess!
Tags: monads, papers, relatives
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