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Okay. I’ve tried to insert the content, now it says “Edit blocked by spam detector”. I hope I’m not that bad at category theory… :p
They worked. Thanks Richard!
Absolutely. But it might go better in a section on “Properties” or something, than in the “Definition” section.
(BTW, it’s helpful to say in the changes box what you did, rather than making the forum reader click on the diff to figure out what “this link” refers to.)
Just a question on the literature:
On slide 4 in his Lecture 2, Uustalu states a way of speaking about algebras over monads that meshes with the Kleisli-style of presenting (strong) monads in FP, and refers to it as being “Kleisli triple algebras” or “Mendler-style algebras”.
What would be a real reference for this “Mendler-style”-presentation of algebras-over-monads-in-FP, or anything along these lines?
What would be a real reference for this “Mendler-style”-presentation of algebras-over-monads-in-FP, or anything along these lines?
There are two references that appeared in 2010: algebras for a relative monad (taking J = 1) in Monads need not be endofunctors, and algebras for extension systems in Monads as Extension Systems - No Iteration is Necessary (though the journal/arXiv version of the former paper is from 2015). An earlier reference is the algebras for a device in Walters’s An alternative approach to universal algebra, though it takes some decoding to see that this is really the same.
Just a playful thought on terminology:
I notice that the term “algebra over a monad” has some justification in classical universal algebra, but in many or most other situations, the term “module for a monad” seems more appropriate.
If, however, ones says “module over a monad”, then it stands to reason that one might just as well refer to the monad as the monoid that it is and stick with the standard terminology of internal algebra.
Conversely, if one does insist on “monad” (as we are bound to do) then it seems natural to make analogous suffix adjustments systematically to all related terms!
Following this logic one will easily be led to say “modale” for “module over a monad”.
But by lucky and fun conincidence, what starts out like a pun happens to make perfect sense with respect to thinking of monads as modalities: Their algebras are then the modal objects.
So where a monoid has modules, we should say that a monad has modales :-)
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