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Isn’t what is described here usually called a “monad morphism”? I’m familiar with the term “monad transformation” being used for a 2-cell in the 2-category of monads, rather than a 1-cell (e.g. this is the usage in The formal theory of monads II).
Thanks. True. Okay, I’ll edit the entry.
(But I find it’s a shame… :-)
Incidentally, the best reason to say “transformation” for the 1-morphisms of monads (besides them having underlying natural transformations) is that the very word “monad” refers to their incarnation as lax 2-functors $\ast \to Cat$ and that the default name for morphisms of such beasts, of course, lax natural transformations. Hence default verbiage would suggest to speak of the 2-category of monads, transformations and modifications.
Besides mentioning this now in the entry, I have added pointer to
(here and at monad)
and to:
I think one could also say that the word “monad” refers to their incarnation as monoids in categories of endofunctors (e.g. as in the quote in the article monad: “It suggested “monoid” of course and it is a monoid in a functor category.”), and the default name for morphisms of such is monoid homomorphisms or simply monoid morphisms.
It’s tautological that they are morphisms, the question is what particular term they want to carry:
We want to non-tautologically say: “The morphisms in the category of monads are the xyz”.
$\,$
Incidentally, for what it’s worth, I am doubtful of the relevance of that quote, given that it shows no sign of interacting with Bénabou’s text or thinking.
Bénabou’s Def. 5.4.1 is quite unmistakably introducing the term monad for lax functors out of the unit $1$ – which even in modern Greek is still called: monad.
(The one time I saw Bénabou in person at Topos à l’IHES I was surprised – despite previous hearsay – at just how grumpy he really was towards the community. Meanwhile I am getting a sense of how that came about. Despite the general praise, there is an immense reluctancy in the community to engage with, let alone reference, his contributions: I have now about half a dozen authors listed at monad transformation who all introduce this concept, all well after Bénabou 1967, and none of them even hints that the definition that they are presenting is already implicit in Bénabou’s definition of monads literally as monads(units) $1 \to \mathcal{K}$.)
It’s tautological that they are morphisms, the question is what particular term they want to carry:
I don’t think it’s tautological. It’s an assertion that there is a canonical choice of morphism between monoids. Prefixing “homo-” isn’t any more enlightening. It’s the same convention as when most authors write $Mon$ for the category of monoids and monoid homomorphisms, even though the choice of morphism is not explicit in the name.
Incidentally, for what it’s worth, I am doubtful of the relevance of that quote, given that it shows no sign of interacting with Bénabou’s text or thinking.
In the footnote on page 40 of Introduction to Bicategories, Bénabou writes:
Our choice of “monad” comes from this example and the definition (5.5).
where “this example” refers to the characterisation of monoids in monoidal categories as monads in one-object bicategories. (5.5) is where Bénabou introduces “polyads”, and so the footnote suggests that “monad” is chosen both because of its similarity with the word “monoid”, and also by following the “polyad” naming convention (though this latter choice of terminology does not appear to be explained).
Despite the general praise, there is an immense reluctancy in the community to engage with, let alone reference, his contributions
Yes, I have the same impression. It is a great pity.
Thanks for pointing out that footnote, I had missed that. Will make a corresponding edit at “monad”…
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