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So, for some examples, closure operators for $\mathbf{2}$, presumably. Anything interesting for metric spaces?
Added the example of closure operator, and linked to enriched adjunction.
Is there a standard reference for the enriched version of things like Kleisli, Eilenberg-Moore categories, monads as lax 2-functors etc.?
Well, from one perspective there’s not much to say, since $V Cat$ is a 2-category and all of that is 2-categorical abstract nonsense. But I think there’s some particular discussion of enriched monads in Dubuc’s book Kan Extensions in Enriched Category Theory.
Incidentally, the references previously provided in the entry seem to have little relation to the current content of the entry. Or else it’s hard to spot.
I have added pointer to:
where the explicit definition of enriched monads in Kleisli form is Def. 5.6.
(It’s not a priori clear that their definition – which is the obvious one – is equivalent to the one in the entry: The entry requires the structural equations to be satisfied on global elements only. Is this intentional?)
[edit: Hm, since the equations appear in revision 3 without any typing, they are probably meant to apply to generalized elements. I’ll edit accordingly, but need to interrupt for the moment…]
expanded on the statement of the relation between enriched and strong monads and added a reference (here)
Since this statement is being alluded to in various other entries, I am thinking of making this an !include
section
relation between strong and enriched functors and monads – section
Not now, but maybe tomorrow…
added pointer to
which has a detailed account of the relation enriched/strong/monoidal monads.
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