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started co-Kleisli category with a minimum of content. Even though its formally dual to Kleisli category, of course, it may be worthwhile to have a separate entry.
It might also be known as the Kleisli category for a comonad (I should check e.g. Lambek and Phil Scott’s book). Some time back I wrote up an application of this concept to functional completeness.
Blute, Cockett and Seely say coKleisli in Differential categories.
I have added more redirects.
I’m pretty sure I’ve seen “Kleisli category for a comonad”.
Google score 4 versus 1020 in favour of coKleisli category.
I do think “coKleisli” is better, even though the other has been used; e.g. we say “category of algebras for a monad” and “category of coalgebras for a comonad” for the dual concepts.
In principle, “opKleisli” might be even better, since both Kleisli and coKleisli are colimit constructions in $Cat$; but probably that isn’t going anywhere. (-:
I don’t think “coKleisli” or “co-Kleisli” is bad, even though I don’t think we say “co-Eilenberg-Moore” for the category of coalgebras (i.e., I think I’ve heard “Eilenberg-Moore category for a comonad”). Anyway, hopefully we have all the redirects we need.
an elementary question on terminology, that will just demonstrate my ignorance:
for a left exact comonad $J$, then every Kleisli morphism from $E$ to $F$ canonically induces a function from the global points of $E$ to those of $F$. What is the name of this construction?
For definitenes, I mean the following map: Since $J$ preserves the terminal object, then the image under $J$ of a global point $\ast \to E$ is a global point $\ast \to J(E)$. Postcomposing this with the given Kleisli morphism $J(E)\to F$ gives a global point of $F$.
I don’t recall hearing of a name for that.
Okay, thanks.
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