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• CommentRowNumber1.
• CommentAuthorUrs
• CommentTimeMay 12th 2015

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.

• CommentRowNumber2.
• CommentAuthorTodd_Trimble
• CommentTimeMay 13th 2015

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.

• CommentRowNumber3.
• CommentAuthorDavid_Corfield
• CommentTimeMay 13th 2015

Blute, Cockett and Seely say coKleisli in Differential categories.

• CommentRowNumber4.
• CommentAuthorUrs
• CommentTimeMay 13th 2015

• CommentRowNumber5.
• CommentAuthorMike Shulman
• CommentTimeMay 13th 2015

I’m pretty sure I’ve seen “Kleisli category for a comonad”.

• CommentRowNumber6.
• CommentAuthorDavid_Corfield
• CommentTimeMay 13th 2015

Google score 4 versus 1020 in favour of coKleisli category.

• CommentRowNumber7.
• CommentAuthorTodd_Trimble
• CommentTimeMay 13th 2015

Confirming the usage in #2 here and here (result 6 of 11) and here.

• CommentRowNumber8.
• CommentAuthorMike Shulman
• CommentTimeMay 13th 2015

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. (-:

• CommentRowNumber9.
• CommentAuthorTodd_Trimble
• CommentTimeMay 14th 2015

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.

• CommentRowNumber10.
• CommentAuthorUrs
• CommentTimeMay 18th 2015

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 $F$.

• CommentRowNumber11.
• CommentAuthorMike Shulman
• CommentTimeMay 18th 2015

I don’t recall hearing of a name for that.

• CommentRowNumber12.
• CommentAuthorUrs
• CommentTimeMay 19th 2015

Okay, thanks.