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• CommentRowNumber1.
• CommentAuthormattecapu
• CommentTimeAug 3rd 2022

Definition and characterization as monoids

• CommentRowNumber2.
• CommentAuthorSam Staton
• CommentTimeAug 5th 2022

mention when the skew monoidal structure is an ordinary monoidal structure

• CommentRowNumber3.
• CommentAuthorSam Staton
• CommentTimeAug 5th 2022

Examples. Hope I got the monads with arities one right.

• CommentRowNumber4.
• CommentAuthorDavid_Corfield
• CommentTimeAug 5th 2022

Put in the spaces, so it renders properly, ’J X’ rather than ’JX’, yields $J X$ rather than $JX$.

• CommentRowNumber5.
• CommentAuthorUrs
• CommentTimeNov 17th 2022

I have spelled out (here) in a fairly elementary fashion the elementary example of “forming linear spans”, namely of sending sets to the $\mathbb{K}$-vector spaces which they span.

In the course of doing so, I have adjusted wording and formatting in the Definition section, specifically in what is now the subsection “Definition – As skew-Kleisli triples” (here), in the hope to improve readability.

(The Idea-section of this entry still needs attention, but I’ll leave it as is for the time being.)

• CommentRowNumber6.
• CommentAuthormaxsnew
• CommentTimeNov 17th 2022

Add a less technical idea section, and add a section that shows $J$ can be generalized to a profunctor.

I hope no one minds I deleted some very technical material from the idea section that is immediately repeated in the Definition section anyway.

• CommentRowNumber7.
• CommentAuthorvarkor
• CommentTimeNov 17th 2022

@maxsnew: I’m curious whether you have some motivating examples for this generalisation?

• CommentRowNumber8.
• CommentAuthormaxsnew
• CommentTimeNov 18th 2022

I gave a talk about relative monads in CBPV recently, and I have some programming examples there: https://www.youtube.com/watch?v=ooj1vJRixEU&list=PLyrlk8Xaylp5hkSMipssQf3QKnj6Nrjg_

To summarize, in CBPV a monad relative to F : val -> comp is a more low-level version of a monad that specifies the stack the computation runs against. It’s natural to consider CBPV where F doesn’t necessarily exist and you can still define relative monads as relative to the “profunctor of computations” which is always present. Additionally, the morphisms of comp (the stacks) are typically not internalized as a data type in CBPV, but the elements of the profunctor are, so even if you have F, the notion of self-enriched relative monad needs to use the profunctor generalization.

I thought it would be a bit too far afield to try to explain those examples on this page.

• CommentRowNumber9.
• CommentAuthorUrs
• CommentTimeNov 18th 2022

I find that a real-world example is just what this entry needs.

Last night I have started watching the talk you pointed to (the one here). It’s nice, but at some point I admittedly missed how the crawling through the stack looking for exceptions is a relative monad. If that could be explained in the entry, it would be interesting.

• CommentRowNumber10.
• CommentAuthorJ-B Vienney
• CommentTimeNov 18th 2022
• (edited Nov 18th 2022)

Maybe could you explain those examples in a new entry “relative monads in CBPV” and link it to the pages CBPV and relative monad?

• CommentRowNumber11.
• CommentAuthorUrs
• CommentTime2 days ago

added missing typing of “$k$” in the associativity clause

• CommentRowNumber12.
• CommentAuthorUrs
• CommentTime1 day ago