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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.)
@maxsnew: I’m curious whether you have some motivating examples for this generalisation?
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.
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.
Maybe could you explain those examples in a new entry “relative monads in CBPV” and link it to the pages CBPV and relative monad?
added pointer to:
added pointer to:
Updated the notation to match that of relative adjunction, cleaned up parts of the entry, and mentioned that relative monads are monoids in skew-multicategories.
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