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Expanded dinatural transformation a little with examples and references.
I’m very glad you did, because it was through the Paré and Román reference that I relocated examples of dinatural transformations (due to Freyd) that I thought were forever lost to buried memory. I may copy them out later.
All I did in my edit was to fix the bad link in:
Here is a blog post inspired by the above discussion that discusses these concepts in the context of the programming language Haskell:
- Dan Piponi, Dinatural transformations and coends
to
Added
Added a yoneda-like correspondence between dinatural transformations and natural transformations between certain hom-sets.
Also, the discussion in “dinaturality vs extranaturality” has some errors. In particular, in the “bending” calculation, $G(x,y)^{F(y,x)}$ doesn’t make sense. Is it supposed to be $D(F(y,x), G(x,y))$ similar to the yoneda-like description I added?
Yes, I think that’s what’s meant. In fact what’s going on here is basically the same as in your two-sided Yoneda lemma: the hom-profunctor is a representing object for extranatural transformations.
I don’t see any other errors, what else do you have in mind?
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