/sci/ - Science & Math

>>12190853
The first is certainly more along the lines of what I'd come up with if you ask me.
(Also in Haskell, what's closest to natural transformations are polymorphic functions that fulfill the defining square, and those are defined over abstract a. The situation there might be loser in several ways, though, since all functors, at least in the standard lib, are endofunctors.)

I'm not used to your notation of writing the quantifier at the end, so what you write is a bit difficult to read - for me at least. And you don't seem to be complete anyway, e.g. in the 1. definition you don't end up with the forall a and forall b.
In those 3 lines you seem to repeat one line, and I'm not sure if you properly disallow that two f's with same domain and codomain have different natural arrows.

Anyway, I think you can either go NBG and work with classes better, or you go the Grothendieck way and force C to be just a set in a universe.
The former option is discussed in some length in this book that I came across last year
https://www.amazon.de/Sets-Functions-Measures-Fundamentals-Mathematics/dp/3110550083

Or, you just don't consider sets of nats. Who needs the Yoneda lemma anyway :^)

>>11915600
While I personally think this would be a cool and good idea, the trigonometric functions that form the matrix elements of orthogonal transformations have transcendence properties that quickly fuck you up. This probably goes for all compact smooth transformation groups.

>>11828007
Yeah they open the talk, stream it and after there's breakout sessions. The talk part goes to youtube as well.

E.g.,

https://youtu.be/RonyrB0kLew?t=1210