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I’m reading the nLab page and Kelly’s book, and I’m unfortunately unable to tell what the issue is with just extending the definition of naturality in the naïve way. Is extranaturality a stronger, weaker, or incomparable condition to honest naturality?
Did you look at the nLab page extranatural transformation (which I recommended to you elsewhere)?
You could say that the notions are “incomparable” but closely related. The archetype is the evaluation map $[x, y] \otimes x \to y$ which is natural in $y$ and extranatural in $x$. A general rule is that in a closed monoidal category, in which $(- \otimes x) \dashv [x, -]$, any transformation of the form $F(x) \to [G(x), a]$, natural in $x$, gives rise to an extranatural transformation $F(x) \otimes G(x) \to a$, and conversely. And any transformation of the form $a \otimes F(x) \to G(x)$ gives rise to an extranatural transformation of the form $a \to [F(x), G(x)]$, and conversely.
An intuitive way of grasping extranaturality is through string diagrams. Instances of naturality correspond to strings or wires running vertically, and instances of extranaturality correspond to cups and caps. In a closed monoidal context, as for example in compact closed categories, the closure allows one to bend “natural wires” into extranatural ones, or unbend extranatural ones into natural ones, if you catch my drift. There are a few lemmas or propositions at extranatural transformation which are usefully visualized in terms of string diagrams, including the “yanking” lemma, of which triangular equations in compact closed categories are a famous instance.
Try reading the nLab page, and ask here again if you have any questions.
I think I’ve got it now, but you should probably include an explanation of the string diagrams. Also, why do we say extranatural in c or b instead of saying source extranatural in b or target extranatural in c?
you should probably include an explanation of the string diagrams.
Sadly, I don’t know how to draw up or deal with pictures like that, and the pictures needed for a full explanation aren’t really there. Maybe you can teach me how to draw. I really need someone to tutor me so that I can continue with my page on surface diagrams. :-)
Also, why do we say extranatural in c or b instead of saying source extranatural in b or target extranatural in c?
(Yes, or say something like that.) Good question; I don’t know. We could try something like that.
Well, if I say that a transformation $F(b,b,c) \to c$ is extranatural in $b$, it seems to me pretty clear that I am referring to the source, since $b$ doesn’t appear in the target.
When I talk about something has property x in a certain variable, I use it to mean “in the coordinate(s) filled by that letter”.
I agree with Mike that it would be pointlessly redundant to say “$F(b, b, c) \to G(c)$ is source-extranatural in $b$”, and I can’t think of a natural situation where you’d need to speak of extranaturality of a transformation where it wasn’t clear which type of extranaturality was involved. So I currently think it’s is an unnecessarily fussy distinction after all. (Should that situation ever arise, we can revisit the issue.)
Max Kelly went further: he would just say “natural” to cover any case, whether ordinary natural or extraordinary natural. And again for exactly the same reason as above: if “natural” means either, then there is an unambiguous interpretation of what “$F(b, b, c) \to G(c)$ is natural in $b$ and $c$” means for either.
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