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Stream: learning: questions

Topic: Question about *-autonomous category definition

Mike Stay (Oct 13 2020 at 18:44):

https://ncatlab.org/nlab/show/star-autonomous+category says

Definition 2.1. A *-autonomous category is a symmetric closed monoidal category $\langle C,\otimes, I,\multimap\rangle$ with a global dualizing object: an object $\bot$ such that the canonical morphism $d_A: A \to (A \multimap \bot) \multimap \bot,$ which is the transpose of the evaluation map $ev_{A,\bot}: (A \multimap \bot) \otimes A \to \bot ,$ is an isomorphism for all $A.$ (Here, $\multimap$ denotes the internal hom.)

Why do they say "for all A"? ev is a natural transformation, so why not just say it's a natural isomorphism? Or is there some subtlety I'm missing?

Reid Barton (Oct 13 2020 at 19:04):

I would guess the thinking is: the "natural" part is automatic, because $d_A$ is always a natural transformation; so why not just say that the required condition is that it be an isomorphism for all $A$?

Reid Barton (Oct 13 2020 at 19:05):

To me that nlab phrasing is the, er, natural one.

Reid Barton (Oct 13 2020 at 19:07):

Maybe a better way to say it is: "something-or-other is an isomorphism for every $A$" is obviously a property, while "something-or-other is a natural isomorphism" is not obviously a property (and perhaps it's not even obvious what it means).

John Baez (Oct 13 2020 at 20:41):

"Why not say it's a natural isomorphism?" - maybe because this phrase comes after "such that". $ev_{A,\perp}$ is obviously a natural transformation, so the only new condition being imposed here is that it's an isomorphism for all $A$. Saying "is a natural isomorphism" might fool people into thinking that naturality is a new requirement.

John Baez (Oct 13 2020 at 20:42):

Oh - I answered before I read Reid's comment. So yeah: I agree with him, and he said it better. "Such that" should precede a statement of a property, not a structure.

John Baez (Oct 13 2020 at 20:44):

But anyway, all this all pretty sophisticated stylistic nitpicking - it's not like there's anything mysterious about the math here.

John Baez (Oct 13 2020 at 20:46):

(Mind you, I have nothing against sophisticated stylistic nitpicking - this is what keeps my style sharp!)