Re: Terminology: Remarks

[Toby Bartels <categories@TobyBartels.name>]

Jean B?nabou wrote in part:

>(i) If F: A -> B  and  G: B -> C are full and faithful essentially surjective functors, so is  GF. How do you compose your equivalences?

Good question!  In this case, we can compose by pullback.
But when I wrote "the obvious thing to do",
maybe I should have written "one obvious thing to try".
After all, it might not work; in this case, it does.

If it didn't work, another obvious thing to try would be zigzags.
In this case, this gives an equivalent 2-groupoid.
So in principle, one could do either, but spans are simpler.

Generalising from the 2-groupoid of categories to the 2-category of them,
we can use spans A <- X -> B where only A <- X needs to be ff eso.
This is what Michael Makkai did, and it allows one to avoid AC
while retaining the usual results about the 2-category of categories.

(Pace Tom Leinster's recent comment under this thread,
  Makkai actually required A <- X to be strictly surjective on objects,
  but again this does not matter; the resulting 2-category is equivalent.)

Beyond this, let me just say that I agree with Marta's answers.


--Toby


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