Related to Marco's point: In order to define the category Set, we have to define Set(X,Y) for all sets X and Y. There are many isomorphic options, but to get a category we have to choose one. Now replace "category" by "category with distinguished binary products and exponentials.". We shall then have to choose a particular implementation of X x Y and X -> Y. Why do some people find this philosophically objectionable? How is it worse than choosing Set(X,Y)? (Personally, I'd be inclined to make the same choice for X -
Y as for Hom(X,Y).)
Paul On 2 Aug 2014, at 11:58, Marco Grandis wrote:
Dear Eduardo,
I agree with many things in your message, but I think you are taking your argument too far. Talking of pullbacks you say:
We precisely teach in category theory courses that you should not work with any particular choice between the choices.
I agree that it is better to avoid such a choice when possible. Yet you cannot define a bicategory of spans without assuming that such a choice has been made; in the same way as you cannot define the (good) monoidal structure of Ab without recurring to a choice of tensor products. Such a situation, we all know, generally arises in non-strict bicategories (and monoidal categories, in particular).
Unless you want to redefine bicategories replacing the composition of arrows with an existence property. I still prefer working with a choice (eg of pullbacks) to such a complicated structure.
Best regards
Marco
-- Paul Blain Levy School of Computer Science, University of Birmingham +44 121 414 4792 http://www.cs.bham.ac.uk/~pbl [For admin and other information see: http://www.mta.ca/~cat-dist/ ]