Dear Marco, supposing you are right (which is probably the case, I have not examined closely the question of the monoidal structure of Ab or the bicategory of spans), this does not contradicts my points. I never said that choices should be banned, I said that they should not be used when not necessary, because of plenty of different reasons. best e.d. On 02/08/14 07:58, 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

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