Re: cleavages and choice

[Paul Levy <P.B.Levy@cs.bham.ac.uk>]

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












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