Re: A challenge to all

[burroni@math.jussieu.fr]

Dear André,

> The test: Can we define a notion of category internal to
> a symmetric monoidal category without using a coalgebra structure
> on the object of objects?

This is possible with this conception :
A "monoid" in a monoidal category is what Benabou call *monad* in his  
paper on bicatégories. The more general notion of "category", in the  
same vein, is the notion of *polyad* (in the same paper of Benabou ---  
apologize me : I have not the reference whith me): the objets are  
given externally.

I don't know if my remark you satisfie, I think not (you have had  
certainly yourself this idea), but, for me, the moral of this fact  
seems to say that the equality of objets have something of  
intrinsically very different of morphisms.
(This meets perhaps the ideas of Bob Paré on index categories)

I am convided of this (If I have the occasion, I will explain this).
Best,
Albert

"Joyal, André" <joyal.andre@uqam.ca> a écrit :

> Dear All,
>
> I cannot imagine a category without an equality relation between the  
>  objects of this category.
> Ok, I may have been brainwashed by my training in mathematics at an   
> early age.
> But more seriously, I think that the equality relation is inseparable
> from the idea of a set. I do not understand what a preset is:
>
> http://ncatlab.org/nlab/show/preset
>
> Two things are equal if they are the same, if they coincide   
> (whatever that mean!).
> Without this notion, an element of a set has no identity, no individuality.

...

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