A challenge to all

["Joyal, André" <joyal.andre@uqam.ca>]

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.
Of course, a set is often constructed from other sets, 
as in arithmetic with congruence classes. 
I am fully aware that the equality relation between the objects of a 
category is not preserved by equivalences in general.
But the art of category theory consists partly in knowing
which construction on the objects and arrows of
a category is invariant under equivalences. 

I would like to propose a test for verifying if the 
notion of category can be freed from the equality relation
on its set of objects. The equality relation on an ordinary 
set S is defined by the diagonal S-->S times S.
The objects of a symmetric monoidal category have no diagonal in general,
ie no coalgebra structure.

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?


Best, André 



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