Re: A challenge to all

[Jeff Egger <jeffegger@yahoo.ca>]

Dear André,

I do not understand the point of your "test".  

What Bob Paré said, and which I agree with, is that equality 
is "okay" for small categories.  And as Paul Taylor wrote, 
by "small" what one really means is "internal".  

So, of course, it makes sense that V-internal categories 
(where V is a not-necessarily-braided monoidal category with 
distributive coreflexive equalisers) should have a comonoid 
of objects.  

But this in no way contradicts the assertion that _large_ 
categories should not have an equality relation between 
objects---internal categories are tautologously small!  

Not being as familiar with indexed categories/fibrations as
I ought to be, I tend to think of large categories in terms 
of enriched category theory.  Here, we see very clearly that 
the collection of objects has nothing whatsoever to do with 
the enriching category V, and this is as it should be.

In fact, I suppose that it probably would make sense to 
generalise enriched categories by taking a (large) groupoid 
of objects (and _canonical_ isos, in the spirit of Paré and 
Schumacher) instead of a mere class.  I don't know if this 
has ever been done.  

My main point is that you are right in asserting that a set 
without an equality relation is not a set.  But the exact 
meaning of large category is one whose objects do not 
necessarily form a set!  

Morally speaking, "set" does mean "collection that has an 
equality predicate", but this leaves open the possibility 
that there are collections which do not have such a 
predicate, and which are therefore not sets.  These suffice 
for the purpose of large category theory---for example, they 
suffice for the purpose of enriched categories; moreover, 
FOLDS is explicitly based on these principles. 

Cheers,
Jeff.

----- Original Message ----
> From: "Joyal, André" <joyal.andre@uqam.ca>
> To: categories@mta.ca
> Sent: Tue, January 12, 2010 4:24:39 PM
> Subject: categories: A challenge to all
> 
> 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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