Re: A challenge to all

[Michael Shulman <shulman@uchicago.edu>]

Dear Andre,

You are absolutely right that the equality relation is inseparable from
the idea of a set.  What is being proposed, however, is that a category
doesn't need to have a *set* of objects.  In fact, the objects of a
category don't need to form an object of any category at all, so I think
your proposed test is misguided.  The formulation of category theory in
dependent type theory which Richard, Toby, I, and others are proposing
makes perfect sense without any equality predicate for the objects.

Best,
Mike

Joyal wrote:
> 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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