Re: A challenge to all

[Steve Vickers <s.j.vickers@cs.bham.ac.uk>]

Dear Andre,

How would you deal with composition?

The standard account of composability - defined by equality between a
domain and a codomain - does seem to presuppose a notion of object equality.

Normally for categories internal in C one therefore takes C to have
pullbacks. Are there obvious ways to generalize that to monoidal C?

Or are you thinking you might want to relax that standard account,
perhaps even losing the well defined domain and codomain? I'm getting a
picture of things like "categories" of smooth curves, where two curves
are composable only if the end of one has an open overlap with the start
of the other. But then "objects" are more nebulous.

Regards,

Steve Vickers.

Joyal wrote:
> ...
> I cannot imagine a category without an equality relation between the objects.
> ...
>
> 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 a set S is
> defined by the diagonal map S-->S times S.
> The diagonal gives a set the structure of a cocommutative coalgebra,
> where the tensor product is the cartesian product.
> The objects of a general symmetric monoidal category have no
> coalgebra structure in general.
>
> The test: Is it possible to define a notion of category internal to
> a symmetric monoidal category without using a coalgebra structure
> on the object of objects?


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