Re: The omega-functor omega-category

[John Baez <baez@math.ucr.edu>]

David Leduc wrote:

>> I'm not sure what [_._] is supposed to mean - an internal
>> hom functor?

> This was supposed to be the "cartesian closed structure" of
> StrictOmegaCat, but since some say it is not a structure I'm not sure
> how to call it...

Just call it the internal hom.

The point is, you can just look at a category and say, yes or no,
whether it's cartesian closed.  So cartesian closedness is a "property"
of a category - not a "structure" that you might equip a category with
in more than one way.

Nonetheless, you can consider properties as a special case of
structures - namely, those structures for which you have at most
one one choice.  And if you do this you're free to speak of a cartesian
closed "structure".

Similarly, you can consider structures as a special case of "stuff".

If you don't know the yoga of "properties, structure and stuff", you
might enjoy this paper where Mike Shulman and I explain it:

http://arxiv.org/PS_cache/math/pdf/0608/0608420v2.pdf#page=15

Best,
jb


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