categories are beautiful

["Eduardo J. Dubuc" <edubuc@dm.uba.ar>]

At the beginning Ehresmann arrived to the notion of category from groupoids,
even topological  groupoids. These are "internal and small". They do not even
have objects (just a partially defined operation with enough neutral elements).

At the beginning Eilemberg-MacLane arrived to the notion of category from the
categories of Sets, Groups, etc. They have objects, and are "external and
large". They were not even aware that groupoids were categories.

They are two very different things, that happen (by chance ?) to satisfy the
same axiomatic definition of category, which is a beautiful definition.

Bob Pare is so much right telling us that the distinction is not of size.
Clearly the small categories of finitely presented rings, of finite groups,
etc, etc, and even the groupoid of finite sets and bijective functions (in
Joyal's theory of species for example) are in spirit Eilember-MacLane's
"large" categories, and not Ehreshmann's "small" categories.

e.d.


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