categories are beautiful

categories are beautiful

[Eduardo J. Dubuc]

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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Re: categories are beautiful

[Vaughan Pratt]

Eduardo J. Dubuc wrote:
> 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.

Let me try out an analogy here.  Atmospheric pressure bears down on our
skin with several tonnes, but we don't notice this because our insides
push back with equal pressure.  Only when we disturb the difference by
swimming ten feet underwater or climbing a 20,000 foot mountain does
pressure come to our attention.

It seems to me that the size of Set is less important than our
day-to-day choices of set-valued functors.  The size of Set is like the
tonnes of atmosphere pressing down on our skin.  The choice of functor
is more like the little variations we make to the equilibrium across our
skin.

Vaughan


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