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 [For admin and other information see: http://www.mta.ca/~cat-dist/ ]