Robert Pare wrote:
Then there are the small categories which are used to study the large ones. These are syntactic in nature.
Don't get me started. Oops, too late.
For these, one can't expect the kinds of universal constructions that large categories have,
Not following. FinSet is an essentially small category, what do you mean that it doesn't enjoy universal constructions? It's even a topos. Then there are the categories enriched in small categories, again subject to cardinality restrictions, which too are perfectly capable of enjoying universal constructions.
but now it's okay, even necessary, to consider equality between objects.
For small as opposed to essentially small categories, yes in some cases. But consider the category of ordinals truncated at say beth_2, certainly a small category when the morphisms are the inequalities. Are you comfortable defining equality on the objects of this category? (PTJ would correctly accuse me of being inconsistent on this point.)
Well, after these ramblings, perhaps my message is lost. So here it is: Small categories -> equality of objects okay Large categories -> equality of objects not okay
I hate to seem argumentative, Bob, but this can't possibly be the difference between small and large.
Small is beautiful, not evil.
Agreed, so long as this is not at the expense of large. Nice to be able to close on a note of consensus. :) Vaughan [For admin and other information see: http://www.mta.ca/~cat-dist/ ]