Re: Small is beautiful

[Vaughan Pratt <pratt@cs.stanford.edu>]

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/ ]