Comments on: Wanted: Theorem about Cocomplete Categories

By: Walt

Walt — Sat, 12 Jul 2008 21:34:09 +0000

Top works something like this: for each space X of n points, introduce an n-ary predicate (n can be infinite) P_X. P_X holds true for a tuple of n points if the obvious map from X to those points is continuous. If P and Q are two such n-ary predicate, P implies Q when identity map on n elements lifts to a continuous map from Q to P. There are no operations required, partial or total (I think).

By: Todd Trimble

Todd Trimble — Thu, 10 Jul 2008 00:06:55 +0000

Your claim about Top is interesting; I wouldn’t mind seeing how it works in that case.

I suspect you might have more success getting a definitive answer if you were to post your query to the categories mailing list. Someone there would surely know!

By: Walt

Walt — Wed, 09 Jul 2008 18:12:14 +0000

Thanks for the info, Todd. I did some reading, and I don’t think it quite falls under that. (I’m not 100% sure, though.) Unfortunately, my local library has neither book about Accessible Categories.

I did find that Top is not an accessible category. Top, even though its not co-well-powered, can be represented by the above construction (or so I’ve convinced myself).

By: Todd Trimble

Todd Trimble — Mon, 07 Jul 2008 03:24:10 +0000

Walt,

I talked about your query with my buddy Jim Dolan for a few minutes, and we both thought that this *kind* of thing (modulo technical details which we didn’t try to figure out then and there) is the kind of thing you might find in the book on Accessible Categories by Makkai and Reyes. There’s also a big literature on the related topic of locally presentable or locally finitely presentable categories.

Sorry that this is a little vague, but if you haven’t looked at these sources yet, you might find something interesting. (Sad to say, I don’t know this stuff as I should.)