I’m pretty sure that a certain theorem about cocomplete categories must be true, and I’m even pretty sure that I know how to write down a proof. (Famous last words, I know.) But I have the feeling that the result is already known, and I just haven’t seen it. I thought I would state the result here (in somewhat vague terms), and hopefully someone can point me to the result, if it already exists.
Every cocomplete category that is co-well-powered and has a set of generators can be constructed explicitly as follows. Each object X can be represented as:
- A family of sets, X_i. This family is always a set. Each set represents a different sort, in the sense of multisorted algebras.
- A family of relations, R_j defined on the X_i. The relations can be of arbitrary arity and signature (so you can have relations on X_1 x X_2, etc.) Infinite arities are allowed. The number of relations of a fixed arity and signature is a set, but the family of all relations can be a proper class.
- A family of partially-defined operations. Each operation has as its domain all tuples that satisfy a certain relation.
- The relations are required to satisfy a collection of specified Horn clauses. The left-hand side of the Horn clauses can contain infinite conjunctions.
The arrows of this category are all families of functions X_i -> X’_i that preserve the R_j and the partial operations.
An easy example of this is the category of small categories. Here X_1 is the set of objects, X_2 is the set of arrows. It has four operations: the id operation that sends an object to its identity element, the dom operation that sends an arrow to its domain, the cod operation that sends an arrow to its codomain, and the partial operation of composition, which is defined for all f and g such that cod f = dom g. The Horn clause it satisfies is the requirement that the identity arrow is an identity under composition. (This example is unusual in that the relation is an equality between two operations; the relations can be arbitrary in general.)
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.)
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).
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!
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).
Arsymmetric
A family of partially-defined operations is of a fixed arity and signature is a set and relation i.i.f a relation is semi-continuous, fixed empty set or an exclusive positive fixed point. This domain of tuple fixed arity and signiture, arsymmetric rate of zero non-empty set of defined limits.