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Ultraproducts the Category Theoretic Way *January 21, 2009*

*Posted by dcorfield in Uncategorized.*

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Following Alexandre’s two posts on ultraproducts of fields (here and here), I was wondering about the category theoretic view on ultraproducts. From Michael Barr’s Models of Sketches we read

Unlike limits and colimits, an ultraproduct is not defined by any universal mapping property. Of course, if the category has limits and (filtered) colimits, then it has ultraproducts constructed as colimits of products…But usually the category of models of a coherent theory (such as the theory of fields) lacks products and hence does not have categorical ultraproducts.

In fact the category theoretic definition is slightly different from the usual one, and comes with an advantage, according to Barr:

The only argument for banning the empty model that has any force comes from the observation that if is a collection of models and is a nonprincipal ultraproduct of the , then one wants and expects that will be empty if and only if the set of for which is null belongs to the ultrafilter. If one takes the traditional definition of an ultraproduct as a quotient of the product, the ultraproduct will be empty as soon as one factor is.

Defined in terms of a colimit of products this problem goes away.

But how to cope with the problem Barr mentions with fields?

Why didn’t I just read on? The term ‘coherent’ was a technical one, referring to the ‘logic’ used in the theory. Barr describes a collection of different strengths of ‘sketch’ (finite product, left exact, regular, finite sum, coherent, geometric). Regular is a stronger condition than coherent is stronger than geometric.

Given a geometric sketch one can form its corresponding regular sketch. Then the category of models of a regular sketch does have products. Categorical ultraproducts can then be formed in that category and if they happen to be models of the geometric sketch, then you say that the category of the latter is closed under ultraproducts.

There’s a nice category theoretic explanation of ultrafilters by Todd Trimble here.

The map from a set to the set of ultrafilters on the Boolean algebra of subsets of , which sends an element of to the principal ultrafilter generated by it, is the unit of an important adjunction.

There’s plenty more wonderfully explained by Todd there.

Thanks, David. There are some other remarkable ways in which ultrafilters arise from adjunctions. Here is one, which I learned about by reading a post by Lawvere to the categories mailing list:

Let 3 denote a 3-element set, say {0, 1, 2}. Let M = hom(3, 3) be the 27-element monoid consisting of functions from 3 to 3, with composition of functions playing the role of monoid multiplication. Then for any set X, M acts naturally on hom(X, 3), which I will abbreviate to 3^X. Thus 3^X lives in the category M-Set. The set

3 = 3^1 in particular carries a structure of M-set.

So, the map X |–> 3^X defines a functor Set –> (M-Set)^{op}. This functor is left adjoint to a functor (M-Set)^{op} –> Set which sends an M-set Y to hom_M(Y, 3), the set of functions Y –> 3 which respect the M-actions. Then, it turns out that the set of ultrafilters on X can be naturally identified with the set hom_M(3^X, 3), and the principal ultrafilter map

X –> hom_M(3^X, 3)

is again the unit of the adjunction. [The object 3 plays a dual role, both as set and as M-set, and the adjunction we speak of here is based on this ambimorphicity of 3. This is parallel to the ambimorphic nature of 2 seen as set or as Boolean algebra, which generates the adjunction David cited.]

One can play a similar game, replacing 3 by any set A, replacing hom(3, 3), by hom(A, A), etc. The monad of the adjunction would then be hom_{A^A} (A^X, A). If A is finite and greater than 2, this monad is again the ultrafilter monad.

Lawvere seemed to be interested in formulating “large cardinal hypotheses” in terms of obstructions to nice duality statements for these and similar monads, where “nice” would mean here that the unit

X –> hom_{A^A} (A^X, A)

is an isomorphism. Again, if A > 2 is finite, and assuming the axiom of choice, this niceness is obstructed by the existence of infinite sets X. But if we take A to be larger, let us say the set of natural numbers N, the existence of obstructions to this niceness then becomes tantamount to the existence of

measurable cardinals. Lawvere cites one or two more examples of this phenomenon, of a more topological nature.At some point I was planning to blog about all this.