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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 (M_i) is a collection of models and M is a nonprincipal ultraproduct of the M_i, then one wants and expects that M(s) will be empty if and only if the set of  i for which M_i(s) 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?

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Born Free January 12, 2009

Posted by dcorfield in Uncategorized.
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Continuing our discussion of free entities, from Fiore and Leinster’s A simple description of Thompson’s group F we read

…many entities of interest can be described as free categories with structure. For instance, the braided monoidal category freely generated by one object is the sequence (B_n)_{n \geq 0} of Artin braid groups; the monoidal category freely generated by a monoid consists of the finite ordinals; the symmetric monoidal category freely generated by a commutative monoid consists of the finite cardinals; the symmetric monoidal category freely generated by a commutative Frobenius algebra consists of 1-dimensional smooth oriented manifolds and diffeomorphism classes of 2-dimensional cobordisms.

You can find definitions of many terms used in higher category theory at the exciting new wiki nLab. For example, see braided monoidal category.

Fiore and Leinster continue,

In this vein, our result is that the monoidal category freely generated by an object A and an isomorphism A \otimes A \longrightarrow A is equivalent to the groupoid 1 + F, where 1 is the trivial group and + is coproduct of groupoids.

and for the related Thompson’s group V,

just replace ‘monoidal category’ by ‘symmetric monoidal category’, or equally ‘finite-product category’.