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Ultrafilters, ultraproducts and ultracategories

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If you have a question about this talk, please contact Julia Goedecke.

Consider the following notions: ultrafilter; ultrapower; product of ultrafilters; ultraproduct; dependent sum of ultrafilters; the category of ultrafilters. Building on an old observation of Reinhard B\”orger, we explain how each of these is forced upon you as soon as you know what it means for a functor to preserve finite coproducts.

If time permits, we go on to describe a bicategory W arising naturally from these considerations, and explain how the category of models and elementary embeddings for any classical first-order theory (i.e., Boolean pretopos) can be seen as a W-enriched category. This seems to be related to Makkai’s “Stone duality for first-order logic”.

This talk is part of the Category Theory Seminar series.

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