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D-ultrafilter monads

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If you have a question about this talk, please contact José Siqueira.

The ultrafilter monad on sets is the codensity monad of the embedding of finite sets into Set, as proved by Kennison and Gildenhuys (1971). In this talk I will present a notion of D-ultrafilter on an object of a category K which generalizes the one of an ultrafilter on a set, where D is a cogenerator of K. Working in a complete, symmetric monoidal closed category, with a ‘nice cogenerator D, the corresponding D-ultrafilter monad is the codensity monad of the embedding of finitely presentable objects of K; moreover, it is a submonad of the double-dualization monad relative to D. This is illustrated by several examples, including commutative varieties and categories of posets and graphs. I will also discuss a generalization with the above embedding replaced by the embedding of a small full subcategory into a complete category, with A containing a cogenerating set of K. This is based on joint work with Jiri Adámek.

This talk is part of the Category Theory Seminar series.

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