Abstract

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.