Codensity and the ultrafilter monad
Abstract
Even a functor without an adjoint induces a monad, namely, its codensity monad; this is subject only to the existence of certain limits. We clarify the sense in which codensity monads act as substitutes for monads induced by adjunctions. We also expand on an undeservedly ignored theorem of Kennison and Gildenhuys: that the codensity monad of the inclusion of (finite sets) into (sets) is the ultrafilter monad. This result is analogous to the correspondence between measures and integrals. So, for example, we can speak of integration against an ultrafilter. Using this language, we show that the codensity monad of the inclusion of (finitedimensional vector spaces) into (vector spaces) is double dualization. From this it follows that compact Hausdorff spaces have a linear analogue: linearly compact vector spaces. Finally, we show that ultraproducts are categorically inevitable: the codensity monad of the inclusion of (finite families of sets) into (families of sets) is the ultraproduct monad.
 Publication:

arXiv eprints
 Pub Date:
 September 2012
 arXiv:
 arXiv:1209.3606
 Bibcode:
 2012arXiv1209.3606L
 Keywords:

 Mathematics  Category Theory;
 Mathematics  General Topology;
 Mathematics  Logic
 EPrint:
 32 pages. Version 3: now includes categorical characterization of ultraproducts. Final journal version