Codensity and the ultrafilter monad

TL;DR

This paper explores the concept of codensity monads, demonstrating their role as substitutes for adjoint-induced monads, and reveals their connections to ultrafilters, dualization, and ultraproducts across various categories.

Contribution

It clarifies the nature of codensity monads, proves their equivalence to ultrafilter monads in specific cases, and establishes their categorical inevitability in various contexts.

Findings

The codensity monad of finite sets inclusion is the ultrafilter monad.

Double dualization arises as the codensity monad for finite-dimensional vector spaces.

Ultraproducts are shown to be categorically inevitable via codensity monads.

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 (finite-dimensional 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.