  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.

Category Theory;General Topology;Logic