Let be a functor between categories. Its codensity monad is the right Kan extension of along itself, if this exists (as it certainly does when is small and is complete).
The name comes because is codense just when its codensity monad is the identity. Thus, in general, the codensity monad “measures the failure of to be codense”.
Revised on September 22, 2012 18:11:59
by David Corfield