Let $j\colon A \to C$ be a functor between categories. Its codensity monad is the right Kan extension$Ran_j j$ of $j$ along itself, if this exists (as it certainly does when $A$ is small and $C$ is complete).

The name comes because $j$ is codense just when its codensity monad is the identity. Thus, in general, the codensity monad “measures the failure of $j$ to be codense”.

Last revised on September 22, 2012 at 18:11:59.
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