Bousfield-Kan completion as a codensity ∞-monad

In this talk we recall the theory of codensity monads in ordinary category theory and tell about its generalization to the ∞-category setting. In particular, we show that the codensity ∞-monad of a full subcategory D of an ∞-category C satisfies a universal property: it is the terminal D-preserving ∞-monad. As an application, we show that the classical Bousfield-Kan R-completion functor can be described as the codensity ∞-monad of the full subcategory K(R) in the ∞-category of spaces spanned by the empty space and the products of Eilenberg-MacLane spaces of R-modules. As a corollary, we obtain that the Bousfield-Kan R-completion is the terminal K(R)-preserving ∞-monad.