2025/07/11 by Farjoun, Emmanuel Dror, Ivanov, Sergei O.
#Algebraic Topology (math.AT) #FOS: Mathematics
paper · doi:10.48550/arxiv.2507.08414
Working in the setting of ∞-categories, we develop a general theory of the codensity monad TD associated with a full subcategory D⊆ C. We show that TD has a canonical monad structure (unique up to a contractible space of choices), and characterize it as a terminal monad preserving all objects of D. For a monad M on an ∞-category C, we consider the M-completion functor defined as the totalization of the cosimplicial resolution associated with M. We show that the M-completion functor is the codensity monad associated with the full subcategory of C spanned by objects that admit a structure of M-algebra. In particular, the M-completion functor is the terminal monad preserving all objects that admit a structure of an M-algebra. This gives a full ∞-categorical characterization of the classical Bousfield-Kan R-completion functor as the terminal monad on the category of spaces preserving the empty space and all products of Eilenberg-MacLane spaces K(A,n), where A is an R-module.