Suppose is an adjunction, with induced monad on . Then we can form the Eilenberg-Moore category , and the comparison functor . If has reflexive coequalizers, then has a left adjoint , with induced monad on , and we can iterate.
If is moreover cocomplete, we can pass to limits and obtain a tower of adjunctions indexed by all ordinal numbers. If this tower converges (which happens, for instance, if is well-copowered), then it factors the original adjunction into an adjunction that is a reflection (i.e. the induced monad on is the identity) followed by an adjunction whose right adjoint is a transfinite composite of monadic functors, hence in particular faithful and conservative. This is called a/the monadic decomposition of the original adjunction, and produces a factorization system on a suitable 2-category.
Applegate and Tierney, Iterated cotriples, springerlink
MacDonald and Stone, The tower and regular decompositions, numdam
Adamek and Herrlich and Tholen, Monadic decompositions, sciencedirect
Last revised on November 6, 2022 at 11:40:11. See the history of this page for a list of all contributions to it.