Suppose $F_1: C_1 \rightleftarrows D : U_1$ is an adjunction, with induced monad$T_1 = U_1 F_1$ on $C_1$. Then we can form the Eilenberg-Moore category$C_2 \coloneqq C_1^{T_1}$, and the comparison functor $U_2 : D \to C_2$. If $D$ has reflexive coequalizers, then $U_2$ has a left adjoint $F_2$, with induced monad $T_2 = U_2 F_2$ on $C_2$, and we can iterate.

If $D$ 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 $D$ is well-copowered), then it factors the original adjunction into an adjunction $F_\infty : C_\infty \rightleftarrows D : U_\infty$ that is a reflection (i.e. the induced monad $T_\infty = U_\infty F_\infty$ on $C_\infty$ is the identity) followed by an adjunction $C_1 \rightleftarrows C_\infty$ whose right adjoint $C_\infty \to C_1$ 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.

References

Applegate and Tierney, Iterated cotriples, springerlink

MacDonald and Stone, The tower and regular decompositions, numdam

Adamek and Herrlich and Tholen, Monadic decompositions, sciencedirect

Created on March 18, 2018 at 14:11:12.
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