nLab monadic decomposition

Idea

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 (co)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

• Harry Applegate, Miles Tierney, Iterated cotriples, In: MacLane, S., et al. Reports of the Midwest Category Seminar IV. Springer LNM 137 (doi)

• John L. MacDonald, Arthur Stone, The tower and regular decompositions, numdam

• Jiří Adámek, Horst Herrlich, Walter Tholen, Monadic decompositions, (doi)

• Noson Yanofsky, The monadic tower for $\infty$-categories, arXiv:2104.01816