nLab monadic decomposition

Monadic decompositions

Monadic decompositions


Suppose F 1:C 1D:U 1F_1: C_1 \rightleftarrows D : U_1 is an adjunction, with induced monad T 1=U 1F 1T_1 = U_1 F_1 on C 1C_1. Then we can form the Eilenberg-Moore category C 2C 1 T 1C_2 \coloneqq C_1^{T_1}, and the comparison functor U 2:DC 2U_2 : D \to C_2. If DD has reflexive coequalizers, then U 2U_2 has a left adjoint F 2F_2, with induced monad T 2=U 2F 2T_2 = U_2 F_2 on C 2C_2, and we can iterate.

If DD 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 DD is well-copowered), then it factors the original adjunction into an adjunction F :C D:U F_\infty : C_\infty \rightleftarrows D : U_\infty that is a reflection (i.e. the induced monad T =U F T_\infty = U_\infty F_\infty on C C_\infty is the identity) followed by an adjunction C 1C C_1 \rightleftarrows C_\infty whose right adjoint C C 1C_\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.


Last revised on July 10, 2024 at 12:54:53. See the history of this page for a list of all contributions to it.