idempotent monad


Let us be in a 22-category KK. Part of the structure of an idempotent monad (C,T,η,μ)(C,T,\eta,\mu) in KK is of course an idempotent morphism T:CCT:C\to C. More precisely (Definition 1.1.9) considers μ\mu as part of the structure such that an idempotent 1-cell has a 2-isomorphism μ:TTT\mu:TT\to T such that μT=Tμ\mu T=T\mu. Equivalently an idempotent morphism is a normalized pseudofunctor from the two object monoid {*,e}\{*,e\} with e 2=ee^2=e to KK.

Recall that a splitting of an idempotent (T,μ)(T,\mu) consists of a pair of 1-cells I:DCI:D\to C and R:CDR:C\to D and a pair of 2-isomorphisms a:RIid Da:RI\to id_D and b:TIRb:T\to IR such that μ=b 1(IAR)(bb)\mu=b^{-1}(I\circ A\circ R)(b\circ b) where \circ denotes horizontal composition of 2-cells. Equivalently a splitting of an idempotent is a limit or a colimit of the defining pseudofunctor. If KK has equalizers or coequalizers, then all its idempotents split.

Now if (I,R,a,b)(I,R,a,b) is a splitting of an idempotemt monad, then RIR\dashv I are adjoint. And in this case the splitting of an idempotent is equivalently an Eilenberg-Moore object for the monad (C,T,η,μ)(C,T,\eta,\mu). In this case DD is called an adjoint retract of CC.


Equivalences (resp. cores) in an allegory are precisely those symmetric idempotents which are idempotent monads (resp. comonads). In an allegory the following statements are equivalent: all symmetric idempotents split, idempotent monads split, idempotent comonads split. A similar statement holds at least for some 2-categories.


  • Peter Johnstone, sketches of an elephant, B 1.1.9, p.248-249

Last revised on May 2, 2013 at 00:32:05. See the history of this page for a list of all contributions to it.