idempotent monad

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

Recall that a *splitting of an idempotent* $(T,\mu)$ consists of a pair of 1-cells $I:D\to C$ and $R:C\to D$ and a pair of 2-isomorphisms $a:RI\to id_D$ and $b:T\to IR$ such that $\mu=b^{-1}(I\circ A\circ R)(b\circ b)$ 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 $K$ has equalizers or coequalizers, then all its idempotents split.

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

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