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\begin{document}

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\section*{lax-idempotent 2-adjunction}

\hypertarget{context}{}\subsubsection*{{Context}}\label{context}

\hypertarget{2category_theory}{}\paragraph*{{2-category theory}}\label{2category_theory}

[[!include 2-category theory - contents]]

\hypertarget{laxidempotent_2adjunctions}{}\section*{{Lax-idempotent 2-adjunctions}}\label{laxidempotent_2adjunctions}

\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{definition}{Definition}\dotfill \pageref*{definition} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak


\hypertarget{idea}{}\subsection*{{Idea}}\label{idea}

A [[lax-idempotent 2-monad]] generalizes the notion of [[idempotent monad]] to 2-categories by replacing inverses with adjoints. A \textbf{lax-idempotent 2-adjunction} (or \textbf{KZ 2-adjunction}) similarly generalizes the notion of [[idempotent adjunction]], and is related to lax-idempotent 2-monads in the same way that idempotent adjunctions are related to idempotent monads.

\hypertarget{definition}{}\subsection*{{Definition}}\label{definition}

We will need to use all three kinds of composition in the 3-category $2 Cat$. We write composition along 0-cells (2-categories) with juxtaposition. We write composition along 1-cells (2-functors) with a dot; this is of course composition along 0-cells \emph{in} a 2-category. And we write composition along 2-cells (transformations) with $\circ$, which is composition along 1-cells \emph{in} a 2-category.

Let $F : C \rightleftarrows D : G$ be a [[2-adjunction]] with unit $\eta: 1_C \to G F$ and counit $\epsilon: F G \to 1_D$. (For simplicity, we will assume it is a strict 2-adjunction, but the same definitions and proofs work in the pseudo case with some equalities replaced by isomorphisms.) This 2-adjunction is said to be \textbf{lax-idempotent} if one (hence all) of the following equivalent conditions hold.

\begin{enumerate}%
\item The triangle identity $1_F = \epsilon F . F\eta$ is the unit of an adjunction $F\eta \dashv \epsilon F$.


\item The induced equality $1_{G F} = G \epsilon F . G F\eta$ is the unit of an adjunction $G F\eta \dashv G \epsilon F$.


\item The induced 2-monad $G F$ is [[lax-idempotent 2-monad|lax-idempotent]].


\item There is a modification $\delta : G F \eta \to \eta G F$ such that $\delta \circ \eta = 1$ and $(G\epsilon F) \circ \delta = 1$.


\item There is a modification $\delta' : G F \eta G \to \eta G F G$ such that $\delta' \circ (\eta G) = 1$ and $(G\epsilon F G) \circ \delta' = 1$.


\item The triangle identity $G \epsilon . \eta G$ is the counit of an adjunction $G \epsilon \dashv \eta G$.


\item The induced equality $F G \epsilon . F \eta G$ is the counit of an adjunction $F G \epsilon \dashv F \eta G$.


\item The induced 2-comonad $F G$ is lax-idempotent.


\item There is a modification $\delta : \epsilon F G \to F G \epsilon$ such that $\epsilon \circ \delta = 1$ and $\delta \circ (F\eta G) = 1$.


\item There is a modification $\delta' : G \epsilon F G \to G F G \epsilon$ such that $(G \epsilon) \circ \delta = 1$ and $\delta \circ (G F\eta G) = 1$.



\end{enumerate}
\begin{proof}
By duality, it suffices to prove that $1\Rightarrow 2 \Rightarrow 3 \Rightarrow 4 \Rightarrow 5 \Rightarrow 6$. Of these, $1\Rightarrow 2$ and $4 \Rightarrow 5$ are obvious by whiskering. And since $G\epsilon F$ is the multiplication of the 2-monad $G F$, a standard fact about [[lax-idempotent 2-monads]] gives $2\Leftrightarrow 3 \Leftrightarrow 4$.

Thus, it remains to show $5 \Rightarrow 6$. We take the unit of the desired adjunction to be

\begin{displaymath}
\begin{aligned}
  1
  &\overset{triangle}{=} G F G \epsilon . G F \eta G\\
  &\xrightarrow{\delta'} G F G \epsilon . \eta G F G\\
  &\overset{naturality}{=} \eta G . G \epsilon
\end{aligned}
\end{displaymath}
The two triangle identities for this putative adjunction follow from the two axioms assumed for $\delta'$.

\end{proof}
Note that when $C$ and $D$ are locally [[discrete category|discrete]], hence just 1-categories, this reduces to the usual characterization of [[idempotent adjunctions]].

In contrast to that situation, however, the lax-idempotent situation is of interest even when the adjunction is monadic or comonadic. In the monadic case, the implication $3\Rightarrow 8$ means that the induced 2-comonad on the 2-category of algebras for a lax-idempotent 2-monad is again lax-idempotent. Its (pseudo) coalgebras are the [[continuous algebras]] for the original 2-monad.

\hypertarget{references}{}\subsection*{{References}}\label{references}

\begin{itemize}%
\item [[Marta Bunge]] and [[Jonathon Funk]], \emph{Singular Coverings of Toposes}. In this book the notion is called a ``KZ adjointness'' and defined by both (1) \emph{and} (6).

\end{itemize}
[[!redirects lax-idempotent 2-adjunction]] [[!redirects lax-idempotent 2-adjunctions]] [[!redirects lax idempotent 2-adjunction]] [[!redirects lax idempotent 2-adjunctions]] [[!redirects KZ 2-adjunction]] [[!redirects KZ 2-adjunctions]] [[!redirects Kock-Zoberlein 2-adjunction]] [[!redirects Kock-Zoberlein 2-adjunctions]]



\end{document}