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\section*{lax-idempotent 2-monad}

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

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

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

\hypertarget{contents}{}\section*{{Contents}}\label{contents}

\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{definition}{Definition}\dotfill \pageref*{definition} \linebreak
\noindent\hyperlink{equivalent_conditions}{Equivalent conditions}\dotfill \pageref*{equivalent_conditions} \linebreak
\noindent\hyperlink{algebras}{Algebras}\dotfill \pageref*{algebras} \linebreak
\noindent\hyperlink{examples}{Examples}\dotfill \pageref*{examples} \linebreak
\noindent\hyperlink{properties}{Properties}\dotfill \pageref*{properties} \linebreak
\noindent\hyperlink{related_concepts}{Related concepts}\dotfill \pageref*{related_concepts} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak


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

A lax-idempotent 2-monad encodes a certain kind of [[stuff, structure, property|property-like structure]] that a [[category]], or more generally an [[object]] of a [[2-category]], can carry.

The archetypal examples are given by [[2-monads]] $T$ on [[Cat]] that take a [[category]] $C$ to the [[free cocompletion]] $T C$ of $C$ under a given class of [[colimits]] -- then an [[algebra of a monad|algebra]] $T C \to C$ is a category $C$ with all such colimits, which are of course essentially unique. Moreover, given thus-cocomplete categories $C$ and $D$, a functor $F \colon C \to D$, and a diagram $S$ in $C$, there is a unique arrow $colim T F S \to F(colim S)$ given by the universal property of the colimit. It is this property that lax-idempotence generalizes.

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

A [[2-monad]] $T$ on a [[2-category]] $K$ is called \textbf{lax-idempotent} if given any two (strict) $T$-algebras $a \colon T A \to A$, $b \colon T B \to B$ and a morphism $f \colon A \to B$, there exists a unique 2-cell $\bar f \colon b \circ T f \Rightarrow f \circ a$ making $(f,\bar f)$ a lax morphism of $T$-algebras:

\begin{displaymath}
\itexarray{
T A & \overset{T f}{\to} & T B \\
a \downarrow & \swArrow \bar f & \downarrow b \\
A & \underset{f}{\to} & B
}
\end{displaymath}
Dually, a 2-monad $T$ is called \textbf{colax-idempotent} if $f \colon A \to B$ gives rise to a colax $T$-morphism $(f,\tilde f)$:

\begin{displaymath}
\itexarray{
T A & \overset{T f}{\to} & T B \\
a \downarrow & \neArrow \tilde f & \downarrow b \\
A & \underset{f}{\to} & B
}
\end{displaymath}
Lax-idempotent monads are also called \textbf{Kock--Z\"o{}berlein} or \textbf{KZ} monads.

\hypertarget{equivalent_conditions}{}\subsubsection*{{Equivalent conditions}}\label{equivalent_conditions}

\begin{theorem}
\label{AlgebraAdjoint}\hypertarget{AlgebraAdjoint}{}
A 2-monad $T$ as above is lax-idempotent if and only if for any $T$-algebra $a \colon T A \to A$ there is a 2-cell $\theta_a \colon 1 \Rightarrow \eta A \circ a$ such that $(\theta_a ,1_{1_A})$ are the unit and counit of an [[adjunction]] $a \dashv \eta_A$.

\end{theorem}
\begin{proof}
\textbf{(Adapted from Kelly--Lack)}. The multiplication $\mu_A \colon T^2 A \to T A$ is a $T$-algebra on $T A$, and $\eta_A \colon A \to T A$ is a morphism from the underlying object of $a$ to that of $\mu_A$. So there is a unique $\bar\eta_A \colon \mu_A \circ T \eta_A = 1_{T A} \Rightarrow \eta_A \circ a$ making $\eta_A$ into a lax $T$-morphism. Set $\theta_a = \bar\eta_A$. The [[adjunction|triangle equalities]] then require that:

\begin{enumerate}%
\item $a \bar\eta_A \colon a \Rightarrow a \circ \eta_A \circ a = a$ is equal to $1_a$. The composite $a \circ \bar\eta_A$ makes $a \circ \eta_A$ a lax $T$-morphism from $a$ to $a$ (paste $\bar\eta_A$ with the identity square $a \circ \mu_A = a \circ T a$). But $a \circ \eta_A = 1_A$, and $1_a$ also makes this into a lax $T$-morphism, so by uniqueness $a \bar\eta_A = 1_a$.


\item $\bar\eta_A \eta_A \colon \eta_A \Rightarrow \eta_A \circ a \circ \eta_A = \eta_A$ is equal to $1_{\eta_A}$. But this follows directly from the unit coherence condition for the lax $T$-morphism $\bar\eta_A$.



\end{enumerate}
Conversely, suppose $\theta_a$, algebras $a,b$ on $A,B$ and $f \colon A \to B$ are given. Take $\bar f$ to be the [[mate]] of $1_f \colon b \circ T f \circ \eta A = f \Rightarrow f$ with respect to the adjunctions $a \dashv \eta_A$ and $1 \dashv 1$, which is given in this case by pasting with $\theta_a$, so we have that $\bar f = b \circ T f \circ \theta_a$. The mate of $\bar f$ in turn is given by $\bar f \circ \eta_A$, which because mates correspond bijectively is equal to $1_f$. So $\bar f$ satisfies the unit condition.

Consider the diagrams expressing the multiplication condition: because $a \circ \mu_A = a \circ T a$ (and the same for $b$), their boundaries are equal, so we have 2-cells $\alpha, \beta \colon b \circ T b \circ T^2 f \Rightarrow f \circ a \circ T a$. Their mates under the adjunction $(T\theta_a, 1) \colon T a \dashv T\eta_A$ are given by pasting with $T \eta_A$. One is $\bar f$ pasted with $T \bar f \circ T \eta_A = T(f \circ \eta_A) = T 1_f = 1_{T f}$, and the other is given by composing $T \eta_A$ with the identity $\mu_B \circ T^2 f = T f \circ \mu_A$ (and then pasting with $\bar f$), but because $\mu_A \circ T \eta_A = 1_{T A}$ this is also equal to $1_{T f}$. The two original 2-cells are hence equal, because their mates are equal, and so $\bar f$ is indeed a lax $T$-morphism.

\end{proof}
Since $T$`s multiplication $\mu$ makes $T$ itself into a (generalized) $T$-algebra, the above implies (and in fact is implied by) the requirement that there exist a [[modification]] $\ell \colon 1_{T^2} \to \eta T \circ \mu$ making $(\ell,1) \colon \mu \dashv \eta T$. Conversely, given an algebra $a \colon T A \to A$, the 2-cell $\theta_a$ is given by $T a \circ \ell_A \circ T \eta_A$.

A different but equivalent condition is that there be a modification $d \colon T \eta \to \eta T$ such that $d \eta = 1$ and $\mu d = 1$; and given $\ell$ as above, $d$ is given by $\ell \circ T \eta$.

These various conditions can also be regarded as ways to say that the [[Eilenberg-Moore category|Eilenberg-Moore adjunction]] for $T$ is a [[lax-idempotent 2-adjunction]]. Thus, $T$ is a lax-idempotent 2-monad exactly when this 2-adjunction is lax-idempotent, and therefore also just when it is the 2-monad induced by \emph{some} lax-idempotent 2-adjunction.

Dually, for $T$ to be colax-idempotent, it is necessary and sufficient that any of the following hold.

\begin{itemize}%
\item For any $T$-algebra $a \colon T A \to A$ there is a 2-cell $\zeta_a \colon \eta_A \circ a \Rightarrow 1$ such that $(1,\zeta_a) \colon \eta_A \dashv a$.


\item There is a modification $m \colon \mu \circ \eta T \to 1$ making $(1,m) \colon \eta T \dashv \mu$.


\item There is a modification $e \colon \eta T \to T\eta$ such that $e\eta = 1$ and $\mu e = 1$.



\end{itemize}
\hypertarget{algebras}{}\subsection*{{Algebras}}\label{algebras}

Theorem \ref{AlgebraAdjoint} gives a necessary condition for an object $A$ to admit a $T$-algebra structure, namely that $\eta_A : A \to T A$ admit a left adjoint with identity counit. In the case of pseudo algebras, this necessary condition is also sufficient.

\begin{theorem}
\label{PseudoAlgebras}\hypertarget{PseudoAlgebras}{}
To give a pseudo $T$-algebra structure on an object $A$ is equivalently to give a left adjoint to $\eta_A : A\to T A$ with invertible counit.

\end{theorem}
In particular, an object admits at most one pseudo $T$-algebra structure, up to unique isomorphism. Thus, $T$-algebra structure is [[property-like structure]].

In many cases it is interesting to consider the pseudo $T$-algebras for which the algebra structure $T A \to A$ has a further left adjoint, forming an [[adjoint triple]]. Algebras of this sort are sometimes called [[continuous algebras]].

\hypertarget{examples}{}\subsection*{{Examples}}\label{examples}

As mentioned above, the standard examples of lax-idempotent 2-monads are those on $Cat$ whose algebras are categories with all colimits of a specified class. In this case, the 2-monad is a [[free cocompletion]] operation. Dually, there are colax-idempotent 2-monads which adjoin limits of a specified class. A converse is given by (\hyperlink{PowerCattaniWinskel}{PowerCattaniWinskel}), who show that a 2-monad is a monad for free cocompletions if and only if it is lax-idempotent and the unit $\eta$ is dense (plus a coherence condition).

Another important example of a colax-idempotent monad is the monad on $Cat/B$ that takes $p \colon E \to B$ to the projection $B/p \to p$ out of the [[comma category]]. The algebras for this monad are [[Grothendieck fibrations]] over $B$; see also [[fibration in a 2-category]]. The monad $p \mapsto p/B$ is lax-idempotent, and its algebras are opfibrations.

This latter is actually a special case of a general situation. If $T$ is a (2-)monad relative to which one can define [[generalized multicategories]], then often it induces a lax-idempotent 2-monad $\tilde{T}$ on the 2-category of such generalized multicategories (aka ``virtual $T$-algebras''), such that (pseudo) $\tilde{T}$-algebras are equivalent to (pseudo) $T$-algebras. When $T$ is the 2-monad whose algebras are strict 2-functors $B\to Cat$ and whose pseudo algebras are pseudofunctors $B\to Cat$, then a virtual $T$-algebra is a category over $B$, and it is a pseudo $\tilde{T}$-algebra just when it is an opfibration. Similarly, there is a lax-idempotent 2-monad on the 2-category of [[multicategories]] whose pseudo algebras are [[monoidal categories]], and so on.

\hypertarget{properties}{}\subsection*{{Properties}}\label{properties}

\begin{itemize}%
\item [[pseudo-distributive laws]] involving lax-idempotent 2-monads have an especially nice form; see \hyperlink{MarmolejoDL}{(Marmolejo)} and \hyperlink{WalkerDL}{(Walker)}.


\item For ordinary 1-monads there exists a presentation due to Manes as ``Kleisli triples'' with primary data a family of unit morphisms and lifts avoiding the iteration of the endofunctor. A similar presentation exists for lax-idempotent 2-monads as shown in Marmolejo-Wood (\hyperlink{MW12}{2012}). It is shown then in Walker \hyperlink{WalkerYS}{(2017)} that provided the units of this presentation are fully faithful (a reflection of the fully-faithfulness of the Yoneda embedding) (almost) all the axioms of a [[Yoneda structure]] are satisfied. In cases where size plays no role like e.g. the ideal completion of posets the two concepts coincide. For further details see at [[Yoneda structure]] or Walker \hyperlink{WalkerYS}{(2017)}.



\end{itemize}
\hypertarget{related_concepts}{}\subsection*{{Related concepts}}\label{related_concepts}

\begin{itemize}%
\item [[idempotent monad]]
\item [[2-monad]]
\item [[stuff, structure, property]]
\item [[property-like structure]]
\item [[continuous algebra]]
\item [[free cocompletion]]
\item [[lax-idempotent 2-adjunction]]

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

Classical references are

\begin{itemize}%
\item [[Max Kelly]], [[Steve Lack]], \emph{On property-like structures}, TAC 3(9), 1997. (\href{http://www.tac.mta.ca/tac/volumes/1997/n9/3-09abs.html}{abstract})


\item [[Anders Kock]], \emph{Monads for which structures are adjoint to units} , Aarhus Preprint 1972/73 No. 35. (\href{http://home.imf.au.dk/kock/msau1.PDF}{pdf})


\item [[Anders Kock]], \emph{Monads for which structures are adjoint to units}, JPAA 104:41--59, 1995.


\item [[Ross Street]], \emph{Fibrations in Bicategories} , Cah. Top. Géom. Diff. \textbf{XXI} no.2 (1980). (\href{http://www.numdam.org/item?id=CTGDC_1980__21_2_111_0}{numdam})


\item Volker Zöberlein, \emph{Doctrines on 2-categories} , Math. Zeitschrift \textbf{148} (1976) pp.267-279. (\href{https://gdz.sub.uni-goettingen.de/id/PPN266833020_0148?tify={%22pages%22:[273],%22view%22:%22info%22}}{gdz})



\end{itemize}
``Textbook'' accounts of the concept can be found in

\begin{itemize}%
\item [[Peter Johnstone]], \emph{Sketches of an elephant vol.1} , Oxford UP 2004. (B1.1.11, pp.250-54)


\item [[Marta Bunge]], [[Jonathon Funk]], \emph{Singular coverings of Toposes} , Springer Heidelberg 2006. (pp.79ff)



\end{itemize}
Special facets of the concept are studied in

\begin{itemize}%
\item [[Marta Bunge]], \emph{Tightly Bounded Completions} , TAC \textbf{28} no.8 (2013) pp.213-240. (\href{http://tac.mta.ca/tac/volumes/28/8/28-8abs.html}{abstract})


\item [[Marta Bunge]], [[Jonathon Funk]], \emph{On a bicomma object condition for KZ-doctrines} , JPAA \textbf{143} (1999) pp.69-105.


\item A. J. Power, G. L. Cattani, G. Winskel, \emph{A representation result for free cocompletions}, JPAA 151:273--286, 2000  



\end{itemize}
Their distributive laws come into focus in

\begin{itemize}%
\item Francisco Marmolejo, \emph{Distributive laws for pseudomonads}, \href{http://tac.mta.ca/tac/volumes/1999/n5/5-05abs.html}{TAC}

\end{itemize}
\begin{itemize}%
\item Francisco Marmolejo, [[Richard J. Wood]], \emph{Kan extensions and lax idempotent pseudomonads} , TAC \textbf{26} no.1 (2012) pp.1-19. (\href{http://www.tac.mta.ca/tac/volumes/26/1/26-01abs.html}{abstract})


\item Charles Walker, \emph{Distributive Laws via Admissibility}, \href{https://arxiv.org/abs/1706.09575}{arXiv}



\end{itemize}
The relation to Yoneda structures is due to

\begin{itemize}%
\item Charles Walker, \emph{Yoneda Structures and KZ Doctrines}, \href{https://arxiv.org/abs/1703.08693}{arxiv}

\end{itemize}
The logical-syntactical side is examined in

\begin{itemize}%
\item [[Jiri Adamek]], Lurdes Sousa, \emph{KZ-monadic categories and their logic}, \href{http://tac.mta.ca/tac/volumes/32/10/32-10abs.html}{tac}

\end{itemize}
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