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\begin{document}

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\section*{lax morphism}

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

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

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

\hypertarget{lax_morphisms}{}\section*{{Lax morphisms}}\label{lax_morphisms}

\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{definition}{Definition}\dotfill \pageref*{definition} \linebreak
\noindent\hyperlink{for_algebras_of_2monads}{For algebras of 2-monads}\dotfill \pageref*{for_algebras_of_2monads} \linebreak
\noindent\hyperlink{for_coalgebras_of_2comonads}{For coalgebras of 2-comonads}\dotfill \pageref*{for_coalgebras_of_2comonads} \linebreak
\noindent\hyperlink{examples}{Examples}\dotfill \pageref*{examples} \linebreak
\noindent\hyperlink{categories_of_lax_morphisms}{Categories of lax morphisms}\dotfill \pageref*{categories_of_lax_morphisms} \linebreak
\noindent\hyperlink{related_pages}{Related pages}\dotfill \pageref*{related_pages} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak


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

In general, if $A$ and $B$ are [[categories]] (or, more generally, any category-like things, such as [[objects]] of some [[2-category]]) equipped with [[algebraic structure]], a \emph{lax morphism} $f\colon A\to B$ is one which ``preserves'' the algebraic structure only up to a not-necessarily invertible transformation.

Of course, this transformation goes in one particular direction; a \emph{colax morphism} is one where the transformation goes in the other direction. The case of 2-monads, below, provides an almost universally applicable way to decide which direction is ``lax'' and which is ``colax''.

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

\hypertarget{for_algebras_of_2monads}{}\subsubsection*{{For algebras of 2-monads}}\label{for_algebras_of_2monads}

Let $T$ be a [[2-monad]] on a 2-category $K$, and let $A$ and $B$ be (strict, pseudo, or even lax or colax) $T$-algebras. A \textbf{lax $T$-morphism} $f\colon A\to B$ is a morphism in $K$ together with a [[2-cell]]

\begin{displaymath}
\itexarray{ T A & \overset{T f}{\to} & T B\\
^{a} \downarrow & \swArrow & \downarrow^{b}\\
A & \underset{f}{\to} & B}
\end{displaymath}
satisfying some axioms.

If the 2-cell goes in the other direction, then we say $f$ is a \textbf{colax $T$-morphism} (or \textbf{oplax $T$-morphism}). Equivalently, a colax $T$-morphism is a lax $T^{co}$-morphism, where $T^{co}$ is the induced 2-monad on the 2-cell dual $K^{co}$ (see [[opposite 2-category]]).

If the 2-cell is invertible, we call $f$ a \textbf{pseudo} or \textbf{strong} $T$-morphism.

\hypertarget{for_coalgebras_of_2comonads}{}\subsubsection*{{For coalgebras of 2-comonads}}\label{for_coalgebras_of_2comonads}

Let $W$ be a 2-comonad on $K$, i.e. a 2-monad on the 1-cell dual $K^{op}$, and let $C$ and $D$ be $W$-coalgebras. A \textbf{lax $W$-morphism} $f\colon C\to D$ is a morphism in $K$ together with a 2-cell

\begin{displaymath}
\itexarray{ C & \overset{T f}{\to} & D\\
^{c} \downarrow & \swArrow & \downarrow^{d}\\
W C & \underset{f}{\to} & W D}
\end{displaymath}
satisfying some axioms.

Note that a lax morphism of algebras for the 2-comonad $W$ is a \emph{colax} morphism of algebras for the 2-monad $W^{op}$. The reason we choose to call this direction for coalgebras ``lax'' is that if $T$ is a 2-monad with a right [[adjoint functor|adjoint]] $T^*$, then $T^*$ automatically becomes a 2-comonad such that $T^*$-coalgebras are the same as $T$-algebras, and with the above definition, lax $T$-morphisms coincide with lax $T^*$-morphisms.

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

\begin{itemize}%
\item A [[lax monoidal functor]] is a lax morphism for the 2-monad on [[Cat]] whose algebras are [[monoidal categories]]. Similarly, an [[oplax monoidal functor]] is a colax morphism for this 2-monad.


\item A [[lax natural transformation]] between [[2-functors]] $C\to D$ is a lax morphism for the 2-monad on $[ob(C),D]$ whose algebras are 2-functors (which exists if $D$ is cocomplete and $C$ is small). Similarly, an oplax natural transformation is a colax morphism for this 2-monad. If $D$ is also complete, then this 2-monad has a right adjoint, which then as usual becomes a 2-comonad whose coalgebras are also 2-functors. The above conventions for lax morphisms between coalgebras mean that a lax natural transformation is unambiguously ``lax'' rather than ``colax'', whether we regard the 2-functors as algebras for a 2-monad or coalgebras for a 2-comonad.

Some authors have tried to change the traditional meanings of ``lax'' and ``colax'' in this case, but the general framework of 2-monads gives a good argument for keeping it this way (even if in this particular case, oplax transformations are more common or useful).


\item A [[lax functor]] between 2-categories is a lax morphism for the 2-monad on Cat-graphs whose algebras are 2-categories.


\item A [[lax algebra for a 2-monad]] $T$ is a lax morphism $T\to \langle A,A\rangle$ for the 2-monad whose algebras are 2-monads, where $\langle A,A\rangle$ is the [[codensity monad]] of the object $A$.


\item If $T$ is a [[lax-idempotent 2-monad]], then (by definition) \emph{every} morphism in the underlying 2-category $K$ between (the objects underlying) $T$-algebras has a unique structure of lax $T$-morphism. For instance, every functor between categories with (some class of) [[colimits]] is a lax morphism for the 2-monad which assigns those colimits; the unique lax structure map is the canonical comparison $colim (F\circ D) \to F(colim D)$. Such a morphism is strong/pseudo exactly when it preserves the colimits in question.


\item For [[probability monads]] on a [[locally posetal 2-category]], such as the [[Radon monad\#the\_ordered\_case|ordered Radon monad]], the lax morphisms of algebras corresponds to [[concave maps]] or a suitable generalization thereof.



\end{itemize}
\hypertarget{categories_of_lax_morphisms}{}\subsection*{{Categories of lax morphisms}}\label{categories_of_lax_morphisms}

For any 2-monad $T$, there are a 2-categories:

\begin{itemize}%
\item $T Alg_l$ of $T$-algebras and lax morphisms
\item $T Alg_c$ of $T$-algebras and colax morphisms
\item $T Alg_p$ (frequently written just $T Alg$) of $T$-algebras and pseudo morphisms
\item (if $T$ is strict) $T Alg_s$ of $T$-algebras and strict morphisms

\end{itemize}
We have obvious 2-functors

\begin{displaymath}
\itexarray{ & & & & T Alg_l \\
  & & & \nearrow\\
  T Alg_s & \to & T Alg_p\\
  & & & \searrow\\
 & & & & T Alg_c }
\end{displaymath}
which are [[bijective on objects functor|bijective on objects]], [[faithful functor|faithful]] on 1-cells, and [[locally fully faithful 2-functor|locally fully faithful]].

Therefore, we can also assemble a number of [[F-categories]] of $T$-algebras and any suitable pair of types of $T$-morphism: strict+pseudo, strict+lax, strict+colax, pseudo+lax, or pseudo+colax.

If we want to consider both lax and colax $T$-morphisms together, the natural structure is a [[double category]]: there is a straightforward definition of the squares in a double category whose vertical arrows are colax $T$-morphisms and whose horizontal arrows are lax ones. We could then, if we wish, add some ``F-ness'' to incorporate pseudo and/or strict morphisms as well.

The 2-category $T Alg_p$ is fairly well-behaved; for strict $T$, it admits all strict [[PIE-limits]] (if the base 2-category does), and therefore all [[2-limits]] (i.e. bilimits). When $T$ is [[accessible functor|accessible]], $T Alg_p$ admits all 2-colimits as well (but not, in general, many strict 2-colimits).

However, the 2-categories $T Alg_l$ and $T Alg_c$ are not so well-behaved; they do not have many limits or colimits. But once we enhance them to [[F-categories]], they admit all [[rigged limits]]. All three 2-categories also admit [[lax morphism classifier|morphism classifiers]]; that is, the inclusions $T Alg_s \to T Alg_*$ have left 2-adjoints.

\hypertarget{related_pages}{}\subsection*{{Related pages}}\label{related_pages}

\begin{itemize}%
\item [[2-monad]]


\item [[lax morphism classifier]]



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

\begin{itemize}%
\item Blackwell, [[Max Kelly|Kelly]], [[John Power|Power]]. ``2-dimensional monad theory''

\end{itemize}
[[!redirects lax morphisms]] [[!redirects colax morphism]] [[!redirects colax morphisms]] [[!redirects oplax morphism]] [[!redirects oplax morphisms]] [[!redirects pseudo morphism]] [[!redirects pseudo morphisms]]



\end{document}