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\begin{document}

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\section*{lax functor}

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

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

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

\hypertarget{higher_category_theory}{}\paragraph*{{Higher category theory}}\label{higher_category_theory}

[[!include higher 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{examples}{Examples}\dotfill \pageref*{examples} \linebreak
\noindent\hyperlink{literature}{Literature}\dotfill \pageref*{literature} \linebreak


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

A \textbf{lax functor} or \textbf{lax $n$-[[n-functor|functor]]} is a morphism of $n$-[[n-category|categories]] that is allowed to have structural cells -- compositors, associators, etc -- that need not be invertible (not even [[weak inverse|weakly]]).

This is to distinguish from [[pseudofunctor]] for which all these cells are required to be [[equivalences]].

This means that the definition of lax functor involves a choice of orientation of these structural cells which is not visible for pseudofunctors. The choice is such that the first example below comes out as stated. With the opposite choice one speaks of an \textbf{oplax functor}.

Often the term lax functor is used for $n$-functors $F : C \to D$ whose domain $C$ is an ordinary [[category]] (regarded as an $n$-category with only trivial higher morphisms), while the codomain $D$ is often taken to be a [[2-category]].

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

See the definition at [[pseudofunctor]], and let the [[natural isomorphisms]] in that definition be merely [[natural transformations]].

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

\begin{itemize}%
\item For $D$ a [[bicategory]], lax functors $F : {*} \to D$ from the [[point]] category to $D$ are equivalent to [[monad]]s in $D$.

The compositor of the lax functor is the monad product, the unitor is the monad unit.

\begin{itemize}%
\item So in particular for $V$ a [[monoidal category]] and $\mathbf{B}V$ its one-object [[delooping]] [[bicategory]], lax functors ${*} \to \mathbf{B}V$ are equivalent to [[monoid]]s in $V$.

\end{itemize}

\item Similarly, oplax functors ${*} \to D$ are equivalent to [[comonad]]s in $D$.


\item If $C$ is the [[codiscrete category]] on a set $S$, and $D$ is a [[bicategory]], lax functors $F : C \to D$ are the same as categories enriched in $D$ having $S$ as their set of objects.

\begin{itemize}%
\item In particular, if $C = {*}$, then this example reduces to the first one.


\item Another special case arises when $D = \mathbf{B}V$ for some monoidal category $V$. Then lax functors $F : C \to D$ are the same as categories enriched in the monoidal category $V$.



\end{itemize}

\item It makes sense to ask that a functor is lax \emph{and} oplax in a compatible way such that ${*} \to D$ yields [[Frobenius algebra|Frobenius]] monads.

This is of relevance in [[conformal field theory]] where Frobenius algebra objects in [[modular tensor category|modular tensor categories]] and bimodules over them play a central role.

Some old remarks on this case are in \href{http://www.math.uni-hamburg.de/home/schreiber/LaxFunc.pdf}{Note on lax functors and bimodules}.

This relation between lax-oplax functors and [[conformal field theory]] was developed in detail in

\begin{itemize}%
\item Liang Kong, Ingo Runkel, \emph{Cardy algebras and sewing constraints, I} (\href{http://arxiv.org/abs/0807.3356}{arXiv})

\end{itemize}
A general discussion of lax-oplax functors is in section 2.1 there.



\end{itemize}
Isn't it odd not to require any extra condition at all on the coherence morphisms? I would have expected a definition where they are required to be split epi, or require that the codomain be a subobject of the domain. Is there a name for something like that?

[[Mike Shulman]]: One could certainly add that as a condition, but I don't think I've ever heard of anyone having a use for it, or giving it a name. The interesting examples listed above (and others) don't use any such condition.

\hypertarget{literature}{}\subsection*{{Literature}}\label{literature}

\begin{itemize}%
\item R. Street, \emph{Two constructions on lax functors}, Cahiers de Topologie et Géométrie Différentielle Catégoriques, Volume 13 (1972) no. 3 , p. 217-264 \href{http://www.numdam.org/item?id=CTGDC_1972__13_3_217_0}{numdam}

\end{itemize}
[[!redirects lax functors]] [[!redirects oplax functor]] [[!redirects oplax functors]] [[!redirects colax functor]] [[!redirects colax functors]] [[!redirects lax n-functor]] [[!redirects lax n-functors]] [[!redirects oplax n-functor]] [[!redirects oplax n-functors]] [[!redirects colax n-functor]] [[!redirects colax n-functors]] [[!redirects lax 2-functor]] [[!redirects lax 2-functors]] [[!redirects oplax 2-functor]] [[!redirects oplax 2-functors]] [[!redirects colax 2-functor]] [[!redirects colax 2-functors]]



\end{document}