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\begin{document}

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\section*{2-functor}

\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{pseudofunctor}{Pseudofunctor}\dotfill \pageref*{pseudofunctor} \linebreak
\noindent\hyperlink{lax_functor}{Lax Functor}\dotfill \pageref*{lax_functor} \linebreak
\noindent\hyperlink{strict_2functor}{Strict 2-Functor}\dotfill \pageref*{strict_2functor} \linebreak
\noindent\hyperlink{discussion}{Discussion}\dotfill \pageref*{discussion} \linebreak
\noindent\hyperlink{related_concepts}{Related concepts}\dotfill \pageref*{related_concepts} \linebreak


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

A \textbf{$2$-functor} is the [[categorification]] of the notion of a [[functor]] to the setting of [[2-category|2-categories]]. At the 2-categorical level there are several possible versions of this notion one might want depending on the given setting, some of which collapse to the standard definition of a functor between categories when considered on $2$-categories with discrete hom-categories (viewed as $1$-categories). The least restrictive of these is a [[lax functor]], and the strictest is (appropriately) called a [[strict 2-functor]].

For the various separate definitions that do collapse to standard functors, see:

\begin{itemize}%
\item [[strict 2-functor]]
\item [[pseudo functor]]

\end{itemize}
There is also a notion of `[[lax functor]]', however this notion does not necessarily yield a standard functor when considered on discrete hom-categories.

For the generalisation of this to higher categories, see [[semistrict higher category]].

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

Here we present explicitly the definition for the middling notion of a pseudofunctor, and comment on alterations that yield the stronger and weaker notions.

\hypertarget{pseudofunctor}{}\paragraph*{{Pseudofunctor}}\label{pseudofunctor}

Let $\mathfrak{C}$ and $\mathfrak{D}$ be [[2-categories]]. A pseudofunctor $F:\mathfrak{C}\to\mathfrak{D}$ consists of

\begin{itemize}%
\item A function $P:Ob_\mathfrak{C}\to Ob_\mathfrak{D}$, and for each pair of objects $A,B\in Ob_\mathfrak{C}$ a functor

\end{itemize}
$\backslash$begin\{centre\}

$P_{A,B}:\mathfrak{C}(A,B)\to\mathfrak{D}(P(A),P(B)).$

$\backslash$end\{centre\}

We will generally write the function and functors as $P$.

\begin{itemize}%
\item For each pair of horizontally composable 1-cells $(f,g)\in Ob_{\mathfrak{C}(B,C)}\times Ob_{\mathfrak{C}(A,B)}$, a $2$-cell isomorphism $\gamma_{f,g}:P(g\circ f)\Rightarrow P(g)\circ P(f)$ called the \emph{associator} as below

\end{itemize}
$\backslash$begin\{centre\}

$\backslash$begin\{xymatrix@C20mm\}

P(A) $\backslash$rtwocell{\tt \symbol{94}}\{P(g$\backslash$circ f)\}\_\{P(g)$\backslash$circ P(f)\}\{$\backslash$;$\backslash$;$\backslash$;$\backslash$;$\backslash$gamma\_\{f,g\}\} \& P(C)

$\backslash$end\{xymatrix\}

$\backslash$end\{centre\}

\begin{itemize}%
\item For each object object $A\in Ob_\mathfrak{C}$, a $2$-cell isomorphism $\iota_A:P(1_A)\Rightarrow1_{P(A)}$ called the \emph{unitor} as below

\end{itemize}
$\backslash$begin\{centre\}

$\backslash$begin\{xymatrix@C20mm\}

P(A) $\backslash$rtwocell{\tt \symbol{94}}\{P(1\_A)\}\_\{1\_\{P(A)\}\}\{$\backslash$;$\backslash$;$\backslash$;$\backslash$;$\backslash$iota\_A\} \& P(A)

$\backslash$end\{xymatrix\}

$\backslash$end\{centre\}

These are subject to the following two axioms:

\begin{enumerate}%
\item For any composable triplet of $1$-cells $(f,g,h)\in Ob_{\mathfrak{C}(C,D)}\times Ob_{\mathfrak{C}(B,C)}\times Ob_{\mathfrak{C}(A,B)}$ we have that

\end{enumerate}
$\backslash$begin\{centre\}

$\gamma_{f\circ g,h}\circ(\gamma_{f,g}\star 1_{P(h)})=\gamma_{f,g\circ h}\circ(1_{P(f)}\star\gamma_{g,h}),$

$\backslash$end\{centre\}

where $\circ$ denotes vertical composition and $\star$ denotes horizontal composition, as illustrated by the following commutative $2$-cell diagram in $\mathfrak{D}(P(A),P(D))$:

$\backslash$begin\{centre\}

$\backslash$begin\{xymatrix@R20mm@C20mm\}

P(f)$\backslash$circ P(g)$\backslash$circ P(h) $\backslash$ar@2\{-{\tt \symbol{62}}\}d\emph{\{$\backslash$gamma}\{f,g\}$\backslash$star 1\_\{P(h)\}\} $\backslash$ar@2\{-{\tt \symbol{62}}\}r{\tt \symbol{94}}\{1\_\{P(f)\}$\backslash$star$\backslash$gamma\_\{g,h\}\} \& P(f)$\backslash$circ P(g$\backslash$circ h)$\backslash$ar@2\{-{\tt \symbol{62}}\}d{\tt \symbol{94}}\{$\backslash$gamma\_\{f,g$\backslash$circ h\}\} $\backslash$ P(f$\backslash$circ g)$\backslash$circ P(h)$\backslash$ar@2\{-{\tt \symbol{62}}\}r\emph{\{$\backslash$gamma}\{f$\backslash$circ g,h\}\} \& P(f$\backslash$circ g$\backslash$circ h)

$\backslash$end\{xymatrix\}

$\backslash$end\{centre\}

\begin{enumerate}%
\item For any composable $1$-cells $(f,g)\in Ob_{\mathfrak{C}(B,C)}\times Ob_{\mathfrak{C}(A,B)}$ we have that

\end{enumerate}
$\backslash$begin\{centre\}

$\iota_B\star 1_{P(g)}=\gamma_{1_B,g},$

$\backslash$end\{centre\}

$\backslash$begin\{centre\}

$1_{P(f)}\star\iota_B=\gamma_{f,1_B},$

$\backslash$end\{centre\}

as illustrated by the commutative $2$-cell diagrams below

$\backslash$begin\{centre\}

$\backslash$begin\{xymatrix@R30mm@C35mm\}

P(A)$\backslash$rtwocell{\tt \symbol{94}}\{P(g)\}\_\{P(g)\}\{$\backslash$;$\backslash$;$\backslash$;$\backslash$;$\backslash$;$\backslash$;1\_\{P(g)\}\} $\backslash$drtwocell

$\backslash$end\{xymatrix\} ~ ~ ~ ~ ~ ~ $\backslash$begin\{xymatrix@R30mm@C35mm\}

P(B) $\backslash$rtwocell{\tt \symbol{94}}\{P(1\_B)\}\_\{1\_\{P(B)\}\}\{$\backslash$;$\backslash$;$\backslash$;$\backslash$;$\backslash$iota\_B\} $\backslash$drtwocell

$\backslash$end\{xymatrix\}

$\backslash$end\{centre\}

\hypertarget{lax_functor}{}\paragraph*{{Lax Functor}}\label{lax_functor}

To obtain the notion of a lax functor we only require that the associators $\gamma_{f,g}$ and unitors $\iota_A$ be $2$-cells, not necessarily $2$-cell isomorphisms -- this prevents us from going back and forth between preimages and images of identity $1$-cells and horizontally composed $1$-cells/$2$-cells.

\hypertarget{strict_2functor}{}\paragraph*{{Strict 2-Functor}}\label{strict_2functor}

To obtain the notion of a strict $2$-functor we require that the associators $\gamma_{f,g}$ and unitors $\iota_A$ be identity arrows, so horizontal composition and $1$-cell identities literally factor through each functor in the same way vertical composition and $2$-cell identities do.

\hypertarget{discussion}{}\subsubsection*{{Discussion}}\label{discussion}

There is a notion of a `weak 2-category', however it usually doesn't make sense to speak of strict $2$-functors between weak $2$-categories\footnote{Although there are certain contexts in which it does. For instance, there is a [[model structure]] on the category of [[bicategories]] and strict 2-functors between them, which models the homotopy theory of bicategories and weak 2-functors.} , but it does make sense to speak of lax (or `weak') $2$-functors between strict $2$-categories. Indeed, the weak $3$-[[3-category|category]] [[Bicat]] of bicategories, pseudofunctors, [[pseudonatural transformations]], and [[modifications]] is [[equivalence of categories|equivalent]] to its full sub-3-category spanned by the strict 2-categories. However, it is not equivalent to the $3$-category [[Str2Cat]] of strict $2$-categories, \emph{strict} $2$-functors, transformations, and modifications. (For discussion of the terminological choice ``$2$-functor'' and $n$-functor in general, see [[higher functor]].)

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

\begin{itemize}%
\item [[function]]


\item [[functor]]


\item \textbf{2-functor} / [[pseudofunctor]] / [[(2,1)-functor]]


\item [[n-functor]]


\item [[(∞,1)-functor]]


\item [[(∞,n)-functor]]



\end{itemize}
[[!redirects 2-functors]] [[!redirects strict 2-functor]] [[!redirects strict 2-functors]]



\end{document}