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\begin{document}

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\section*{vertical transformation}

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

\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{definition}{Definition}\dotfill \pageref*{definition} \linebreak
\noindent\hyperlink{the_double_category_of_double_functors}{The double category of double functors}\dotfill \pageref*{the_double_category_of_double_functors} \linebreak
\noindent\hyperlink{generalizations}{Generalizations}\dotfill \pageref*{generalizations} \linebreak


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

A \textbf{vertical transformation} is an analogue of a [[natural transformation]] which goes between [[double functors]] of [[double categories]], and whose components are vertical arrows and squares. There is a dual notion of \textbf{horizontal transformation.}

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

If $C$ and $D$ are strict double categories regarded as [[internal categories]] in $Cat$ and $F,G\colon C\to D$ are double functors regarded as internal functors in $Cat$, then a transformation between them is simply an internal natural transformation in $Cat$. Whether this is a vertical or horizontal transformation depends on how we identify double categories with internal categories in $Cat$ (there being two ways).

More explicitly, a vertical transformation $\alpha\colon F\to G$ consists of

\begin{itemize}%
\item For every object $c\in C$, a vertical arrow

\begin{displaymath}
\itexarray{ F c \\ \downarrow ^{\alpha_c} \\ G c}
\end{displaymath}
in $D$, which are natural with respect to vertical composition of vertical arrows in $C$.


\item For every horizontal arrow $p\colon c_1 \to c_2$ in $C$, a square

\begin{displaymath}
\itexarray{F c_1 & \overset{F p}{\to} & F c_2\\
  ^{\alpha_{c_1}}\downarrow & \Downarrow^{\alpha_p}& \downarrow^{\alpha_{c_2}}\\
  G c_1& \underset{G p}{\to} & G c_2}
\end{displaymath}
in $D$, which are natural with respect to vertical composition of squares in $C$.


\item For each $c\in C$, if $1_c\colon c\to c$ is its horizontal identity, then the square $\alpha_{1_c}$ is equal to $1_{\alpha_c}$, the identity square on the arrow $\alpha_c$.


\item For $p\colon c_1\to c_2$ and $q\colon c_2\to c_3$, the horizontal composite of $\alpha_p$ and $\alpha_q$ is equal to $\alpha_{q p}$.



\end{itemize}
The notion of horizontal transformation is dual.

\hypertarget{the_double_category_of_double_functors}{}\subsection*{{The double category of double functors}}\label{the_double_category_of_double_functors}

Another characterization of transformations between double categories comes from observing that the 1-category $DblCat$ is [[cartesian closed category|cartesian closed]], and so any two double categories have an exponential $D^C$. The objects of $D^C$ are double functors, its vertical arrows are vertical transformations, and its horizontal arrows are horizontal transformations. Its squares are a sort of ``square modification'' relating a pair of vertical and a pair of horizontal transformations.

\hypertarget{generalizations}{}\subsection*{{Generalizations}}\label{generalizations}

It is easy to modify the explicit definition to handle the cases when $C$ and $D$ are weak in one direction or the other, and/or when $F$ and $G$ are pseudo functors in one direction or the other, by composing with appropriate coherence constraints. In this way, we obtain many [[2-categories]] of double categories.

It is also easy to define \emph{vertical} transformations between double functors which are \emph{horizontally} lax or colax, and dually. In fact, given double categories $C,D,C',D'$, lax functors $F\colon C\to D$ and $F'\colon C'\to D'$, and colax functors $G\colon C\to C'$ and $G'\colon D\to D'$, we can define a vertical transformation having the shape

\begin{displaymath}
\itexarray{C & \overset{F}{\to} & D \\
  ^G\downarrow & \Downarrow& \downarrow^{G'}\\
  C'& \underset{F'}{\to} & D'}
\end{displaymath}
despite the fact that the composites $G' \circ F$and $F'\circ G$ do not exist as double functors of any sort. Such transformations are the squares of a large double category $Dbl$ whose objects are double categories, whose horizontal arrows are lax functors, and whose vertical arrows are colax functors. This, in turn, is a special case of a construction which works for algebras over any [[2-monad]].

Finally, we can also define vertical transformations between functors of (horizontally) [[virtual double categories]].

[[!redirects horizontal transformation]] [[!redirects vertical transformations]] [[!redirects horizontal transformations]]



\end{document}