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\begin{document}
\subsection{Background}
Charles Ehresmann defined in 1963 a {\em double category} $\mathcal{D}$ as an internal \htmladdnormallink{category}{http://planetphysics.us/encyclopedia/Cod.html} in the category of \htmladdnormallink{small categories}{http://planetphysics.us/encyclopedia/Cod.html} $\bf{Cat}$.
\subsection{Double category definition}
\begin{definition}
A double category $\mathcal{D}$ consists of:
\begin{itemize}
\item a set of \htmladdnormallink{objects}{http://planetphysics.us/encyclopedia/TrivialGroupoid.html},
\item a set of horizontal \htmladdnormallink{morphisms}{http://planetphysics.us/encyclopedia/TrivialGroupoid.html} $$f: A \to B,$$
\item a set of vertical morphisms $$j: A \to C,$$ and
\item a class of \htmladdnormallink{squares}{http://planetphysics.us/encyclopedia/PiecewiseLinear.html} with source and target as shown in the following \htmladdnormallink{diagrams}{http://planetphysics.us/encyclopedia/TrivialGroupoid.html}: $$\begin{xy}
*!C\xybox{
\xymatrix{
{A}\ar[r]^{f}\ar[d]_{k}&{B}\ar[d]^{g}\\
{C}\ar[r]_{h}&{D}
} }\end{xy}$$
\end{itemize}
with \htmladdnormallink{compositions}{http://planetphysics.us/encyclopedia/Cod.html} and units of the double category that satisfy the following axioms:
\begin{itemize}
\item \emph{i.} Horizontal:
\[
A\buildrel f_1 \over \longrightarrow
B \buildrel f_2 \over \longrightarrow
C = [f_1, f_2]= f_2 \circ f_1
\]
\[
A\buildrel 1^h_A \over \longrightarrow
A \buildrel f_1 \over \longrightarrow
B = A\buildrel f_1 \over \longrightarrow
B = A \buildrel f_1 \over \longrightarrow
B \buildrel 1^h_B \over \longrightarrow
B
\]
\item \emph{ii.} Vertical:
\[
[A\buildrel j_1 \over \longrightarrow
B \buildrel j_2 \over \longrightarrow
C]_{vert} = [j_1, j_2]_{vert.}= j_2 \circ j_1
\]
\[
[A\buildrel 1^v_A \over \longrightarrow
A \buildrel j_1 \over \longrightarrow
B = A\buildrel j_1 \over \longrightarrow
B = A \buildrel j_1 \over \longrightarrow
B \buildrel 1^v_B \over \longrightarrow
B]_{vert.}
\]
\emph{Compositions for \htmladdnormallink{square diagrams}{http://planetphysics.us/encyclopedia/Commutativity.html} in a double category $\mathcal{D}$:}
\item \emph{iii.} Horizontal composition:
$$\xymatrix{
{A}\ar[r]^{f_1}\ar[d]_{j}&{B}\ar[d]^{k}\\
{D}\ar[r]_{g_1}&{E}}[[User:MaintenanceBot|MaintenanceBot]] ([[User talk:MaintenanceBot|discuss]] • [[Special:Contributions/MaintenanceBot|contribs]]) 20:49, 25 June 2015 (UTC)[\alpha]``\circ'' \xymatrix{
{B}\ar[r]^{f_2}\ar[d]_{k}&{C}\ar[d]^{l}\\
{E}\ar[r]_{g_2}&{F}}[[User:MaintenanceBot|MaintenanceBot]] ([[User talk:MaintenanceBot|discuss]] • [[Special:Contributions/MaintenanceBot|contribs]]) 20:49, 25 June 2015 (UTC)[\beta] = \xymatrix{
{A}\ar[r]^{[f_1f_2]}\ar[d]_{j}&{C}\ar[d]^{l}\\
{D}\ar[r]_{g_1g_2}&{F}} [[User:MaintenanceBot|MaintenanceBot]] ([[User talk:MaintenanceBot|discuss]] • [[Special:Contributions/MaintenanceBot|contribs]]) 20:49, 25 June 2015 (UTC)[\alpha \beta].$$
\item \emph{iv.} Vertical composition of squares in $\mathcal{D}$:
${[\alpha \beta]}_{vert.}$ is expressed as
$$\xymatrix{
{A}\ar[r]^{f}\ar[d]_{[j_1 j_2]_v}&{B}\ar[d]^{[k_1 k_2]_v}\\
{E}\ar[r]_{h}&{F}}[[User:MaintenanceBot|MaintenanceBot]] ([[User talk:MaintenanceBot|discuss]] • [[Special:Contributions/MaintenanceBot|contribs]]) 20:49, 25 June 2015 (UTC)[\alpha \beta]_v.$$
\end{itemize}
\end{definition}
Moreover, all compositions are associative and unital, and also subject to the Interchange Law:
$$\xymatrix{
{[\alpha]}\ar[r]^{--}\ar[d]_{|}&{[\beta]}\ar[d]^{|}\\
{[\gamma]}\ar[r]_{--}&{[\delta]}
} =
{[ [\alpha \beta] ~~over~~ [\gamma \delta]]}_{vert.} = [\alpha \gamma]_v \circ [\beta \delta]_v.$$
Unit morphisms are also subject to the axioms of the double category. For further details on double categories and examples please see the related \htmladdnormallink{free download PDF file}{http://www.math.uchicago.edu/~fiore/1/fiorefolding.pdf}.
\end{document}
