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\begin{document}
\subsection{Definition 0.1}
A small 2-category, $\mathcal{C}_2$, is the first of higher order \htmladdnormallink{categories}{http://planetphysics.us/encyclopedia/Cod.html} constructed as follows.
\begin{enumerate}
\item define \textbf{Cat} as the category of \htmladdnormallink{small categories}{http://planetphysics.us/encyclopedia/Cod.html} and \htmladdnormallink{functors}{http://planetphysics.us/encyclopedia/TrivialGroupoid.html} \item define a class of \htmladdnormallink{objects}{http://planetphysics.us/encyclopedia/TrivialGroupoid.html} $A, B,...$ in $\mathcal{C}_2$ called `$0$- \emph{cells}'
\item for all `$0$-cells' $A$, $B$, consider a set denoted as ``$\mathcal{C}_2 (A,B)$'' that is defined as
$\hom_{\mathcal{C}_2}(A,B)$, with the elements of the latter set being the functors between the $0$-cells $A$ and $B$; the latter is then organized as a small category whose \htmladdnormallink{$2$-`morphisms'}{http://planetphysics.us/encyclopedia/FunctorCategories.html}, or `$1$-cells' are defined by the \htmladdnormallink{natural transformations}{http://planetphysics.us/encyclopedia/VariableCategory2.html} $\eta: F \to G$ for any two \htmladdnormallink{morphisms}{http://planetphysics.us/encyclopedia/TrivialGroupoid.html} of $\mathcal{C}at$, (with $F$ and $G$ being functors between the `$0$-cells' $A$ and $B$, that is, $F,G: A \to B$); as the `$2$-cells' can be considered as `2-morphisms' between 1-morphisms, they are also written as: $\eta : F \Rightarrow G$, and are depicted as labelled faces in the plane determined by their \htmladdnormallink{domains}{http://planetphysics.us/encyclopedia/Bijective.html} and \htmladdnormallink{codomains}{http://planetphysics.us/encyclopedia/Bijective.html} \item the $2$-categorical \htmladdnormallink{composition}{http://planetphysics.us/encyclopedia/Cod.html} of $2$-morphisms is denoted as ``$\bullet$'' and is called the \emph{\htmladdnormallink{vertical composition}{http://planetphysics.us/encyclopedia/HorizontalIdentities.html}}
\item a \emph{\htmladdnormallink{horizontal composition}{http://planetphysics.us/encyclopedia/HorizontalIdentities.html}}, ``$\circ$'', is also defined for all triples of $0$-cells, $A$, $B$ and
$C$ in $\mathcal{C}at$ as the functor $$\circ: \mathcal{C}_2(B,C) \times \mathcal{C}_2(A,B) = \mathcal{C}_2(A,C),$$
which is \emph{associative}
\item the \htmladdnormallink{identities}{http://planetphysics.us/encyclopedia/Cod.html} under horizontal composition are the identities of the $2$-cells of $1_X$
for any $X$ in $\mathcal{C}at$
\item for any object $A$ in $\mathcal{C}at$ there is a functor from the one-object/one-arrow category
$\textbf{1}$ (terminal object) to $\mathcal{C}_2(A,A)$.
\end{enumerate}
\subsubsection{Examples of 2-categories}
\begin{enumerate}
\item The $2$-category $\mathcal{C}at$ of small categories, functors, and natural transformations;
\item The $2$-category $\mathcal{C}at(\mathcal{E})$ of \emph{internal categories in any category $\mathcal{E}$ with
finite limits}, together with the internal functors and the internal natural transformations between such internal
functors;
\item When $\mathcal{E} = \mathcal{S}et$, this yields again the category $\mathcal{C}at$, but if
$\mathcal{E} = \mathcal{C}at$, then one obtains the 2-category of small \emph{\htmladdnormallink{double categories}{http://planetphysics.us/encyclopedia/HorizontalIdentities.html}};
\item When $\mathcal{E} = \textbf{Group}$, one obtains the \emph{$2$-category of \htmladdnormallink{crossed modules}{http://planetphysics.us/encyclopedia/CubicalHigherHomotopyGroupoid.html}}.
\end{enumerate}
\textbf{Remarks:}
\begin{itemize}
\item In a manner similar to the (alternative) definition of small categories, one can
describe $2$-categories in terms of $2$-arrows. Thus, let us consider a set with two defined \htmladdnormallink{operations}{http://planetphysics.us/encyclopedia/Cod.html} $\otimes$, $\circ$, and also with units such that each operation endows the set with the structure of a
(strict) category. Moreover, one needs to assume that all $\otimes$-units are also $\circ$-units, and that an associativity \htmladdnormallink{relation}{http://planetphysics.us/encyclopedia/Bijective.html} holds for the two products:
$$(S \otimes T ) \circ (S \otimes T) = (S \circ S) \otimes (T \circ T)$$
\item A $2$-category is an example of a \htmladdnormallink{supercategory}{http://planetphysics.us/encyclopedia/SuperCategory6.html} with just two \htmladdnormallink{composition laws}{http://planetphysics.us/encyclopedia/Identity2.html}, and it
is therefore an $\S_1$-supercategory, because the $\S_0$ supercategory is defined as a standard `$1$'-category subject only to the \htmladdnormallink{ETAC axioms}{http://planetphysics.us/encyclopedia/ETACAxioms.html}.
\end{itemize}
\end{document}
