2-category2Category2008-08-10 04:35:532009-02-03 05:30:36Definitionsmall $Cat$-category$0$-cell$2$-cell$3$-cell$2$-morphism2-categorical compositionhorizontal compositionvertical composition2-morphism0-cell2-categorysmall $Cat$-categoryfunctor categorieshigher order categorieshigher dimensional algebra2- and 3- cells and morphisms% this is the default PlanetMath preamble. as your knowledge
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\newcommand{\hr}{{\hookrightarrow}} \begin{definition}
A small $2$-category, $\mathcal{C}_2$, is the first of higher order n-categories
constructed as follows.
\begin{enumerate}
\item Define $\mathcal{C}at$ as the category of small categories and functors
\item Define a class of objects $A, B,...$ in $\mathcal{C}at$ 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
\PMlinkname{$\hom_{\mathcal{C}_2}(A,B)$}{Multifunctor}, 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 \PMlinkname{$2$-`morphisms'}{FunctorCategories}, or `$1$-cells' are defined by the natural transformations $\eta: F \to G$ for any two morphisms 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 domains and codomains
\item The $2$-categorical composition of $2$-morphisms is denoted as ``$\bullet$'' and is called the \emph{vertical composition}
\item A \emph{horizontal composition}, ``$\circ$'', 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 identities 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}
\end{definition}
\subsection{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{double categories};
\item When $\mathcal{E} = \textbf{Group}$, one obtains the \emph{$2$-category of crossed modules}.
\end{enumerate}
\subsection{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 operations
$\otimes$, $\circ$, 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 relation 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 supercategory with just two composition laws, and it
is therefore an $\S_1$-supercategory, because the $\S_0$ supercategory is defined as a standard `$1$'-category subject only to the ETAC axioms.
\end{itemize}