PlanetMath: n-category

Definition 0.1 a small $n$ -category , $\mathcal{C}_n$ , is the $n$ -th order category of (small) $n$ -categories $n$ -$\mathcal{C}at$ constructed by induction on $n$ in two main stages:

define the category $0$ -Cat as the category $\mathcal{S}et$ of sets and functions;
define the category $(n+1)-\mathcal{C}at$ as the category of ($n$ ) categories enriched over the category $\mathcal{C}_n$ . The construction is simplified by beginning with the definition of the 2-category.
The following, more detailed recursive construction of $n-\mathcal{C}at$ utilizes the fact that if a category $\mathcal{C}$ has finite products, the category of $\mathcal{C}$ -enriched categories also has finite products.

define $\mathcal{C}at$ , or category $1-\mathcal{C}at$ as the category of small categories and functors;
define a class of objects $A, B,...$ in $\mathcal{C}at$ called `$0$ -cells';
for all `$0$ -cells' $A$ , $B$ , consider the set $Hom_{\mathcal{C}_2}(A, B)$ , or $\mathcal{C}_2(A,B)$ , organized as a small category, whose $2$ -morphisms, or `$1$ -cells', are defined as natural transformations called `$2$ -cells', $\eta: F \to G$ for any two `morphisms' of $\mathcal{C}at$ , with $F$ and $G$ being functors between the `$0$ -cells' $A$ and $B$ , $F,G: A \to B$ );
the 2-categorical composition is denoted as ``$ \bullet$ " and is called the vertical composition;
a horizontal composition, ``$\circ$ ", is 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 associative;
the identities under horizontal composition are the identities of the $2$ -cells of $1_X$ for any $X$ in $\mathcal{C}at$ ;
for any object $A$ in $\mathcal{C}at$ there is a functor from the one-object/one-arrow category $1$ (terminal object) to $\mathcal{C}_2(A,A)$ .
repeat the last $(n-1)$ steps to define `3'-cells, ..., to $n$ -cells; the resulting structure is called an $n$ -category, but it is in fact a metagraph, metacategory, or more generally, a $\S_{n-1}$ -supercategory with $n$ composition laws and it is also called more recently a higher order category or a higher dimensional algebra.

Note Because 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.