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2-category

(Definition)

Definition 0.1

A small 2-category, $\mathcal{C}_2$ , is the first of higher order categories constructed as follows.

define Cat as the category of small categories and functors
define a class of objects in $\mathcal{C}_2$ called `0- cells'
for all `0-cells' , , 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 and ; the latter is then organized as a small category whose -`morphisms', or `-cells' are defined by the natural transformations $\eta: F \to G$ for any two morphisms of $\mathcal{C}at$ , (with and being functors between the `0-cells' and , that is, $F,G: A \to B$ ); as the `-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
the -categorical composition of -morphisms is denoted as “ $\bullet$ ” and is called the vertical composition
a horizontal composition, “ $\circ$ ”, is also defined for all triples of 0-cells, , and in $\mathcal{C}at$ as the functor

$\displaystyle \circ: \mathcal{C}_2(B,C) \times \mathcal{C}_2(A,B) = \mathcal{C}_2(A,C),$
which is associative
the identities under horizontal composition are the identities of the -cells of for any in $\mathcal{C}at$
for any object in $\mathcal{C}at$ there is a functor from the one-object/one-arrow category $\textbf{1}$ (terminal object) to $\mathcal{C}_2(A,A)$ .

The -category $\mathcal{C}at$ of small categories, functors, and natural transformations;
The -category $\mathcal{C}at(\mathcal{E})$ of internal categories in any category $\mathcal{E}$ with finite limits, together with the internal functors and the internal natural transformations between such internal functors;
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 double categories;
When $\mathcal{E} = \textbf{Group}$ , one obtains the -category of crossed modules.

Remarks:

In a manner similar to the (alternative) definition of small categories, one can describe -categories in terms of -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:

$\displaystyle (S \otimes T ) \circ (S \otimes T) = (S \circ S) \otimes (T \circ T)$
A -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 `'-category subject only to the ETAC axioms.