# PlanetPhysics/2 Category

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### Definition 0.1

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

1. define Cat as the category of small categories and functors #define a class of objects $A,B,...$  in ${\mathcal {C}}_{2}$  called $0$ - cells '
2. 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 $2$ -morphisms', 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 #the $2$ -categorical composition of $2$ -morphisms is denoted as "$\bullet$ " and is called the vertical composition

1. a 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 associative

1. the identities under horizontal composition are the identities of the $2$ -cells of $1_{X}$

for any $X$  in ${\mathcal {C}}at$

1. for any object $A$  in ${\mathcal {C}}at$  there is a functor from the one-object/one-arrow category

$'''1'''$  (terminal object) to ${\mathcal {C}}_{2}(A,A)$ .

#### Examples of 2-categories

1. The $2$ -category ${\mathcal {C}}at$  of small categories, functors, and natural transformations;
2. The $2$ -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;

1. 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;
2. When ${\mathcal {E}}='''Group'''$ , one obtains the $2$ -category of crossed modules.

Remarks:

• 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: $(S\otimes T)\circ (S\otimes T)=(S\circ S)\otimes (T\circ T)$
• 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.