# PlanetPhysics/2 Category

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### Definition 0.1

A small 2-category, ${\displaystyle {\mathcal {C}}_{2}}$ , 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 ${\displaystyle A,B,...}$  in ${\displaystyle {\mathcal {C}}_{2}}$  called ${\displaystyle 0}$ - cells '
2. for all ${\displaystyle 0}$ -cells' ${\displaystyle A}$ , ${\displaystyle B}$ , consider a set denoted as "${\displaystyle {\mathcal {C}}_{2}(A,B)}$ " that is defined as

${\displaystyle \hom _{{\mathcal {C}}_{2}}(A,B)}$ , with the elements of the latter set being the functors between the ${\displaystyle 0}$ -cells ${\displaystyle A}$  and ${\displaystyle B}$ ; the latter is then organized as a small category whose ${\displaystyle 2}$ -morphisms', or ${\displaystyle 1}$ -cells' are defined by the natural transformations ${\displaystyle \eta :F\to G}$  for any two morphisms of ${\displaystyle {\mathcal {C}}at}$ , (with ${\displaystyle F}$  and ${\displaystyle G}$  being functors between the ${\displaystyle 0}$ -cells' ${\displaystyle A}$  and ${\displaystyle B}$ , that is, ${\displaystyle F,G:A\to B}$ ); as the ${\displaystyle 2}$ -cells' can be considered as 2-morphisms' between 1-morphisms, they are also written as: ${\displaystyle \eta :F\Rightarrow G}$ , and are depicted as labelled faces in the plane determined by their domains and codomains #the ${\displaystyle 2}$ -categorical composition of ${\displaystyle 2}$ -morphisms is denoted as "${\displaystyle \bullet }$ " and is called the vertical composition

1. a horizontal composition, "${\displaystyle \circ }$ ", is also defined for all triples of ${\displaystyle 0}$ -cells, ${\displaystyle A}$ , ${\displaystyle B}$  and

${\displaystyle C}$  in ${\displaystyle {\mathcal {C}}at}$  as the functor ${\displaystyle \circ :{\mathcal {C}}_{2}(B,C)\times {\mathcal {C}}_{2}(A,B)={\mathcal {C}}_{2}(A,C),}$  which is associative

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

for any ${\displaystyle X}$  in ${\displaystyle {\mathcal {C}}at}$

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

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

#### Examples of 2-categories

1. The ${\displaystyle 2}$ -category ${\displaystyle {\mathcal {C}}at}$  of small categories, functors, and natural transformations;
2. The ${\displaystyle 2}$ -category ${\displaystyle {\mathcal {C}}at({\mathcal {E}})}$  of internal categories in any category ${\displaystyle {\mathcal {E}}}$  with

finite limits, together with the internal functors and the internal natural transformations between such internal functors;

1. When ${\displaystyle {\mathcal {E}}={\mathcal {S}}et}$ , this yields again the category ${\displaystyle {\mathcal {C}}at}$ , but if ${\displaystyle {\mathcal {E}}={\mathcal {C}}at}$ , then one obtains the 2-category of small double categories;
2. When ${\displaystyle {\mathcal {E}}='''Group'''}$ , one obtains the ${\displaystyle 2}$ -category of crossed modules.

Remarks:

• In a manner similar to the (alternative) definition of small categories, one can describe ${\displaystyle 2}$ -categories in terms of ${\displaystyle 2}$ -arrows. Thus, let us consider a set with two defined operations ${\displaystyle \otimes }$ , ${\displaystyle \circ }$ , 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 ${\displaystyle \otimes }$ -units are also ${\displaystyle \circ }$ -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 ${\displaystyle 2}$ -category is an example of a supercategory with just two composition laws, and it is therefore an ${\displaystyle \S _{1}}$ -supercategory, because the ${\displaystyle \S _{0}}$  supercategory is defined as a standard ${\displaystyle 1}$ '-category subject only to the ETAC axioms.