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.