PlanetPhysics/2 Category

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Definition 0.1

A small 2-category, ${𝒞}_2$, is the first of higher order [[../Cod/|categories]] constructed as follows.

1. define Cat as the category of [[../Cod/|small categories]] and [[../TrivialGroupoid/|functors]] #define a class of [[../TrivialGroupoid/|objects]] $A,B,...$ in ${𝒞}_2$ called `$0$- cells '
2. for all `$0$-cells' $A$, $B$, consider a set denoted as "${𝒞}_2(A,B)$" that is defined as

${\mathrm{hom}}_{{𝒞}_2}(A,B)$, 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 [[../FunctorCategories/|$2$-`morphisms']], or `$1$-cells' are defined by the [[../VariableCategory2/|natural transformations]] $\eta :F\to G$ for any two [[../TrivialGroupoid/|morphisms]] of $𝒞at$, (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 [[../Bijective/|domains]] and [[../Bijective/|codomains]] #the $2$-categorical [[../Cod/|composition]] of $2$-morphisms is denoted as "$\bullet$" and is called the [[../HorizontalIdentities/|vertical composition]]

1. a [[../HorizontalIdentities/|horizontal composition]], "$\circ$", is also defined for all triples of $0$-cells, $A$, $B$ and

$C$ in $𝒞at$ as the functor $\circ :{𝒞}_2(B,C)\times {𝒞}_2(A,B)={𝒞}_2(A,C),$ which is associative

1. the [[../Cod/|identities]] under horizontal composition are the identities of the $2$-cells of $1_X$

for any $X$ in $𝒞at$

1. for any object $A$ in $𝒞at$ there is a functor from the one-object/one-arrow category

$1^{‴}$ (terminal object) to ${𝒞}_2(A,A)$.

Examples of 2-categories

1. The $2$-category $𝒞at$ of small categories, functors, and natural transformations;
2. The $2$-category $𝒞at(ℰ)$ of internal categories in any category $ℰ$ with

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

1. When $ℰ=𝒮et$, this yields again the category $𝒞at$, but if $ℰ=𝒞at$, then one obtains the 2-category of small [[../HorizontalIdentities/|double categories]];
2. When $ℰ{=}^{‴}Grou{p}^{‴}$, one obtains the $2$-category of [[../CubicalHigherHomotopyGroupoid/|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 [[../Cod/|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 [[../Bijective/|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 [[../SuperCategory6/|supercategory]] with just two [[../Identity2/|composition laws]], and it is therefore an ${\mathrm{\S }}_1$-supercategory, because the ${\mathrm{\S }}_0$ supercategory is defined as a standard `$1$'-category subject only to the [[../ETACAxioms/|ETAC axioms]].