0.1 Definition 0.1

A small 2-category, 𝒞₂, is the first of higher order categories constructed as follows.

1.

define Cat as the category of small categories and functors

2.

define a class of objects A,B,... in 𝒞₂ called ‘0- cells’

3.

for all ‘0-cells’ A, B, consider a set denoted as “𝒞₂(A,B)” that is defined as hom _𝒞2(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 η : F → G for any two morphisms of 𝒞at, (with F and G being functors between the ‘0-cells’ A and B, that is, F,G : A → B); as the ‘2-cells’ can be considered as ‘2-morphisms’ between 1-morphisms, they are also written as: η : F ⇒ G, and are depicted as labelled faces in the plane determined by their domains and codomains

4.

the 2-categorical composition of 2-morphisms is denoted as “∙” and is called the vertical composition

5.

a horizontal composition, “∘”, is also defined for all triples of 0-cells, A, B and C in 𝒞at as the functor

which is associative

6.

the identities under horizontal composition are the identities of the 2-cells of 1_X for any X in 𝒞at

7.

for any object A in 𝒞at there is a functor from the one-object/one-arrow category 1 (terminal object) to 𝒞₂(A,A).

0.1.1 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;

3.

When ℰ = 𝒮et, this yields again the category 𝒞at, but if ℰ = 𝒞at, then one obtains the 2-category of small double categories;

4.

When 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 ⊗, ∘, 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 ⊗-units are also ∘-units, and that an associativity relation holds for the two products:
  
• A 2-category is an example of a supercategory with just two composition laws, and it is therefore an §₁-supercategory, because the §₀ supercategory is defined as a standard ‘1’-category subject only to the ETAC axioms.