# PlanetPhysics/2 Category

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### Definition 0.1Edit

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

- define
**Cat**as the category of small categories and functors #define a class of objects in called ` -*cells*' - for all ` -cells' , , consider a set denoted as " " that is defined as

, with the elements of the latter set being the functors between the -cells and ; the latter is then organized as a small category whose -`morphisms', or ` -cells' are defined by the natural transformations for any two morphisms of , (with and being functors between the ` -cells' and , that is, ); as the ` -cells' can be considered as `2-morphisms' between 1-morphisms, they are also written as: , and are depicted as labelled faces in the plane determined by their domains and codomains #the -categorical composition of -morphisms is denoted as " " and is called the *vertical composition*

- a
*horizontal composition*, " ", is also defined for all triples of -cells, , and

in as the functor
which is *associative*

- the identities under horizontal composition are the identities of the -cells of

for any in

- for any object in there is a functor from the one-object/one-arrow category

(terminal object) to .

#### Examples of 2-categoriesEdit

- The -category of small categories, functors, and natural transformations;
- The -category of internal categories in any category with

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

- When , this yields again the category , but if , then one obtains the 2-category of small
*double categories*; - When , one obtains the
*-category of crossed modules*.

**Remarks:**

- In a manner similar to the (alternative) definition of small categories, one can describe -categories in terms of -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 -category is an example of a supercategory with just two composition laws, and it is therefore an -supercategory, because the supercategory is defined as a standard ` '-category subject only to the ETAC axioms.