PlanetPhysics/2 Category

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Definition 0.1Edit

A small 2-category,  , 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   in   called ` - cells '
  2. 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

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

  in   as the functor   which is associative

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

for any   in  

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

  (terminal object) to  .

Examples of 2-categoriesEdit

  1. The  -category   of small categories, functors, and natural transformations;
  2. 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;

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