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PlanetPhysics/2 Category

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Definition 0.1

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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-categories

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  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.