PlanetPhysics/2 Category
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Definition 0.1
[edit | edit source]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-categories
[edit | edit source]- 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.