PlanetPhysics/2 Category

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Definition 0.1
A small 2-category, $$\mathcal{C}_2$$, is the first of higher order categories constructed as follows.

$$\hom_{\mathcal{C}_2}(A,B)$$, 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 $$\eta: F \to G$$ for any two morphisms of $$\mathcal{C}at$$, (with $$F$$ and $$G$$ being functors between the `$$0$$-cells' $$A$$ and $$B$$, that is, $$F,G: A \to B$$); as the `$$2$$-cells' can be considered as `2-morphisms' between 1-morphisms, they are also written as: $$\eta : F \Rightarrow G$$, and are depicted as labelled faces in the plane determined by their domains and codomains #the $$2$$-categorical composition of $$2$$-morphisms is denoted as "$$\bullet$$" and is called the vertical composition $$C$$ in $$\mathcal{C}at$$ as the functor $$\circ: \mathcal{C}_2(B,C) \times \mathcal{C}_2(A,B) = \mathcal{C}_2(A,C),$$ which is associative for any $$X$$ in $$\mathcal{C}at$$ $$1 $$ (terminal object) to $$\mathcal{C}_2(A,A)$$.
 * 1) define Cat as the category of small categories and functors #define a class of objects $$A, B,...$$ in $$\mathcal{C}_2$$ called `$$0$$- cells '
 * 2) for all `$$0$$-cells' $$A$$, $$B$$, consider a set denoted as "$$\mathcal{C}_2 (A,B)$$" that is defined as
 * 1) a horizontal composition, "$$\circ$$", is also defined for all triples of $$0$$-cells, $$A$$, $$B$$ and
 * 1) the identities under horizontal composition are the identities of the $$2$$-cells of $$1_X$$
 * 1) for any object $$A$$ in $$\mathcal{C}at$$ there is a functor from the one-object/one-arrow category

Examples of 2-categories
finite limits, together with the internal functors and the internal natural transformations between such internal functors;
 * 1) The $$2$$-category $$\mathcal{C}at$$ of small categories, functors, and natural transformations;
 * 2) The $$2$$-category $$\mathcal{C}at(\mathcal{E})$$ of internal categories in any category $$\mathcal{E}$$ with
 * 1) When $$\mathcal{E} = \mathcal{S}et$$, this yields again the category $$\mathcal{C}at$$, but if $$\mathcal{E} = \mathcal{C}at$$, then one obtains the 2-category of small double categories;
 * 2) When $$\mathcal{E} = Group $$, 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 $$\otimes$$, $$\circ$$, 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 $$\otimes$$-units are also $$\circ$$-units, and that an associativity relation holds for the two products: $$(S \otimes T ) \circ (S \otimes T) = (S \circ S) \otimes (T \circ T)$$
 * A $$2$$-category is an example of a supercategory with just two composition laws, and it is therefore an $$\S_1$$-supercategory, because the $$\S_0$$ supercategory is defined as a standard `$$1$$'-category subject only to the ETAC axioms.