2-category

Definition 0.1

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

  1. define Cat as the category of small categories and functors
  2. define a class of objects $ A, B,...$ in $ \mathcal{C}_2$ called `0 - cells'
  3. for all `0 -cells' $ A$ , $ B$ , consider a set denoted as “ $ \mathcal{C}_2 (A,B)$ ” that is defined as $ \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
  4. the $ 2$ -categorical composition of $ 2$ -morphisms is denoted as “$ \bullet$ ” and is called the vertical composition
  5. a horizontal composition, “$ \circ$ ”, is also defined for all triples of 0 -cells, $ A$ , $ B$ and $ C$ in $ \mathcal{C}at$ as the functor

    $\displaystyle \circ: \mathcal{C}_2(B,C) \times \mathcal{C}_2(A,B) = \mathcal{C}_2(A,C),$

    which is associative
  6. the identities under horizontal composition are the identities of the $ 2$ -cells of $ 1_X$ for any $ X$ in $ \mathcal{C}at$
  7. for any object $ A$ in $ \mathcal{C}at$ there is a functor from the one-object/one-arrow category $ \textbf{1}$ (terminal object) to $ \mathcal{C}_2(A,A)$ .

Examples of 2-categories

  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 finite limits, together with the internal functors and the internal natural transformations between such internal functors;
  3. 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;
  4. When $ \mathcal{E} = \textbf{Group}$ , one obtains the $ 2$ -category of crossed modules.

Remarks:



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