2-category

From Wikipedia, the free encyclopedia

In category theory, a 2-category is a category with "morphisms between morphisms". It can be formally defined as a category enriched over Cat (the category of categories and functors, with the monoidal structure given by products).

More explicitly, a 2-category C consists of:

A class of 0-cells (or objects) A, B, ....
For all objects A and B, a category $\mathbf{C}(A,B)$ . The objects $f:A\to B$ of this category are called 1-cells and its morphisms $\alpha:f\Rightarrow g$ are called 2-cells; the composition in this category is written $\bullet$ and called vertical composition.
For all objects A, B and C, there is a functor $\circ : \mathbf{C}(B,C)\times\mathbf{C}(A,B)\to\mathbf{C}(A,C)$ , called horizontal composition, which is associative and admits the identity 2-cells of id_A as identities.
For any object "A" there is a functor from the terminal category (with one object and one arrow) to $\mathbf{C}(A,A)$ .

The notion of 2-category differs from the more general notion of a bicategory in that composition of (1-)morphisms is required to be strictly associative, whereas in a bicategory it need only be associative up to a 2-isomorphism.