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# 2-category

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 $mathbf\left\{C\right\}\left(A,B\right)$. The objects $f:Ato B$ of this category are called 1-cells and its morphisms $alpha:fRightarrow g$ 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 $circ : mathbf\left\{C\right\}\left(B,C\right)timesmathbf\left\{C\right\}\left(A,B\right)tomathbf\left\{C\right\}\left(A,C\right)$, called horizontal composition, which is associative and admits the identity 2-cells of idA as identities.
• For any object "A" there is a functor from the terminal category (with one object and one arrow) to $mathbf\left\{C\right\}\left(A,A\right)$.

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.