2-morphism

In category theory, a (strict) 2-category is a category with "morphisms between morphisms", that is, where each hom-set itself carries the structure of a category. It can be formally defined as a category enriched over Cat (the category of categories and functors, with the monoidal structure given by product of categories).

The concept of 2-category was first introduced by Charles Ehresmann in his work on enriched categories, in 1965. The more general concept of bicategory, where composition of morphisms is associative only up to a 2-isomorphism, was invented in 1968 by Jean Bénabou

A 2-category C consists of:

The notion of 2-category differs from the more general notion of a bicategory in that composition of 1-cells (horizontal composition) is required to be strictly associative, whereas in a bicategory it needs only be associative up to a 2-isomorphism. The axioms of a 2-category are consequences of their definition as Cat-enriched categories:

The interchange law follows from the fact that ${\displaystyle \circ _{0}}$ is a functor between hom categories. It can be drawn as a pasting diagram as follows:

Here the left-hand diagram denotes the vertical composition of horizontal composites, the right-hand diagram denotes the horizontal composition of vertical composites, and the diagram in the centre is the customary representation of both.

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