Page "Category theory" Paragraph 63

For example, a ( strict ) 2-category is a category together with " morphisms between morphisms ", i. e., processes which allow us to transform one morphism into another.

We can then " compose " these " bimorphisms " both horizontally and vertically, and we require a 2-dimensional " exchange law " to hold, relating the two composition laws.

In this context, the standard example is Cat, the 2-category of all ( small ) categories, and in this example, bimorphisms of morphisms are simply natural transformations of morphisms in the usual sense.

Another basic example is to consider a 2-category with a single object ; these are essentially monoidal categories.

Bicategories are a weaker notion of 2-dimensional categories in which the composition of morphisms is not strictly associative, but only associative " up to " an isomorphism.