Editing Category theory (section)

=== Higher-dimensional categories ===
{{Main|Higher category theory}}
Many of the above concepts, especially equivalence of categories, adjoint functor pairs, and functor categories, can be situated into the context of ''higher-dimensional categories''. Briefly, if we consider a morphism between two objects as a "process taking us from one object to another", then higher-dimensional categories allow us to profitably generalize this by considering "higher-dimensional processes".

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 transformation]]s of morphisms in the usual sense. Another basic example is to consider a 2-category with a single object; these are essentially [[monoidal category|monoidal categories]]. [[bicategory|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.

This process can be extended for all [[natural number]]s ''n'', and these are called [[n-category|''n''-categories]]. There is even a notion of ''[[quasi-category|ω-category]]'' corresponding to the [[ordinal number]] [[ω (ordinal number)|ω]].

Higher-dimensional categories are part of the broader mathematical field of [[higher-dimensional algebra]], a concept introduced by [[Ronald Brown (mathematician)|Ronald Brown]].  For a conversational introduction to these ideas, see [http://math.ucr.edu/home/baez/week73.html John Baez, 'A Tale of ''n''-categories' (1996).]