Editing Category theory (section)

== Other concepts ==

=== Universal constructions, limits, and colimits ===
{{Main|Universal property|Limit (category theory)}}

Using the language of category theory, many areas of mathematical study can be categorized. Categories include sets, groups and topologies.

Each category is distinguished by properties that all its objects have in common, such as the [[empty set]] or the [[product topology|product of two topologies]], yet in the definition of a category, objects are considered atomic, i.e., we ''do not know'' whether an object ''A'' is a set, a topology, or any other abstract concept. Hence, the challenge is to define special objects without referring to the internal structure of those objects. To define the empty set without referring to elements, or the product topology without referring to open sets, one can characterize these objects in terms of their relations to other objects, as given by the morphisms of the respective categories. Thus, the task is to find ''[[universal property|universal properties]]'' that uniquely determine the objects of interest.

Numerous important constructions can be described in a purely categorical way if the ''category limit'' can be developed and dualized to yield the notion of a ''colimit''.

=== Equivalent categories ===
{{Main|Equivalence of categories|Isomorphism of categories}}

It is a natural question to ask: under which conditions can two categories be considered ''essentially the same'', in the sense that theorems about one category can readily be transformed into theorems about the other category? The major tool one employs to describe such a situation is called ''equivalence of categories'', which is given by appropriate functors between two categories. Categorical equivalence has found [[Equivalence of categories#Examples|numerous applications]] in mathematics.

=== Further concepts and results ===
The definitions of categories and functors provide only the very basics of categorical algebra; additional important topics are listed below. Although there are strong interrelations between all of these topics, the given order can be considered as a guideline for further reading.
* The [[functor category]] ''D''<sup>''C''</sup> has as objects the functors from ''C'' to ''D'' and as morphisms the natural transformations of such functors. The [[Yoneda lemma]] is one of the most famous basic results of category theory; it describes representable functors in functor categories.
* [[Dual (category theory)|Duality]]: Every statement, theorem, or definition in category theory has a ''dual'' which is essentially obtained by "reversing all the arrows". If one statement is true in a category ''C'' then its dual is true in the dual category ''C''<sup>op</sup>. This duality, which is transparent at the level of category theory, is often obscured in applications and can lead to surprising relationships.
* [[Adjoint functors]]: A functor can be left (or right) adjoint to another functor that maps in the opposite direction. Such a pair of adjoint functors typically arises from a construction defined by a universal property; this can be seen as a more abstract and powerful view on universal properties.

=== 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).]