Editing Limit (category theory) (section)

==Examples==

===Limits===
{{anchor|Examples of limits}}
The definition of limits is general enough to subsume several constructions useful in practical settings. In the following we will consider the limit (''L'', ''φ'') of a diagram ''F'' : ''J'' → ''C''.
*'''[[Terminal object]]s'''. If ''J'' is the empty category there is only one diagram of shape ''J'': the empty one (similar to the [[empty function]] in set theory). A cone to the empty diagram is essentially just an object of ''C''. The limit of ''F'' is any object that is uniquely factored through by every other object. This is just the definition of a ''terminal object''.
*'''[[Product (category theory)|Products]]'''. If ''J'' is a [[discrete category]] then a diagram ''F'' is essentially nothing but a [[indexed family|family]] of objects of ''C'', indexed by ''J''. The limit ''L'' of ''F'' is called the ''product'' of these objects. The cone ''φ'' consists of a family of morphisms ''φ''<sub>''X''</sub> : ''L'' → ''F''(''X'') called the ''projections'' of the product. In the [[category of sets]], for instance, the products are given by [[Cartesian product]]s and the projections are just the natural projections onto the various factors.
**'''Powers'''. A special case of a product is when the diagram ''F'' is a [[constant functor]] to an object ''X'' of ''C''. The limit of this diagram is called the ''J<sup>th</sup> power'' of ''X'' and denoted ''X''<sup>''J''</sup>.
*'''[[Equalizer (mathematics)|Equalizer]]s'''. If ''J'' is a category with two objects and two parallel morphisms from one object to the other, then a diagram of shape ''J'' is a pair of parallel morphisms in ''C''. The limit ''L'' of such a diagram is called an ''equalizer'' of those morphisms.
**'''[[Kernel (category theory)|Kernel]]s'''. A ''kernel'' is a special case of an equalizer where one of the morphisms is a [[zero morphism]].
*'''[[Pullback (category theory)|Pullbacks]]'''. Let ''F'' be a diagram that picks out three objects ''X'', ''Y'', and ''Z'' in ''C'', where the only non-identity morphisms are ''f'' : ''X'' → ''Z'' and ''g'' : ''Y'' → ''Z''. The limit ''L'' of ''F'' is called a ''pullback'' or a ''fiber product''. It can nicely be visualized as a [[commutative diagram|commutative square]]:
[[Image:Pullback_categories.svg|center]]
*'''[[Inverse limit]]s'''. Let ''J'' be a [[directed set]] (considered as a small category by adding arrows ''i'' → ''j'' if and only if ''i'' ≥ ''j'') and let ''F'' : ''J''<sup>op</sup> → ''C'' be a diagram. The limit of ''F'' is called an ''inverse limit'' or ''projective limit''.
*If ''J'' = '''1''', the category with a single object and morphism, then a diagram of shape ''J'' is essentially just an object ''X'' of ''C''. A cone to an object ''X'' is just a morphism with codomain ''X''. A morphism ''f'' : ''Y'' → ''X'' is a limit of the diagram ''X'' if and only if ''f'' is an [[isomorphism]]. More generally, if ''J'' is any category with an [[initial object]] ''i'', then any diagram of shape ''J'' has a limit, namely any object isomorphic to ''F''(''i''). Such an isomorphism uniquely determines a universal cone to ''F''.
*'''Topological limits'''. Limits of functions are a special case of [[Filters in topology|limits of filters]], which are related to categorical limits as follows. Given a [[topological space]] ''X'', denote by ''F'' the set of filters on ''X'', ''x'' ∈ ''X'' a point, ''V''(''x'') ∈ ''F'' the [[Filter (mathematics)#Neighbourhood bases|neighborhood filter]] of ''x'', ''A'' ∈ ''F'' a particular filter and <math>F_{x,A}=\{G\in F \mid V(x)\cup A \subset G\} </math> the set of filters finer than ''A'' and that converge to ''x''. The filters ''F'' are given a small and thin category structure by adding an arrow ''A'' → ''B'' if and only if ''A'' ⊆ ''B''. The injection <math>I_{x,A}:F_{x,A}\to F</math> becomes a functor and the following equivalence holds :

:: ''x'' is a topological limit of ''A'' if and only if ''A'' is a categorical limit of <math>I_{x,A}</math>

===Colimits===
{{anchor|Examples of colimits}}
Examples of colimits are given by the dual versions of the examples above:
*'''[[Initial object]]s''' are colimits of empty diagrams.
*'''[[Coproduct]]s''' are colimits of diagrams indexed by discrete categories.
**'''Copowers''' are colimits of constant diagrams from discrete categories.
*'''[[Coequalizer]]s''' are colimits of a parallel pair of morphisms.
**'''[[Cokernel]]s''' are coequalizers of a morphism and a parallel zero morphism.
*'''[[Pushout (category theory)|Pushouts]]''' are colimits of a pair of morphisms with common domain.
*'''[[Direct limit]]s''' are colimits of diagrams indexed by directed sets.