Editing Limit (category theory) (section)

== Definition ==
Limits and colimits in a [[category (mathematics)|category]] <math>C</math> are defined by means of diagrams in <math>C</math>. Formally, a '''[[diagram (category theory)|diagram]]''' of shape <math>J</math> in <math>C</math> is a [[functor]] from <math>J</math> to <math>C</math>:
:<math>F:J\to C.</math>
The category <math>J</math> is thought of as an [[index category]], and the diagram <math>F</math> is thought of as indexing a collection of objects and [[morphism]]s in <math>C</math> patterned on <math>J</math>.

One is most often interested in the case where the category <math>J</math> is a [[small category|small]] or even [[Finite set|finite]] category. A diagram is said to be '''small''' or '''finite''' whenever <math>J</math> is.

===Limits===
{{see also|Inverse limit}}
Let <math>F : J \to C</math> be a diagram of shape <math>J</math> in a category <math>C</math>. A '''[[cone (category theory)|cone]]''' to <math>F</math> is an object <math>N</math> of <math>C</math> together with a family <math>\psi_X:N\to F(X)</math> of morphisms indexed by the objects <math>X</math> of <math>J</math>, such that for every morphism <math>f: X \to Y</math> in <math>J</math>, we have <math>F(f)\circ\psi_X=\psi_Y</math>.

A '''limit''' of the diagram <math>F:J\to C</math> is a cone <math>(L, \phi)</math> to <math>F</math> such that for every cone <math>(N, \psi)</math> to <math>F</math> there exists a ''unique'' morphism <math>u:N\to L</math> such that <math>\phi_X\circ u=\psi_X</math> for all <math>X</math> in <math>J</math>.
[[File:Functor cone (extended).svg|center|A universal cone]]
One says that the cone <math>(N, \psi)</math> factors through the cone <math>(L, \phi)</math> with
the unique factorization <math>u</math>. The morphism <math>u</math> is sometimes called the '''mediating morphism'''.

Limits are also referred to as ''[[universal cone]]s'', since they are characterized by a [[universal property]] (see below for more information). As with every universal property, the above definition describes a balanced state of generality: The limit object <math>L</math> has to be general enough to allow any cone to factor through it; on the other hand, <math>L</math> has to be sufficiently specific, so that only ''one'' such factorization is possible for every cone.

Limits may also be characterized as [[terminal object]]s in the [[category of cones]] to ''F''.

It is possible that a diagram does not have a limit at all. However, if a diagram does have a limit then this limit is essentially unique: it is unique [[up to]] a unique [[isomorphism]]. For this reason one often speaks of ''the'' limit of ''F''.

===Colimits===
{{see also|Direct limit}}
The [[Dual (category theory)|dual notions]] of limits and cones are colimits and co-cones. Although it is straightforward to obtain the definitions of these by inverting all morphisms in the above definitions, we will explicitly state them here:

A '''[[co-cone]]''' of a diagram <math>F:J\to C</math> is an object <math>N</math> of <math>C</math> together with a family of morphisms
:<math>\psi_X:F(X) \to N</math>

for every object <math>X</math> of <math>J</math>, such that for every morphism <math>f:X\to Y</math> in <math>J</math>, we have <math>\psi_Y\circ F(f)=\psi_X</math>.

A '''colimit''' of a diagram <math>F:J\to C</math> is a co-cone <math>(L, \phi)</math> of <math>F</math> such that for any other co-cone <math>(N, \psi)</math> of <math>F</math> there exists a unique morphism <math>u:L\to N</math> such that <math>u\circ \phi_X = \psi_X</math> for all <math>X</math> in <math>J</math>.

[[File:Functor co-cone (extended).svg|center|A universal co-cone]]

Colimits are also referred to as ''[[universal co-cone]]s''. They can be characterized as [[initial object]]s in the [[category of co-cones]] from <math>F</math>.

As with limits, if a diagram <math>F</math> has a colimit then this colimit is unique up to a unique isomorphism.

===Variations===
Limits and colimits can also be defined for collections of objects and morphisms without the use of diagrams. The definitions are the same (note that in definitions above we never needed to use composition of morphisms in <math>J</math>). This variation, however, adds no new information. Any collection of objects and morphisms defines a (possibly large) [[directed graph]] <math>G</math>. If we let <math>J</math> be the [[free category]] generated by <math>G</math>, there is a universal diagram <math>F:J\to C</math> whose image contains <math>G</math>. The limit (or colimit) of this diagram is the same as the limit (or colimit) of the original collection of objects and morphisms.

'''Weak limit''' and '''weak colimits''' are defined like limits and colimits, except that the uniqueness property of the mediating morphism is dropped.