Editing Limit (category theory) (section)

===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''.