Editing Limit (category theory) (section)

==Functors and limits==
If ''F'' : ''J'' → ''C'' is a diagram in ''C'' and ''G'' : ''C'' → ''D'' is a [[functor]] then by composition (recall that a diagram is just a functor) one obtains a diagram ''GF'' : ''J'' → ''D''. A natural question is then:
:“How are the limits of ''GF'' related to those of ''F''?”

===Preservation of limits===
A functor ''G'' : ''C'' → ''D'' induces a map from Cone(''F'') to Cone(''GF''): if ''Ψ'' is a cone from ''N'' to ''F'' then ''GΨ'' is a cone from ''GN'' to ''GF''. The functor ''G'' is said to '''preserve the limits of ''F''''' if (''GL'', ''Gφ'') is a limit of ''GF'' whenever (''L'', ''φ'') is a limit of ''F''. (Note that if the limit of ''F'' does not exist, then ''G'' [[vacuous truth|vacuously]] preserves the limits of ''F''.)

A functor ''G'' is said to '''preserve all limits of shape ''J''''' if it preserves the limits of all diagrams ''F'' : ''J'' → ''C''. For example, one can say that ''G'' preserves products, equalizers, pullbacks, etc. A '''continuous functor''' is one that preserves all ''small'' limits.

One can make analogous definitions for colimits. For instance, a functor ''G'' preserves the colimits of ''F'' if ''G''(''L'', ''φ'') is a colimit of ''GF'' whenever (''L'', ''φ'') is a colimit of ''F''. A '''cocontinuous functor''' is one that preserves all ''small'' colimits.

If ''C'' is a [[complete category]], then, by the above existence theorem for limits, a functor ''G'' : ''C'' → ''D'' is continuous if and only if it preserves (small) products and equalizers. Dually, ''G'' is cocontinuous if and only if it preserves (small) coproducts and coequalizers.

An important property of [[adjoint functors]] is that every right adjoint functor is continuous and every left adjoint functor is cocontinuous. Since adjoint functors exist in abundance, this gives numerous examples of continuous and cocontinuous functors.

For a given diagram ''F'' : ''J'' → ''C'' and functor ''G'' : ''C'' → ''D'', if both ''F'' and ''GF'' have specified limits there is a unique canonical morphism
:<math>\tau_F : G \lim F \to \lim GF</math>
which respects the corresponding limit cones. The functor ''G'' preserves the limits of ''F'' if and only if this map is an isomorphism. If the categories ''C'' and ''D'' have all limits of shape ''J'' then lim is a functor and the morphisms τ<sub>''F''</sub> form the components of a [[natural transformation]]
:<math>\tau:G \lim \to \lim G^J.</math>
The functor ''G'' preserves all limits of shape ''J'' if and only if τ is a natural isomorphism. In this sense, the functor ''G'' can be said to ''commute with limits'' ([[up to]] a canonical natural isomorphism).

Preservation of limits and colimits is a concept that only applies to ''[[covariant functor|covariant]]'' functors. For [[contravariant functor]]s the corresponding notions would be a functor that takes colimits to limits, or one that takes limits to colimits.

===Lifting of limits===
A functor ''G'' : ''C'' → ''D'' is said to '''lift limits''' for a diagram ''F'' : ''J'' → ''C'' if whenever (''L'', ''φ'') is a limit of ''GF'' there exists a limit (''L''&prime;, ''φ''&prime;) of ''F'' such that ''G''(''L''&prime;, ''φ''&prime;) = (''L'', ''φ''). A functor ''G'' '''lifts limits of shape ''J''''' if it lifts limits for all diagrams of shape ''J''. One can therefore talk about lifting products, equalizers, pullbacks, etc. Finally, one says that ''G'' '''lifts limits''' if it lifts all limits. There are dual definitions for the lifting of colimits.

A functor ''G'' '''lifts limits uniquely''' for a diagram ''F'' if there is a unique preimage cone (''L''&prime;, ''φ''&prime;) such that (''L''&prime;, ''φ''&prime;) is a limit of ''F'' and ''G''(''L''&prime;, ''φ''&prime;) = (''L'', ''φ''). One can show that ''G'' lifts limits uniquely if and only if it lifts limits and is [[amnestic functor|amnestic]].

Lifting of limits is clearly related to preservation of limits. If ''G'' lifts limits for a diagram ''F'' and ''GF'' has a limit, then ''F'' also has a limit and ''G'' preserves the limits of ''F''. It follows that:
*If ''G'' lifts limits of all shape ''J'' and ''D'' has all limits of shape ''J'', then ''C'' also has all limits of shape ''J'' and ''G'' preserves these limits.
*If ''G'' lifts all small limits and ''D'' is complete, then ''C'' is also complete and ''G'' is continuous.
The dual statements for colimits are equally valid.

===Creation and reflection of limits===
Let ''F'' : ''J'' → ''C'' be a diagram. A functor ''G'' : ''C'' → ''D'' is said to
*'''create limits''' for ''F'' if whenever (''L'', ''φ'') is a limit of ''GF'' there exists a unique cone (''L''&prime;, ''φ''&prime;) to ''F'' such that ''G''(''L''&prime;, ''φ''&prime;) = (''L'', ''φ''), and furthermore, this cone is a limit of ''F''.
*'''reflect limits''' for ''F'' if each cone to ''F'' whose image under ''G'' is a limit of ''GF'' is already a limit of ''F''.
Dually, one can define creation and reflection of colimits.

The following statements are easily seen to be equivalent:
*The functor ''G'' creates limits.
*The functor ''G'' lifts limits uniquely and reflects limits.
There are examples of functors which lift limits uniquely but neither create nor reflect them.

===Examples===
* Every [[representable functor]] ''C'' → '''Set''' preserves limits (but not necessarily colimits).  In particular, for any object ''A'' of ''C'', this is true of the covariant [[Hom functor]] Hom(''A'',&ndash;) : ''C'' → '''Set'''.
* The [[forgetful functor]] ''U'' : '''Grp''' → '''Set''' creates (and preserves) all small limits and [[filtered colimit]]s; however, ''U'' does not preserve coproducts. This situation is typical of algebraic forgetful functors.
* The [[free functor]] ''F'' : '''Set''' → '''Grp''' (which assigns to every set ''S'' the [[free group]] over ''S'') is left adjoint to forgetful functor ''U'' and is, therefore, cocontinuous. This explains why the [[free product]] of two free groups ''G'' and ''H'' is the free group generated by the [[disjoint union]] of the generators of ''G'' and ''H''.
* The inclusion functor '''Ab''' → '''Grp''' creates limits but does not preserve coproducts (the coproduct of two abelian groups being the [[Direct sum of abelian groups|direct sum]]).
* The forgetful functor '''Top''' → '''Set''' lifts limits and colimits uniquely but creates neither.
* Let '''Met'''<sub>''c''</sub> be the category of [[metric space]]s with [[continuous function]]s for morphisms. The forgetful functor '''Met'''<sub>''c''</sub> → '''Set''' lifts finite limits but does not lift them uniquely.