Editing Limit (category theory) (section)

== Properties ==

=== Existence of limits ===
A given diagram ''F'' : ''J'' → ''C'' may or may not have a limit (or colimit) in ''C''. Indeed, there may not even be a cone to ''F'', let alone a universal cone.

A category ''C'' is said to '''have limits of shape ''J''''' if every diagram of shape ''J'' has a limit in ''C''. Specifically, a category ''C'' is said to
*'''have products''' if it has limits of shape ''J'' for every ''small'' discrete category ''J'' (it need not have large products),
*'''have equalizers''' if it has limits of shape <math>\bullet\rightrightarrows\bullet</math> (i.e. every parallel pair of morphisms has an equalizer),
*'''have pullbacks''' if it has limits of shape <math>\bullet\rightarrow\bullet\leftarrow\bullet</math> (i.e. every pair of morphisms with common codomain has a pullback).
A '''[[complete category]]''' is a category that has all small limits (i.e. all limits of shape ''J'' for every small category ''J'').

One can also make the dual definitions. A category '''has colimits of shape ''J''''' if every diagram of shape ''J'' has a colimit in ''C''. A '''[[cocomplete category]]''' is one that has all small colimits.

The '''existence theorem for limits''' states that if a category ''C'' has equalizers and all products indexed by the classes Ob(''J'') and Hom(''J''), then ''C'' has all limits of shape ''J''.{{r|Mac Lane|r={{cite book | first = Saunders | last = Mac Lane | author-link = Saunders Mac Lane | year = 1998 | title = [[Categories for the Working Mathematician]] | series = [[Graduate Texts in Mathematics]] | volume=5 | edition = 2nd | publisher = [[Springer-Verlag]] | isbn = 0-387-98403-8 | zbl=0906.18001 }}|at=§V.2 Thm.1}} In this case, the limit of a diagram ''F'' : ''J'' → ''C'' can be constructed as the equalizer of the two morphisms{{r|Mac Lane|at=§V.2 Thm.2}}
:<math>s,t : \prod_{i\in\operatorname{Ob}(J)}F(i) \rightrightarrows \prod_{f\in\operatorname{Hom}(J)} F(\operatorname{cod}(f))</math>
given (in component form) by
:<math>\begin{align}
s &= \bigl( F(f)\circ\pi_{\operatorname{dom}(f)}\bigr)_{f\in\operatorname{Hom}(J)} \\
t &= \bigl( \pi_{\operatorname{cod}(f)}\bigr)_{f\in\operatorname{Hom}(J)}.
\end{align}</math>
There is a dual '''existence theorem for colimits''' in terms of coequalizers and coproducts. Both of these theorems give sufficient and necessary conditions for the existence of all (co)limits of shape ''J''.

=== Universal property ===
Limits and colimits are important special cases of [[universal construction]]s.

Let ''C'' be a category and let ''J'' be a small index category. The [[functor category]] ''C''<sup>''J''</sup> may be thought of as the category of all diagrams of shape ''J'' in ''C''. The ''[[diagonal functor]]''
:<math>\Delta : \mathcal C \to \mathcal C^{\mathcal J}</math>
is the functor that maps each object ''N'' in ''C'' to the constant functor Δ(''N'') : ''J'' → ''C'' to ''N''. That is, Δ(''N'')(''X'') = ''N'' for each object ''X'' in ''J'' and Δ(''N'')(''f'') = id<sub>''N''</sub> for each morphism ''f'' in ''J''.

Given a diagram ''F'': ''J'' → ''C'' (thought of as an object in ''C''<sup>''J''</sup>), a [[natural transformation]] ''ψ'' : Δ(''N'') → ''F'' (which is just a morphism in the category ''C''<sup>''J''</sup>) is the same thing as a cone from ''N'' to ''F''. To see this, first note that Δ(''N'')(''X'') = ''N'' for all X implies that the components of ''ψ'' are morphisms ''ψ''<sub>''X''</sub> : ''N'' → ''F''(''X''), which all share the domain ''N''. Moreover, the requirement that the cone's diagrams commute is true simply because this ''ψ'' is a natural transformation. (Dually, a natural transformation ''ψ'' : ''F'' → Δ(''N'') is the same thing as a co-cone from ''F'' to ''N''.)

Therefore, the definitions of limits and colimits can then be restated in the form:
*A limit of ''F'' is a universal morphism from Δ to ''F''.
*A colimit of ''F'' is a universal morphism from ''F'' to Δ.

=== Adjunctions ===
Like all universal constructions, the formation of limits and colimits is functorial in nature. In other words, if every diagram of shape ''J'' has a limit in ''C'' (for ''J'' small) there exists a '''limit functor'''
:<math>\lim : \mathcal{C}^\mathcal{J} \to \mathcal{C}</math>
which assigns each diagram its limit and each [[natural transformation]] η : ''F'' → ''G'' the unique morphism lim η : lim ''F'' → lim ''G'' commuting with the corresponding universal cones. This functor is [[right adjoint]] to the diagonal functor Δ : ''C'' → ''C''<sup>''J''</sup>.
This adjunction gives a [[bijection]] between the set of all morphisms from ''N'' to lim ''F'' and the set of all cones from ''N'' to ''F''
:<math>\operatorname{Hom}(N,\lim F) \cong \operatorname{Cone}(N,F)</math>
which is natural in the variables ''N'' and ''F''. The counit of this adjunction is simply the universal cone from lim ''F'' to ''F''. If the index category ''J'' is [[connected category|connected]] (and nonempty) then the unit of the adjunction is an isomorphism so that lim is a left inverse of Δ. This fails if ''J'' is not connected. For example, if ''J'' is a discrete category, the components of the unit are the [[diagonal morphism]]s δ : ''N'' → ''N''<sup>''J''</sup>.

Dually, if every diagram of shape ''J'' has a colimit in ''C'' (for ''J'' small) there exists a '''colimit functor'''
:<math>\operatorname{colim} : \mathcal{C}^\mathcal{J} \to \mathcal{C}</math>
which assigns each diagram its colimit. This functor is [[left adjoint]] to the diagonal functor Δ : ''C'' → ''C''<sup>''J''</sup>, and one has a natural isomorphism
:<math>\operatorname{Hom}(\operatorname{colim}F,N) \cong \operatorname{Cocone}(F,N).</math>
The unit of this adjunction is the universal cocone from ''F'' to colim ''F''. If ''J'' is connected (and nonempty) then the counit is an isomorphism, so that colim is a left inverse of Δ.

Note that both the limit and the colimit functors are [[covariant functor|''covariant'']] functors.

=== As representations of functors ===
{{see also|Limit and colimit of presheaves}}
One can use [[Hom functor]]s to relate limits and colimits in a category ''C'' to limits in '''Set''', the [[category of sets]]. This follows, in part, from the fact the covariant Hom functor Hom(''N'', &ndash;) : ''C'' → '''Set''' [[#Preservation of limits|preserves all limits]] in ''C''. By duality, the contravariant Hom functor must take colimits to limits.

If a diagram ''F'' : ''J'' → ''C'' has a limit in ''C'', denoted by lim ''F'', there is a [[canonical isomorphism]]
:<math>\operatorname{Hom}(N,\lim F)\cong \lim\operatorname{Hom}(N,F-)</math>
which is natural in the variable ''N''. Here the functor Hom(''N'', ''F''&ndash;) is the composition of the Hom functor Hom(''N'', &ndash;) with ''F''. This isomorphism is the unique one which respects the limiting cones.

One can use the above relationship to define the limit of ''F'' in ''C''. The first step is to observe that the limit of the functor Hom(''N'', ''F''&ndash;) can be identified with the set of all cones from ''N'' to ''F'':
:<math>\lim\operatorname{Hom}(N,F-) = \operatorname{Cone}(N,F).</math>
The limiting cone is given by the family of maps π<sub>''X''</sub> : Cone(''N'', ''F'') → Hom(''N'', ''FX'') where {{pi}}<sub>''X''</sub>(''ψ'') = ''ψ''<sub>''X''</sub>. If one is given an object ''L'' of ''C'' together with a [[natural isomorphism]] ''Φ'' : Hom(''L'', &ndash;) → Cone(&ndash;, ''F''), the object ''L'' will be a limit of ''F'' with the limiting cone given by ''Φ''<sub>''L''</sub>(id<sub>''L''</sub>). In fancy language, this amounts to saying that a limit of ''F'' is a [[representable functor|representation]] of the functor Cone(&ndash;, ''F'') : ''C'' → '''Set'''.

Dually, if a diagram ''F'' : ''J'' → ''C'' has a colimit in ''C'', denoted colim ''F'', there is a unique canonical isomorphism
:<math>\operatorname{Hom}(\operatorname{colim} F, N)\cong\lim\operatorname{Hom}(F-,N)</math>
which is natural in the variable ''N'' and respects the colimiting cones. Identifying the limit of Hom(''F''&ndash;, ''N'') with the set Cocone(''F'', ''N''), this relationship can be used to define the colimit of the diagram ''F'' as a representation of the functor Cocone(''F'', &ndash;).

=== Interchange of limits and colimits of sets ===
Let ''I'' be a finite category and ''J'' be a small [[filtered category]]. For any [[bifunctor]]

:<math>F : I\times J \to \mathbf{Set},</math>

there is a [[natural isomorphism]]

:<math>\operatorname{colim}\limits_J \lim_I F(i, j) \rightarrow \lim_I\operatorname{colim}\limits_J F(i, j).</math>

In words, filtered colimits in '''Set''' commute with finite limits. It also holds that small colimits commute with small limits.<ref> {{nlab|id=commutativity+of+limits+and+colimits|title=commutativity of limits and colimits}}</ref>