Editing Limit (category theory) (section)

=== 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''.