Editing Direct limit (section)

== Related constructions and generalizations ==
We note that a direct system in a category <math>\mathcal{C}</math> admits an alternative description in terms of [[functor]]s. Any directed set <math>\langle I,\le \rangle</math> can be considered as a [[small category]] <math>\mathcal{I}</math> whose objects are the elements <math>I</math> and there is a morphisms  <math>i\rightarrow j</math> [[if and only if]] <math>i\le j</math>. A direct system over <math>I</math> is then the same as a [[covariant functor]] <math>\mathcal{I}\rightarrow \mathcal{C}</math>. The [[Limit (category theory)|colimit]] of this functor is the same as the direct limit of the original direct system.

A notion closely related to direct limits are the [[Filtered category|filtered colimits]]. Here we start with a covariant functor <math>\mathcal J \to \mathcal C</math> from a [[filtered category]] <math>\mathcal J</math> to some category <math>\mathcal{C}</math> and form the colimit of this functor. One can show that a category has all directed limits if and only if it has all filtered colimits, and a functor defined on such a category commutes with all direct limits if and only if it commutes with all filtered colimits.<ref>{{Cite book|url=https://books.google.com/books?id=iXh6rOd7of0C|title=Locally Presentable and Accessible Categories|last1=Adamek|first1=J.|last2=Rosicky|first2=J.|publisher=Cambridge University Press|year=1994|location=|pages=15|isbn=9780521422611|language=en}}</ref>

Given an arbitrary category <math>\mathcal{C}</math>, there may be direct systems in <math>\mathcal{C}</math> that don't have a direct limit in <math>\mathcal{C}</math> (consider for example the category of finite sets, or the category of [[finitely generated abelian group]]s). In this case, we can always embed <math>\mathcal{C}</math> into a category <math>\text{Ind}(\mathcal{C})</math> in which all direct limits exist; the objects of  <math>\text{Ind}(\mathcal{C})</math> are called [[Ind-object|ind-objects]] of  <math>\mathcal{C}</math>.  

The [[Dual (category theory)|categorical dual]] of the direct limit is called the [[inverse limit]]. As above, inverse limits can be viewed as limits of certain functors and are closely related to limits over cofiltered categories.