Editing Direct limit (section)

{{Short description|Special case of colimit in category theory}}
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In [[mathematics]], a '''direct limit''' is a way to construct a (typically large) object from many (typically smaller) objects that are put together in a specific way. These objects may be [[Group (mathematics)|groups]], [[Ring (mathematics)|rings]], [[Vector space|vector spaces]] or in general objects from any [[Category (mathematics)|category]].  The way they are put together is specified by a system of [[Homomorphism|homomorphisms]] ([[group homomorphism]], [[ring homomorphism]], or in general [[morphism]]s in the category) between those smaller objects. The direct limit of the objects <math>A_i</math>, where <math>i</math> ranges over some [[directed set]] <math>I</math>, is denoted by <math>\varinjlim A_i </math>. This notation suppresses the system of homomorphisms; however, the limit depends on the system of homomorphisms.

Direct limits are a special case of the concept of [[limit (category theory)|colimit]] in [[category theory]]. Direct limits are [[Dual (category theory)|dual]] to [[Inverse limit|inverse limits]], which are a special case of [[Limit (category theory)|limits]] in category theory.