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== Definition ==
[[File:Proj-map.svg|thumb|191x191px|The commutativity of this diagram is the universality of the projection {{mvar|π}}, for any map
Generally, a mapping where the [[Domain of a function|domain]] and [[codomain]] are the same [[set (mathematics)|set]] (or [[mathematical structure]]) is a projection if the mapping is [[idempotent]], which means that a projection is equal to its [[Function composition|composition]] with itself. A projection may also refer to a mapping which has a [[Inverse_function#Right_inverses|right inverse]]. Both notions are strongly related, as follows. Let {{mvar|p}} be an idempotent mapping from a set {{mvar|A}} into itself (thus {{math|1=''p'' ∘ ''p'' = ''p''}}) and {{math|1=''B'' = ''p''(''A'')}} be the image of {{mvar|p}}. If we denote by {{mvar|π}} the map {{mvar|p}} viewed as a map from {{mvar|A}} onto {{mvar|B}} and by {{mvar|i}} the [[injective function|injection]] of {{mvar|B}} into {{mvar|A}} (so that {{math|1=''p'' = ''i'' ∘ ''π''}}), then we have {{math|1=''π'' ∘ ''i'' = Id<sub>''B''</sub>}} (so that {{mvar|π}} has a right inverse). Conversely, if {{mvar|π}} has a right inverse {{mvar|i}}, then {{math|1=''π'' ∘ ''i'' = Id<sub>''B''</sub>}} implies that {{math|1=''i'' ∘ ''π'' ∘ ''i'' ∘ ''π'' = ''i'' ∘ Id<sub>''B''</sub> ∘ ''π'' = ''i'' ∘ ''π''}}; that is, {{math|1=''p'' = ''i'' ∘ ''π''}} is idempotent.{{Citation needed|date=August 2021}}
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