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In this way, the collection of all subsets of ''S'', denoted by '''''P'''''(''S''), and the collection of all maps from ''S'' to Ω = {0,1}, denoted by Ω<sup>''S''</sup>, are [[isomorphic]].
To categorize this notion, recall that, in category theory, a subobject is actually a pair consisting of an object and a [[monomorphism|monic arrow]] (interpreted as the inclusion into another object). Accordingly, '''true''' refers to the
[[Image:SubobjectClassifier-01.png|center]]
Defined that way, χ is a morphism ''Sub''<sub>C</sub>(''S'') → Hom<sub>C</sub>(S, Ω). By definition, Ω is a '''subobject classifier''' if this morphism is an isomorphism.
== Definition ==
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