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To categorize this notion, recall that, in category theory, a subobject is actually represented by a pair consisting of an object ''A'' and a [[monomorphism|monic arrow]] ''A → S'' (interpreted as the inclusion into another object ''S''). Accordingly, '''true''' refers to the element 1, which is selected by the arrow: '''true''': {0} → {0, 1} that maps 0 to 1. The subset ''A'' of ''S'' can now be defined as the [[pullback (category theory)|pullback]] of '''true''' along the characteristic function χ<sub>''A''</sub>, shown on the following diagram:
[[Image:SubobjectClassifier-01.
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 χ is an isomorphism.
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with the following property:
:For each [[monomorphism]] ''j'': ''U'' → ''X'' there is a unique morphism ''χ<sub>j</sub>'': ''X'' → Ω such that the following [[commutative diagram]]
[[Image:SubobjectClassifier-02.
:is a [[pullback diagram]]—that is, ''U'' is the [[limit (category theory)|limit]] of the diagram:
[[Image:SubobjectClassifier-03.
The morphism ''χ<sub> j</sub>'' is then called the '''classifying morphism''' for the subobject represented by ''j''.
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| year = 1983
| isbn = 0-444-85207-7
| url =
*{{cite book
| last = Johnstone
| |||