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(Here we interpret 1 as true and 0 as false.) The role of the characteristic function is to determine which elements belong or not to a certain subset.
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 object 1 and 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''' and the characteristic function χ<sub>''A''</sub>, also written ''A'' = χ<sub>''A''</sub><sup>−1</sup>(1); the following diagram commutes:
[[Image:SubobjectClassifier-01.png|center]]
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