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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 to the subset ''A''. In fact, χ<sub>''A''</sub> is true precisely on the elements of ''A''.
In this way, the collection of all subsets of ''S'', denoted by
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 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:
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