Subobject classifier: Difference between revisions

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== Introductory example ==

As an example, the set Ω = {0,1} is a subobject classifier in the [[category of sets]] and functions: to every subset ''A'' of ''S'' defined by the inclusion function   '' j '' : ''A'' → ''XS'' we can assign the function ''χ_j'' from ''XS'' to Ω that maps precisely the elements of ''A'' to 1 (see [[indicator function|characteristic function]]). Every function from ''XS'' to Ω arises in this fashion from precisely one subset ''A''.

To be clearer, consider a [[subset]] ''A'' of ''S'' (''A'' ⊆ ''S''), where ''S'' is a set. The notion of being a subset can be expressed mathematically using the so-called characteristic function χ_''A'' : S → {0,1}, which is defined as follows: