definition. subobject classifier [kostecki2011introduction, 6.4] [tt-004I]

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A subobject classifier is an object \(\Omega \) in \({\cal C}\), together with an arrow \(\top : \mathbf {1} \to \Omega \), called the true arrow, such that for each monic arrow \(m: Y \mapsto X\) there is a unique arrow \(\chi : X \to \Omega \), called the characteristic arrow of \(m\) (or of \(Y\)), such that the diagram is a pullback, where \(!\) is the unique functor.

\(\Omega \) is also called a generalized truth-value object.

The arrow \((! \mathbin {\bullet } T) : Y \stackrel {!}{\to } \mathbf {1} \rightarrowtail \Omega \) is often denoted as \(\top _Y: Y \to \Omega \).