# Subobject classifiers as internalizations

I recently read the article on internalizations on nlab, but I am not quite sure what falls under that description.

Is it fair to say, that subobjects are internalizations of subsets and that the subobject classifier internalizes characteristic functions?

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The subobject classifier is the monomorphism $\mathrm{true} \colon 1 \to \Omega$, not just $\Omega$. So it would be more accurate to say that it is the internalization of the the inclusion $\{0\} \subset \{0,1\}$. But I'm being a little nitpicking... – Pece Jun 13 '13 at 17:00