Editing Subobject classifier (section)

== Definition ==

For the general definition, we start with a category '''C''' that has a [[terminal object]], which we denote by 1. The object Ω of '''C''' is a subobject classifier for '''C''' if there exists a morphism 
:1 → Ω

with the following property:
:For each [[monomorphism]] ''j'': ''U'' → ''X'' there is a unique morphism ''χ<sub> j</sub>'': ''X'' → Ω such that the following [[commutative diagram]]
[[Image:SubobjectClassifier-02.png|center]]
:is a [[pullback diagram]]—that is, ''U'' is the [[limit (category theory)|limit]] of the diagram:
[[Image:SubobjectClassifier-03.png|center]]

The morphism ''χ<sub> j</sub>'' is then called the '''classifying morphism''' for the subobject represented by ''j''.