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In category theory, a subobject classifier is a special object Ω of a category; intuitively, the subobjects of an object X correspond to the morphisms from X to Ω.
Introductory example As an example, the set Ω = is a subobject classifier in the category of sets and functions: to every subset j:U → X we can assign the function χj from X to Ω that maps precisely the elements of U to 1 (see characteristic function). Every function from X to Ω arises in this fashion from precisely one subset U. 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 χj: X -> Ω such that the following commutative diagram U -> 1 | | v v X -> Ω is a pullback diagram - that is, U is the limit of the diagram: 1 | v χj: X -> Ω The morphism χj is then called the classifying morphism for the subobject represented by j. Further examples Every topos has a subobject classifier. For the topos of sheaves of sets on a topological space X, it can be described in these terms: take the disjoint union Ω of all the open sets U of X, and its natural mapping π to X coming from all the inclusion maps. Then π is a local homeomorphism, and the corresponding sheaf is the required subobject classifier (in other words the construction of Ω is by means of its espace étalé). One can also consider Ω to be, in a (tautological) sense, the graph of the membership relation between points x and open sets U of X. | ||||||||
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