Topoi

Subobject Classifier

Let be a category with all finite limits. A subobject classifier in consists of an object together with an arrow that is a “universal subobject”, i.e. Given any object and any subobject , there is a unique arrow making the following diagram a pullback: The arrow is called the classifying arrow of the subobject . e.g. The most familiar example of a subobject classifier is of course the set with a selected element as . The fact that every subset of any set has a unique characteristic function is then exactly the subobject classifier condition.

Proposition

A subobject classifier is unique up to isomorphism. Proof The pullback condition is clearly equivalent to requiring the contravariant subobject functor to be representable:The required isomorphism is just the pullback condition stated in the definition of a subobject classifier.

Prop For any poset category , the subobject classifier in is the functor: that is, is the set of all upper sets above . Proof

Topos

A topos is a category such that

• has all finite limits;
• has a subobject classifier;
• has all exponentials.

Proposition

Prop For any small category , the category of diagrams is a topos.