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 = {0,1} is a subobject classifier in the category of sets and functions: to every subset U of X we can assign the function 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 g: 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
     g: X -> Ω

The morphism g is then called the classifying morphism for the subobject 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 sheaf corresponding 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 obtaining between points x of X and open sets U