Table of contents |

2 Definition 3 Further examples |

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*.

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 vX-> Ω

*
*

1 | vg:X-> Ω

The morphism *g* is then called the **classifying morphism** for the subobject *j*.