Sub-object classifier
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.
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