Subobject classifier
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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.
[edit] Introductory example
As an example, the set Ω = {0,1} 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.
[edit] 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.
[edit] 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.
