Sub-Object Classifier

In category theory, a subobject classifier is a special object Ω in a category; intuitively, the subobjects of an object X correspond to equivalence classes (via iso) of the monomorphisms of X to Ω.

Introductory Example

As an example, within the category of finite sets and applications among them we can consider the set with only two elements Ω Ω {displaystyle Omega } = {0.1} and turns out to be a sub-object sorter: to each subset U of X we can assign the function of X towards Ω Ω {displaystyle Omega } send the elements U a 1 (see feature). Each of these features (of X Al Ω Ω {displaystyle Omega }) are presented this way for exactly a subset U.

Definition

For the general definition, we begin with a category C that has terminal object, that we denote by 1. The object Ω Ω {displaystyle Omega } of C is a sub-object sorter for C if there is a morphism 1 → → Ω Ω {displaystyle rightarrow Omega } with the following property:

for every monomorphism j: U → → {displaystyle rightarrow } X There is a unique morphism g: X → → Ω Ω {displaystyle rightarrow Omega } such that the following commutative diagram

 U - 2005
 j..
v
 X - 2005 Ω

A pullback diagram - that is, U is the limit of the diagram:

 1
日本語
v
 g: X - 2005 Ω

Morphism g Then it's called the Classifying morphosism for the sub-object j.

Additional Examples

Each topos has a sub-object classifier.