Subobject Classifier - Introductory Example

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 χ_j 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.

To be clearer, consider a subset A of S (A ⊆ S), where S is a set. The notion of being a subset can be expressed mathematically using the so-called characteristic function χ_A : S → {0,1}, which is defined as follows:

(Here we interpret 1 as true and 0 as false.) The role of the characteristic function is to determine which elements belong or not to a certain subset. Since in any category subobjects are identified as monic arrows, we identify the value true with the arrow: true: {0} → {0, 1} which maps 0 to 1. Given this definition, the subset A can be uniquely defined through the characteristic function A = χ_A−1(1). Therefore the diagram

is a pullback.

The above example of subobject classifier in Set is very useful because it enables us to easily prove the following axiom:

Axiom: Given a category C, then there exists an isomorphism,

y: Sub_C(X) ≅ Hom_C(X, Ω) ∀ X ∈ C

In Set this axiom can be restated as follows:

Axiom: The collection of all subsets of S denoted by, and the collection of all maps from S to the set {0, 1} = 2 denoted by 2S are isomorphic i.e. the function, which in terms of single elements of is A → χ_A, is a bijection.

The above axiom implies the alternative definition of a subobject classifier:

Definition: Ω is a subobject classifier iff there is a one to one correspondence between subobjects of X and morphisms from X to Ω.