Subobject 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 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.

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 χ_j: 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 χ_j is then called the classifying morphism for the subobject represented by j.

The above example of subobject classifier in Sets is very usefull because it enables us to easliy prove the following axiom:

Axiom: Given a category C, then there exists an Isomorphisms,${\displaystyle y:Sub_{C}(X)\cong Hom_{C}(X,\Omega )}$  ${\displaystyle \forall X\in C}$ 

In Set this axiom can be restated as follows:

Axiom: The collection of all subsets of S denoted by ${\displaystyle {\mathcal {P}}(S)}$ , and the collection of all maps from S to the set ${\displaystyle \{0,1\}=2}$  denoted by ${\displaystyle 2^{S}}$  are isomorphic i.e. the function ${\displaystyle y:{\mathcal {P}}(S)\rightarrow 2^{S}}$ , which in terms of single elements of ${\displaystyle {\mathcal {P}}(S)}$  is ${\displaystyle A\rightarrow \chi _{A}}$ , it is a bijection.

The above axiom implyes the alternative definition of a subobject calssifier:

Definition: ${\displaystyle \Omega }$  is a Subobject classifier iff there is a ``one to one" correspondence between subobject of X and morphisms from X to ${\displaystyle \Omega }$ .

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.

Let us consider an example of a subobject classifer in the Topos of presheafs ${\displaystyle {\mathcal {S}}^{C^{op}}}$ . The formal definition goes as follows

Definition: A Subobject Classifier ${\displaystyle \Omega }$  is a presheaf ${\displaystyle \Omega :C\rightarrow {\mathcal {S}}^{C^{op}}}$  such that to each object ${\displaystyle A\in C}$  there corresponds an object ${\displaystyle \Omega (A)\in {\mathcal {S}}^{C^{op}}}$  which represents the set of all sieves on A, and to each ${\displaystyle C}$ -arrow ${\displaystyle f:B\rightarrow A}$  there corresponds an ${\displaystyle {\mathcal {S}}^{C^{op}}}$ -arrow ${\displaystyle \Omega (f):\Omega (A)\rightarrow \Omega (B)}$  such that ${\displaystyle \Omega (f)(S):=\{h:C\rightarrow B|foh\in S\}}$  is a sieve on B, where ${\displaystyle \Omega (f)(S)\equiv f^{*}(S)}$