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.

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 ${\displaystyle \Omega }$ .

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.

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 sievesThe 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 ${\displaystyle \Omega }$ .

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 ${\displaystyle \Omega (f)(S)\equiv f^{*}(S)}$ 

References

• Cecilia-Flori: Topos-physics, [1]

An explanation of Topos theory and its implementation in Physics