Subobject classifiers in topoi are key to internalizing logic within categories. They represent truth values and enable a bijection between subobjects and morphisms, allowing for logical operations and reasoning within the topos structure.

Constructing subobject classifiers varies across categories. In Set, it's the two-element set {true, false}. In presheaves, it's defined by sieves. In sheaves on a topological space, it's the sheaf of open subsets.

Subobject Classifier in a Topos

Subobject classifier in topoi

• A topos is a category that behaves similarly to the category of sets and functions, possessing a terminal object, finite limits, exponentials, and a subobject classifier
• The subobject classifier, denoted as $\Omega$, represents the set of all truth values in a topos
  • In the category of sets (Set), $\Omega$ is the two-element set ${true, false}$
• There exists a bijection between subobjects of any object $A$ in the topos and morphisms from $A$ to $\Omega$
  • This bijection enables the internalization of logic within the topos
• The subobject classifier is crucial as it allows the definition of subobjects and characteristic functions within the topos
  • It provides a means to express logical statements and perform reasoning inside the topos (intuitionistic logic, higher-order logic)

Construction of subobject classifiers

• In the category of sets (Set)
  • The subobject classifier is the two-element set $\Omega = {true, false}$
  • The true morphism $true: 1 \to \Omega$ maps the single element of the terminal object to $true$
• In the category of presheaves (SetC^op)
  • The subobject classifier is the presheaf $\Omega: C^{op} \to Set$ defined by $\Omega(c) = {S \subseteq Hom_C(c, -)}$ for each object $c$ in $C$
  • $\Omega(c)$ is the set of all sieves on $c$
• In the category of sheaves on a topological space $X$ (Sh(X))
  • The subobject classifier is the sheaf of open subsets of $X$
  • For each open set $U \subseteq X$, $\Omega(U)$ is the set of all open subsets of $U$ (topology on $X$)

Power Objects and the Subobject Classifier

Subobject classifier vs power objects

• In a topos, the power object of an object $A$, denoted as $P(A)$, is the object of all subobjects of $A$
• A natural bijection exists between morphisms $A \to P(B)$ and subobjects of $A \times B$
  • This bijection is given by the exponential adjunction $Hom(A, P(B)) \cong Hom(A \times B, \Omega)$
• The subobject classifier $\Omega$ is isomorphic to the power object of the terminal object, $P(1)$
  • The isomorphism is given by the bijection $Hom(A, \Omega) \cong Hom(A \times 1, \Omega) \cong Hom(A, P(1))$
• The power object can be defined using the subobject classifier as $P(A) = \Omega^A$
  • This demonstrates that the subobject classifier is a fundamental object in a topos, from which power objects can be derived (exponential objects)

Applications of subobject classifiers

• Given an object $A$ in a topos and a subobject $S \hookrightarrow A$, there exists a unique morphism $\chi_S: A \to \Omega$ called the characteristic function of $S$
  • For each element $a \in A$, $\chi_S(a) = true$ if $a \in S$ and $\chi_S(a) = false$ if $a \notin S$
• Conversely, given a morphism $f: A \to \Omega$, there exists a unique subobject $S_f \hookrightarrow A$ corresponding to $f$
  • The subobject $S_f$ is defined as the pullback of the true morphism $true: 1 \to \Omega$ along $f$
• This bijective correspondence between subobjects of $A$ and morphisms $A \to \Omega$ allows the definition of subobjects using the subobject classifier
  • Subobjects can be represented by their characteristic functions, which are morphisms into $\Omega$ (indicator functions)
• Logical operations on subobjects can be defined using the internal logic of the topos
  • The intersection of two subobjects $S, T \hookrightarrow A$ corresponds to the logical conjunction of their characteristic functions $\chi_{S \cap T} = \chi_S \wedge \chi_T$
  • The union of two subobjects corresponds to the logical disjunction of their characteristic functions $\chi_{S \cup T} = \chi_S \vee \chi_T$
  • The complement of a subobject corresponds to the logical negation of its characteristic function $\chi_{\neg S} = \neg \chi_S$