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 $\chi_{\neg S} = \neg \chi_S$