Topoi

Subobject Classifier

Let $\mathbf{E}$ be a category with all finite limits. A subobject classifier in $\mathbf{E}$ consists of an object $\Omega$ together with an arrow $t:1\to \Omega$ that is a “universal subobject”, i.e. Given any object $E$ and any subobject $U\rightarrowtail E$, there is a unique arrow $u:E\to \Omega$ making the following diagram a pullback: The arrow $u$ is called the classifying arrow of the subobject $U\rightarrowtail E$. e.g. The most familiar example of a subobject classifier is of course the set $2=\{0,1\}$ with a selected element as $t:1\to 2$. The fact that every subset $U\subseteq S$ of any set $S$ has a unique characteristic function $u:S\to 2$ is then exactly the subobject classifier condition.

Proposition

A subobject classifier is unique up to isomorphism. Proof The pullback condition is clearly equivalent to requiring the contravariant subobject functor ${\mathrm{Sub}}_{\mathbf{E}}(-):{\mathbf{E}}^{\mathrm{op}}\to \mathbf{Sets}$to be representable:${\mathrm{Sub}}_{\mathbf{E}}(-)\cong {\mathrm{Hom}}_{\mathbf{E}}(-,\Omega )$The required isomorphism is just the pullback condition stated in the definition of a subobject classifier.

Prop For any poset category $\mathbf{P}$, the subobject classifier $\Omega$ in ${\mathbf{Sets}}^{\mathbf{P}}$ is the functor: $\Omega (p)=\{F\subseteq \mathbf{P}\mid (x\in F\Rightarrow p\le x)\wedge (x\in F\wedge x\le y\Rightarrow y\in F)\}$that is, $\Omega (p)$ is the set of all upper sets above $p$. Proof

Topos

A topos is a category $\mathbf{E}$ such that

• $\mathbf{E}$ has all finite limits;
• $\mathbf{E}$ has a subobject classifier;
• $\mathbf{E}$ has all exponentials.

Proposition

Prop For any small category $\mathbf{C}$, the category of diagrams ${\mathbf{Sets}}^{{\mathbf{C}}^{\mathrm{op}}}$ is a topos.