Elements of Subobject Classifier

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Elements of Subobject Classifier

The elements of the subobject classifier $ \Omega$ (see 1.7) in a Topos are derived from the following Theorem

Theorem 1.1   Subpresheaves 1.10 can be identified with sieves (see definition 1.7.1)

In order to prove the above theorem we need the follwing lemma:

Lemma 1.1   Preliminary: if $ \mathscr{C}$ is a locally small category 1.1, then each object A of $ \mathscr{C}$ induces a natural contravariant functor 3.2 to Set called a hom-functor, i.e there exists a map $ y:C\rightarrow Sets^{\mathscr{C}^{op}}$ which in terms of individual elements is defined as $ A\rightarrow Hom_C(-,A)$.
Yoneda lemma: Given an arbitrary presheaf P 3.6 on $ \mathscr{C}$ there exists a bijective correspondence between natural transformations 3.5 $ y(A)\rightarrow P$ and elements of the set P(A) defined as an arrow $ \theta:Nat_C(y(A),P)\buildrel\sim\over\rightarrow P(A)$ which in terms of individual elements $ \alpha:y(A)\rightarrow P$ is $ \theta(\alpha)=\alpha_A(1_A)$

We have now the right tools to prove the above theorem.

Proof. Let's consider $ \Omega$ to be a subobject classifier of $ \hat{C}=Sets^{C^{op}}$, i.e. we want $ \Omega$ to classify subobjects in $ Sets^{C^{op}}$.
Consider now a presheaf $ y(C)=Hom_{\mathscr{C}}(-,C)\in\hat{C}$. We know from axiom 1.1 that $ Sub_{\hat{C}}(Hom_\mathscr{C}(-,C))\cong Hom_{\hat{C}}(Hom_\mathscr{C}(-,C),\Omega)$
form Yonedas lemma it follows that
$ Hom_{\hat{C}}(Hom_{\mathscr{C}}(-,C),\Omega)=\Omega(C)$, therefore the subobject classifier $ \Omega$ must be a presheaf $ \Omega:\mathscr{C}\rightarrow Set$ such that

$\displaystyle \Omega(C)=$ $\displaystyle Sub_{\hat{C}}(Hom_{\mathscr{C}}(-,C))$    
  $\displaystyle =\{S\vert S\hspace{.1in}a\hspace{.1in}subfunctor\hspace{.1in}of\hspace{.1in}Hom_{\mathscr{C}}(-,C)\}$    

Now if $ Q\subset Hom_{\mathscr{C}}(-,C)$ is a subfunctor of $ Hom_{\mathscr{C}}(-,C)$, then the set
$ S=\{f\vert\hspace{.1in}for\hspace{.1in}some\hspace{.1in}object\hspace{.1in}A,\hspace{.1in}f:A\rightarrow C
 \hspace{.1in}and\hspace{.1in}f\in Q(A)\}$
is a sieve on C. Conversely given a sieve S on C we define
$ Q(A)=\{f\vert f:A\rightarrow C\hspace{.1in}and\hspace{.1in}f\in S\}\subseteq Hom_{\mathscr{C}}(A,C)$
which produces a preshaef $ Q:\mathscr{C}\rightarrow Set$ which is a subfunctor of $ Hom_{\mathscr{C}}(-,C)$, i.e to each object $ A\in\mathscr{C}$ Q assigns the set $ Q(A)\subseteq Hom_{\mathscr{C}}(A,C)$. Since the transformation function from Q to S is a bijection (as can be seen from the above definition) we can conclude the following:

$ Sieve\hspace{.05in}on\hspace{.05in}A=\hspace{.05in}subfunctor\hspace{.05in}of\hspace{.05in}Hom_{\mathscr{C}}(-,C)$
$ \qedsymbol$

Next: Example of Yoneda's Lemma Up: Subobject Classifier in Topos Previous: Sieve
Cecilia Flori 2007-01-02