Subobject Classifier in the Topos of Presheaves


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Definition 1.14   A Subobject Classifier $ \Omega$ is a presheaf $ \Omega:\mathscr{C}\rightarrow\mathcal{S}$ such that to each object $ A\in\mathscr{C}$ there corresponds an object $ \Omega(A)\in\mathcal{S}$ which represents the set of all sieves 1.9.1 on A, and to each $ \mathscr{C}$-arrow $ f:B\rightarrow A$ there corresponds an $ \mathcal{S}$-arrow $ \Omega(f):\Omega(A)\rightarrow\Omega(B)$ such that $ \Omega(f)(S):=\{h:C\rightarrow B\vert f o h\in S\}$ is a sieve on B, where $ \Omega(f)(S)\equiv f^*(S)$

We now want to show that this definition of subobject classifier is in agreement with definition 1.12. In order to do that we need to define the analog of arrow true (T) and the characteristic function (see section 1.8.1) in Topos.

Definition 1.15   $ \textbf{T}:1\rightarrow\Omega$ is the natural transformation 3.5 that has components $ T_A:\{0\}\rightarrow\Omega(A)$ given by $ T_A(0)=\downarrow A$ = principal sieve on A

To understand how T works, let us consider a monic arrow $ f:F\rightarrow X$ in $ \mathcal{S}^{\mathscr{C}^{op}}$ which is defined component-wise by $ f_A:F(A)\rightarrow X(A)$ and represents subset inclusion. Now we define the character $ \chi^f:X\rightarrow\Omega$ of f which is a natural transformation in the category of presheaves such that the components $ \chi_A^f$ represent functions from X(A) to $ \Omega(A)$, as shown in the following diagram.

$\displaystyle \xymatrix{
 *++{F(A)}\ar@{^{(}->}[rr]^{f_A}\ar[dd]&&X(A)\ar[dd]^{\chi^f_A}\\
 &&\\
 \{0\}\ar[rr]^T&&\Omega(A)\\
 }$

where $ \{0\}\equiv 1$. From the above diagram we can see that $ \chi^f_A$ assigns to each element x of X(A) a sieve $ \Omega(A)$ on A. For a function to belong to the sieve $ \Omega(A)$ on A we require that the following diagram commutes

$\displaystyle \xymatrix{
 *++{F(A)}\ar@{^{(}->}[rr]\ar[dd]_{F(f)}&&X(A)\ar[dd]^{X(f)}\\
 &&\\
 *++{F(B)}\ar@{^{(}->}[rr]&&X(B)\\
 }$

therefore

$\displaystyle \chi_A^F(x):=\{f:B\rightarrow A\vert X(f)(x)\in F(B)\}$ (1.2)

What equation 1.2 means is that we require F(f) to be the restriction of X(f) to F(A). This condition is expressed by the following diagram

$\displaystyle \begin{xy} 0;/r15mm/:
(0,0)=

i.e. f belongs to $ \Omega(A)$ iff X(f) maps x into F(B).
$ \chi_{A}^F(x)$, as defined by equation 1.2, represents a sieve on A.

Proof. Consider the following commuting diagram which represents subobject F of the presheaf X.

$\displaystyle \xymatrix{
 F(A)\ar[rr]^{F(f)}\ar@{^{(}->}[dd]&&F(B)\ar[rr]^{F(g)}\ar@{^{(}->}[dd]&&F(C)\ar@{^{(}->}[dd]\\
 &&&&\\
 X(A)\ar[rr]^{X(f)}&&X(B)\ar[rr]^{X(g)}&&X(C)\\
 }$

If $ f:B\rightarrow{A}$ belongs to $ \chi_{A}^F(x)$ then given $ g:C\rightarrow B$ it follows that f o g belongs to $ \chi_{A}^F(x)$ since from the above diagram it can be deduced that $ X(f o g)(x)\in F(C)$. This is precisely the definition of a sieve 1.9.1 so we have proved that $ \chi_{A}^F(x):=\{f:B\rightarrow A\vert X(f)(x)\in F(B)\}$ is a sieve. $ \qedsymbol$

As a consequence of axiom 1.1 the condition of being a subobject classifier can be restated in the following way.

Definition 1.16   $ \Omega$ is a subobject classifier iff there is a ``one to one" correspondence between subobject of X and morphisms from X to $ \Omega$.

Given this alternative definition of a subobject classifier, it is easy to prove that $ \Omega$ is a subobject classifier. In fact, from equation 1.2, we can see that indeed, there is a 1:2:1 correspondence between subobject of X and characteristic morphism (character) $ \chi$.
Moreover for each morphism $ \chi:X\rightarrow\Omega$ we have

$\displaystyle F^{\chi}(A):$ $\displaystyle =\chi^{-1}_{A}\{1_{\Omega(A)}\}$    
  $\displaystyle =\{x\in X(A)\vert\chi_{A}(x)=\downarrow A\}$    
  $\displaystyle =$subobject of X    



Next: Global sections Up: Topos Previous: Example of Yoneda's Lemma
Cecilia Flori 2007-03-17