Local sections


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Local sections

Definition 1.17   A local or partial section of a presheaf X in $ \mathcal{S}^{\mathscr{C}^{op}}$ is a map $ \rho:U\rightarrow X$ where U is a subobject of the terminal object 1.

In a presheaf, a subobject U of 1 can either be the empty set $ \emptyset$, or a singleton $ \{*\}$. From the above definition, it is clear that a local section is an assignment of an element of an object of X to the corresponding subobject U of 1 in $ \mathscr{C}$. This assignment is said to be ``closed downwards", i.e. given a subobject U(A)=$ \{*\}$ of 1 and a $ \mathscr{C}$-morphisms $ f:B\rightarrow A$ then we have U(B)=$ \{*\}$. To better explain the above let us consider a category with 4 elements $ \{A,B,C,D\}$ such that the following relations hold between the elements:

$\displaystyle \xymatrix{ A\ar[rr]^f\ar[dd]_i&&B\ar[dd]^g\\ &&\\ D\ar[rr]_p&&C\\ }$

Given a subobject U of 1 we then have the following relations

$\displaystyle \xymatrix{ U(A)\ar[rr]^{U(f)}\ar[dd]_{U(i)}&&U(B)\ar[dd]^{U(g)}\\ &&\\ U(D)\ar[rr]_{U(p)}&&U(C)\\ }$

If U(A)=$ \emptyset$ then U(f) is either the unique function $ \emptyset\rightarrow\{*\}$ iff $ U(B)=\{*\}$ or $ \emptyset\rightarrow\emptyset$ iff $ U(B)=\emptyset$. If instead $ U(A)=\{*\}$ then the only possibility is that $ U(B)=\{*\}$ since there does not exist a function $ \{*\}\rightarrow\emptyset$. Therefore $ \rho$ assigns to particular subsets of objects $ A\in X$, elements $ \rho_A$. These objects A are called the domain of $ \rho$ $ (dom\hspace{.05in}\rho)$ and are such that the following conditions are satisfied:
  • The domain is closed downwards i.e. if $ A\in dom\hspace{.05in}\rho$ and if there exists a map $ f:B\rightarrow A$ then $ B\in dom\hspace{.05in}\rho$


  • If $ A\in dom\rho$ and if there exists a map $ f:B\rightarrow A$, then the following condition is satisfied:

    $\displaystyle X(f)(\rho_A)=\rho_B$    


Cecilia Flori 2007-01-02