Global Sections


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Other important features of Topos theory are the global sections which I will define below, and the local section which are defined in 1.10.

Definition 1.16   A global section or global element of a presheaf X in $ \mathcal{S}^{\mathscr{C}^{op}}$ is a map $ k:1\rightarrow X$ from the terminal object 1 to the presheaf X.

What k does is to assign to each object A in $ \mathscr{C}$ an element $ k_A\in X(A)$ in the corresponding object of the presheaf X. The assignment is such that, given a function $ f:B\rightarrow A$ the following relation holds

$\displaystyle X(f)(k_A)=k_B$ (1.3)

What 1.3 uncovers, is that the elements of X(A) assigned by the global section k, are mapped into each other by the morphisms in X. Presheaves with a local or partial section can exist even if they do not have a global section.
Cecilia Flori 2007-01-02