## Sheaf

### Collection

context | $\langle X,\mathcal T\rangle$ … topological space |

definiendum | $F$ in it |

inclusion | $F$ … seperated presheaf |

for all | $U\in \mathcal T$ |

for all | $C_U$ … open cover$(U)$ |

for all | $S:\prod_{V\in C} FV$ |

postulate | $\left(\forall(V,W\in C_U).\ S(V)|_{V\cap W}=S(W)|_{V\cap W}\right) \implies \exists (s\in FU).\ \forall V.\ S(V)=s|_V$ |

### Discussion

As in seperated presheaf, $t|_V$ denotes the image of $t$ under $F(i)$ with $i:V\to W$.

#### Elaboration

Previously, we defined the 'Locality axiom' which makes a presheaf into a seperated presheaf. The postulate in this entry is the second axiom which makes a seperated presheaf into a sheaf.

**Gluing axiom**: If $V\in C_U$ are elements of a cover $C_U$ of an open set $U$, the function $S$ selects one section $S(V)$ from every $FV$.
A seperated presheaf is a sheaf if such a collection of selected to-be-partions-of-a-section $S(V)$, which locally agree with each other, indeed come from a globally defined section $s$ which is an element of the big $FU$.
So the postulate says that the image of $F$ contains all sections which can arise from gluing together other available sections.

#### Example

From the Wikipedia page below:

“Any continuous map of topological spaces determines a sheaf of sets. Let f : Y → X be a continuous map. We define a sheaf Γ(Y/X) on X by setting Γ(Y/X)(U) equal to the sections U → Y, that is, Γ(Y/X)(U) is the set of all continuous functions s : U → Y such that f ∘ s = id_U. Restriction is given by restriction of functions. This sheaf is called the sheaf of sections of f, and it is especially important when f is the projection of a fiber bundle onto its base space. Notice that if the image of f does not contain U, then Γ(Y/X)(U) is empty. For a concrete example, take X = C \ {0}, Y = C, and f(z) = exp(z). Γ(Y/X)(U) is the set of branches of the logarithm on U.”

#### Reference

Wikipedia: Sheaf