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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
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.
In the above, if $V\in C$ are elements of a cover of an open set, the function $S$ selects one section $S(V)$ from every $FV$.
Gluing axiom: Let $V,U$ with $V\subset U$ be open sets. A seperated presheaf is a sheaf if a bunch of chosen to-be-partions-of-a-section $S(V)\in FV$, which locally agree with each other, indeed come from a globally defined section $s$ which is an element of the big $FU$. The postulate says that the image of $F$ contains contain all sections which can arise from gluing together other available sections.
Reference
Wikipedia: Sheaf
Parents
Context
Subset of
