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--- sheaf [2014/10/29 10:10]
nikolaj
+++ sheaf [2014/10/29 10:12]
nikolaj
@@ Line 13: / Line 13: @@
 Previously, we defined the 'Locality axiom' which makes a [[presheaf . topology|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**: 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 $FU$.
-**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 $FU$.
+So the postulate says that the image of $F$ contains all sections which can arise from gluing together other available sections.
-The postulate says that the image of $F$ contains contain all sections which can arise from gluing together other available sections.
 === Reference ===