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seperated_presheaf [2014/10/29 10:16] nikolaj |
seperated_presheaf [2014/10/29 14:29] nikolaj |
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| ===Elaboration=== | ===Elaboration=== | ||
| - | A presheaf in topology assigns a set $FU$ to each open set $U$ of a topological space $\langle X,\mathcal T\rangle$. A [[sheaf]] is a presheaf fulfilling two axioms ('locality' and 'gluing') and it's intended to capture the case where $FU$ are sets of sections over a topological space (e.g. the section of a fibre bundle $p:E\to X$, e.g. vector fields) and the arrows $F(i)$ are restrictions of those section to smaller open sets. The definition of a seperated presheaf is the first step towards the definition of a sheaf. | + | A presheaf in topology assigns a set $FU$ to each open set $U$ of a topological space $\langle X,\mathcal T\rangle$. A [[sheaf]] is a presheaf fulfilling two axioms ('locality' and 'gluing') and it's intended to capture the case where $FU$ are sets of sections over a topological space (e.g. the section of a fibre bundle $p:E\to X$, e.g. vector fields, e.g. all 1-forms) and the arrows $F(i)$ are restrictions of those section to smaller open sets. The definition of a seperated presheaf is the first step towards the definition of a sheaf. |
| **Locality axiom**: To understand the postulate, recall that for sections $s,t:U\to Y$ (or function in general) we trivially have that | **Locality axiom**: To understand the postulate, recall that for sections $s,t:U\to Y$ (or function in general) we trivially have that | ||
