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seperated_presheaf [2014/10/29 09:55] nikolaj |
seperated_presheaf [2014/12/11 17:27] nikolaj |
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| | @#FFFDDD: for all | @#FFFDDD: $s,t\in FU$ | | | @#FFFDDD: for all | @#FFFDDD: $s,t\in FU$ | | ||
| | @#FFFDDD: for all | @#FFFDDD: $C_U$ ... open cover$(U)$ | | | @#FFFDDD: for all | @#FFFDDD: $C_U$ ... open cover$(U)$ | | ||
| - | | @#55EE55: postulate | @#55EE55: $\left(\forall (V\in C_U).\ F(i)(s)=F(i)(t)\right) \implies s=t$ | | + | | @#55EE55: postulate | @#55EE55: $\left(\forall (V\in C_U).\ s|_V=t|_V\right) \implies s=t$ | |
| ==== Discussion ==== | ==== Discussion ==== | ||
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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 | ||
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| $s=t\implies \forall(x\in U).\,s(x)=t(x)$ | $s=t\implies \forall(x\in U).\,s(x)=t(x)$ | ||
| and as corollary we have | and as corollary we have | ||
| - | $s=t\implies \forall(V\subseteq U).\,s|_V=t|_V.$. | + | $s=t\implies \forall(V\subseteq U).\,s|_V=t|_V$ |
| + | . | ||
| Sections also fulfill function extensionality, which goes in the reverse direction | Sections also fulfill function extensionality, which goes in the reverse direction | ||
| - | $\left(\forall(x\in U).\,s(x)=(x)\right)\implies s=t$ | + | $\left(\forall(x\in U).\,s(x)=t(x)\right)\implies s=t$ |
| and consequently, if $C_U$ is a covering of $U$, then | and consequently, if $C_U$ is a covering of $U$, then | ||
| $\left(\forall(V\in C_U).\,s|_V=t|_V\right)\implies s=t$ | $\left(\forall(V\in C_U).\,s|_V=t|_V\right)\implies s=t$ | ||
| + | . | ||
| Now if the $FU$'s of a sheaf are to be a sets of such sections over a topological space, then it should be that the maps $F(i)$ between them can be seen as restriction of the function domains. To this end, if $i:V\to U$ is the inclusion of a small open set $V$ in another $U$, require that if $s,t\in FU$ agree on all restricted domains which make up the cover of $U$, i.e. $F(i)(s)=F(i)(t)$, then there can be no other way in which they can differ, so $s=t$. | Now if the $FU$'s of a sheaf are to be a sets of such sections over a topological space, then it should be that the maps $F(i)$ between them can be seen as restriction of the function domains. To this end, if $i:V\to U$ is the inclusion of a small open set $V$ in another $U$, require that if $s,t\in FU$ agree on all restricted domains which make up the cover of $U$, i.e. $F(i)(s)=F(i)(t)$, then there can be no other way in which they can differ, so $s=t$. | ||
