# Differences

This shows you the differences between two versions of the page.

Both sides previous revision Previous revision Next revision | Previous revision | ||

seperated_presheaf [2014/10/29 10:16] nikolaj |
seperated_presheaf [2014/12/11 17:27] nikolaj |
||
---|---|---|---|

Line 13: | Line 13: | ||

===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 | ||

Line 24: | Line 24: | ||

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 | ||

$\left(\forall(V\in C_U).\,s|_V=t|_V\right)\implies s=t$ | $\left(\forall(V\in C_U).\,s|_V=t|_V\right)\implies s=t$ |