===== Seperated presheaf =====
==== Collection ====
| @#55CCEE: context     | @#55CCEE: $\langle X,\mathcal T_X\rangle$ ... topological space |
| @#FFBB00: definiendum | @#FFBB00: $F$ in it |
| @#AAFFAA: inclusion   | @#AAFFAA: $F$ in ${\bf Set}^{\mathrm{Op}(X)^{\mathrm{op}}}$ |
| @#FFFDDD: for all     | @#FFFDDD: $U\in \mathcal T_X$ |
| @#FFFDDD: for all     | @#FFFDDD: $s,t\in FU$ |
| @#FFFDDD: for all     | @#FFFDDD: $C_U$ ... open cover$(U)$ |
| @#55EE55: postulate   | @#55EE55: $\left(\forall (V\in C_U).\ s|_V=t|_V\right) \implies s=t$ |

==== Discussion ====
Here we use the notation discussed in [[Presheaf . topology|presheaf]]. I.e. in the last line, "$s|_V=t|_V$" is notation for "$F(i)(s)=F(i)(t)$", where $i:V\to U$.

===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, 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

$s=t\implies \forall(x\in U).\,s(x)=t(x)$ 
and as corollary we have 
$s=t\implies \forall(V\subseteq U).\,s|_V=t|_V$
.

Sections also fulfill function extensionality, which goes in the reverse direction

$\left(\forall(x\in U).\,s(x)=t(x)\right)\implies s=t$
and consequently, if $C_U$ is a covering of $U$, then
$\left(\forall(V\in C_U).\,s|_V=t|_V\right)\implies s=t$
.

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 $s=t$.

=== Reference ===
Wikipedia: [[http://en.wikipedia.org/wiki/Sheaf_%28mathematics%29|Sheaf]]

==== Parents ====
=== Context ===
[[Topological space]]
=== Subset of ===
[[Presheaf . topology]]
=== Requirements ===
[[Open cover]]