Seperated presheaf


context $\langle X,\mathcal T_X\rangle$ … topological space
definiendum $F$ in it
inclusion $F$ in ${\bf Set}^{\mathrm{Op}(X)^{\mathrm{op}}}$
for all $U\in \mathcal T_X$
for all $s,t\in FU$
for all $C_U$ … open cover$(U)$
postulate $\left(\forall (V\in C_U).\ s|_V=t|_V\right) \implies s=t$


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


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 $s=t$.


Wikipedia: Sheaf



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