This is an old revision of the document!


Sheaf

Collection

context $\langle X,\mathcal T\rangle$ … topological space
definiendum $F$ in it
inclusion $F$ … seperated presheaf
for all $U\in \mathcal T$
for all $C_U$ … open cover$(U)$
for all $S:\prod_{V\in C} FV$
postulate $\left(\forall(V,W\in C_U).\ S(V)|_{V\cap W}=S(W)|_{V\cap W}\right) \implies \exists (s\in FU).\ \forall V.\ S(V)=s|_V$

Discussion

As in seperated presheaf, $t|_V$ denotes the image of $t$ under $F(i)$ with $i:V\to W$.

Elaboration

Previously, we defined the 'Locality axiom' which makes a presheaf into a seperated presheaf. The postulate in this entry is the second axiom which makes a seperated presheaf into a sheaf.

Gluing axiom: If $V\in C_U$ are elements of a cover $C_U$ of an open set $U$, the function $S$ selects one section $S(V)$ from every $FV$. A seperated presheaf is a sheaf if such a collection of selected to-be-partions-of-a-section $S(V)$, which locally agree with each other, indeed come from a globally defined section $s$ which is an element of the big $FU$. So the postulate says that the image of $F$ contains all sections which can arise from gluing together other available sections.

Reference

Wikipedia: Sheaf

Parents

Context

Subset of

Link to graph
Log In
Improvements of the human condition