Loopholes in Bell test experiments

Let X be a topological space, and $\mathrm {Sh} (X,{\mathcal {C}})$ denote the category of sheaves with values in ${\mathcal {C}}$. Then the map that associates to a sheaf ${\mathcal {F}}$ its global sections $\Gamma (X,{\mathcal {F}})$ is a covariant functor to ${\mathcal {C}}$.

If ${\mathcal {C}}$ is the category of abelian groups, then this functor is left exact. This important remark leads to the notion of sheaf cohomology, via derived functors.

Examples

• Let ${\underline {\mathbb {Z} }}_{X}$ be the locally constant sheaf. Then its global sections are given by $\Gamma (X,{\underline {\mathbb {Z} }}_{X})=\mathbb {Z} ^{\pi _{0}(X)}$, i.e. the direct sum indexed by connected components of X

• Let ${\mathcal {O}}_{X}$ be the sheaf of holomorphic functions on the compact connected complex manifold X, then by the maximum principle, global sections are constant, ie. $\Gamma (X,{\mathcal {O}}_{X})=\mathbb {C}$

• Let ${\mathcal {O}}(i),i\in \mathbb {Z}$ denote the twisting sheaves on the projective space $\mathbb {P} _{k}^{n}$, then $\Gamma (X,{\mathcal {O}}(d))=k_{d}[X_{0},\ldots ,X_{n}]$ for $d\geq 0$, and 0 for $d<0$.

See also

• Sheaf (mathematics)
• Sheaf cohomology