Loopholes in Bell test experiments

Let X be a topological space, and $\mathrm{S}\mathrm{h}(X,\mathcal{𝒞})$ denote the category of sheaves with values in $\mathcal{𝒞}$. Then the map that associates to a sheaf $\mathcal{ℱ}$ its global sections $\mathrm{\Gamma }(X,\mathcal{ℱ})$ 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 ${\underset{\_}{\mathbb{\mathbb{Z}}}}_X$ be the locally constant sheaf. Then its global sections are given by $\mathrm{\Gamma }(X,{\underset{\_}{\mathbb{\mathbb{Z}}}}_X)={\mathbb{\mathbb{Z}}}^{{\pi }_0(X)}$, 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. $\mathrm{\Gamma }(X,{\mathcal{𝒪}}_X)=\mathbb{ℂ}$

• Let $\mathcal{𝒪}(i),i\in \mathbb{\mathbb{Z}}$ denote the twisting sheaves on the projective space ${\mathbb{\mathbb{P}}}_k^n$, then $\mathrm{\Gamma }(X,\mathcal{𝒪}(d))=k_d[X_0,\dots ,X_n]$ for $d\ge 0$, and 0 for $d<0$.

See also

• Sheaf (mathematics)
• Sheaf cohomology