Why are the quantum observables defined on opens sets a presheaf and not a sheaf? In local quantum field theory or AQFT one can mathematically describe over each open set $U$ of a spacetime $M$ the quantum states or observables of the theory. This structure is commonly referred as a presheaf or a copresheaf.
Why are the states (or observables) over the open sets not a sheaf (cosheaf) structure?
This question is motivated by the following considerations:
The net of local observables which can be roughly described as a copresheaf of (C-star algebras) on pieces of spacetime such that algebras, $A(U) $, assigned to causally disconnected regions commute inside the algebra assigned to any joint neighbourhood.
Up to this point we have by definition a copresheaf.
In order to have a sheaf we need to verify the following two conditions:


*

*(Locality) If ($U_{i}$) is an open covering of an open set $U$, and if $s,t ∈ A(U)$ are such that $s|U_{i} = t|U_{i}$ for each set $U_{i}$ of the covering, then $s = t$

*(Gluing) If ($U_{i}$) is an open covering of an open set $U$, and if for each $i$ a section $s_{i} ∈ A(U_{i})$ is given such that for each pair $U_{i},U_{j}$ of the covering sets the restrictions of $s_{i}$ and $s_{j} $ agree on the overlaps: $s_{i}|U_{i}∩U_{j} = s_{j}|U_{i}∩U_{j}$, then there is a section $s ∈ A(U)$ such that $s|U_{i} = s_{i}$ for each $i$.
The gluing condition guarantee the existence of a section $s$ which the locality condition shows it is unique. 
Evidently, one of this conditions or both fail in general. 
I would be interested in a physical picture of why the sheaf conditions are not satisfied.
To me, the locality condition is stating intuitively that if the observables coincide in every region that form an open cover, then the observables (and the qft) are the same in the open cover. The gluing condition, in the other hand establish that one is able to construct the theory just by gluing local pieces of the theory. Is there then some non local restriction that perhaps avoid us constructing the theory just from local pieces? 
Are these intuitions correct?
 A: As you mention ncatlab I would bet that you have already revised all this... Looking in the net for old discussions and papers, it seems that the an open question was about defining the open sets beyond 1+1 dimensions. Of course (1+1) has a lot of niceties, I remember Borcherds -with 'd'- exploited very well them. 
The net of open sets must be consistent with the "causal diamonds" of Haag et al.  Particularly this is discussed in this thread https://golem.ph.utexas.edu/~distler/blog/archives/000987.html where Urs finishes telling that

In summary: it is not clear to me if the answer to “Should
  Haag-Kastler nets be taken to satisfy the co-sheaf condition?” is
  really “No.”

Later in https://golem.ph.utexas.edu/category/2008/11/local_nets_and_cosheaves.html someone points to the paper   Generally covariant quantum field theory and scaling limits. Comm. Math. Phys. 108 (1987), no. 1, 91--115. http://projecteuclid.org/euclid.cmp/1104116359 to try to use for for a gluing property. Abstracting this paper Urs mentions than 

b) It seems that for A the net of Borchers algebras, A is a co-sheaf

but still the answer is inconclusive
