I have been looking for references on sheaves that take value in the category of Boolean rings (e.g. about cohomology, etc). Would someone be able to give me some?
Or are they interesting at all? Thanks!
This is not exactly what you're asking for, but there are some results about cohomology groups of sheaves on Stone spaces. But it is probably related to your question since sheaves with values in Boolean rings are only interesting when the space is disconnected.
The paper "Sheaf cohomology of locally compact totally disconnected spaces" by Roger Wiegand shows that cohomology on a Stone space may be computed via Cech cohomology with respect to a cover consisting of compact open subsets. The paper "Some topological invariants of Stone spaces" by the same author discusses the relationship between the covering dimension and the cohomological dimension of Stone spaces.
Sheaves on Stone spaces are useful in the topological characterization of algebraic rings. See for example "Topological representation of algebras" by Arens, Kaplansky and "Modules over commutative regular rings" (Section "m-rings") by Pierce.