Categories of Boolean sheaves of simple algebras
This book takes the axiomatic point of view toward sheaf representation theorems. There are three chapters: Categories of Boolean sheaves of algebras; Categories of Boolean sheaves of simple algebras (including various Galois theories); and Categories of locally Boolean sheaves of simple algebras. Details of the development must be left to the work itself, but here is a bit of the flavor. A category is locally indecomposable if it is locally finitely presentable, the product of the initial object with itself is finitely presentable, and the finite products are codisjoint and couniversal. A direct factor of an object \(A\) is a quotient \(f: A\to B\) for which there exists \(g: A\to C\) so that \((f,g)\) makes \(A\) a product of \(B\) and \(C\). An object is indecomposable if it has exactly two direct factors. Theorem 1.7.3: Any locally indecomposable category is equivalent to the category of sheaves of indecomposable objects on Boolean spaces. Those readers interested in such results will want to compare and contrast the paper by \textit{M. Coste} [Applications of sheaves, Proc. Res. Symp., Durham 1977, Lect. Notes Math. 753, 212--238 (1979; Zbl 0422.18007)].
scientific article; zbMATH DE number 3955180