On some sheaves of special groups

We consider sheaves of special groups (mainly over Boolean spaces). These are connected to the sheaves of abstract Witt rings considered by Marshall in [9], used therein in particular to classify spaces of orderings with a finite number of accumulation points. Our approach allows us to show that the so-called “question 1” (see [11, 1]) has a positive answer for these spaces. We conclude these notes by computing the behaviour of the Boolean hull functor when applied to a sheaf of special groups. The author would like to thank the referee for some very helpful comments. Our references are [3] for special groups and the Boolean hull functor, and [10] for spaces of orderings. The functorial link between (reduced) special groups and spaces of orderings –a categorical duality– can be found in [3, Chapter 3]. If G is a (reduced) special group we denote by (XG, G) (or simply XG) its associated space of orderings. Conversely, if Y is a space of orderings, we denote by GY its associated reduced special group.