Essential pure monomorphisms of sheaves of group actions

• DOI: 10.1007/s00233-010-9212-7
• Publication Date: 2010-02-11T19:33:36Z
• Authors: M. M. Ebrahimi; M. Haddadi
• Keywords: 0101 mathematics; 01 natural sciences

Nowadays purity plays a role in at least four branches of mathematics: Module Theory, Theory of Acts over semigroups, Model Theory, and Category Theory. Adamek and Rosciký have studied these notions categorically, and Rothmaler model-theoretically. Some authors including Banaschewski, Gould, and Normak have studied purity on G-acts, acts over a monoid or a group G. In this paper we take both the group G and the set A in the definition of a G-act to be sheaves (multigroups and multisets) over a topological space X to get the category G-ShX of G-sheaves, and mainly study different types of essentiality with respect to the different classes ℳ of pure monomorphisms in G-ShX. We give some necessary and sometimes sufficient conditions for the different types of essentiality. We will also see that these notions of essentiality are local, in the sense that an ℳ-monomorphism f is ℳ-essential if and only if there exists a cover of disjoint open sets of X such that the restriction of f to every component is essential.