Essential pure monomorphisms of sheaves of group actions. (Q982207)

Here are studied various notions of essentiality in the category \(\mathbf G\text{-}\mathbf{ShX}\) of sheaves -- multigroup \(G\) acts on multisets on a topological space \(X\). Let \(\mathcal M\) be the class of affine, stalk or quasi-pure \(G\)-monomorphisms in \(\mathbf G\text{-}\mathbf{ShX}\); \(f\in\mathcal M\) is called \(\mathcal M\)-essential if \(gf\in\mathcal M\) implies \(g\in\mathcal M\); weak \(\mathcal M\)-essential if \(gf\in\mathcal M\) implies that \(g\) is monic; essential \(\mathcal M\)-monomorphism if \(g\) is a monomorphism whenever \(gf\) is a monomorphism. Properties of these types of essentiality are investigated; it is shown that an \(\mathcal M\)-monomorphism \(f\) is \(\mathcal M\)-essential iff there exists a cover of disjoint open sets of \(X\) such that the restriction of \(f\) to components is \(\mathcal M\)-essential.