\(L\)-Presheaves and Their Stalks

Abstract

The main aim of this paper is to present the idea of \(L\)-presheaves on a topological space \(X\). Categorical properties of \(L\)-presheaves are studied. The nature of \(L\)-presheaves locally in the neighbourhood of some point is summarized. This aim required constructing the notions of category of \(L\)-sets, \(L\)-direct systems and their \(L\)-limits and \(L\)-functors with their \(L\)-natural transformations. We prove that the ”\(L\)-stalk” is an \(L\)-functor from the category of \(L\)-presheaves to the category of \(L\)-sets.