local epimorphism

Given a site , then a local epimorphism is a morphism in the category of presheaves over the site which becomes an epimorphism under sheafification. More abstractly, for a small category, one says axiomatically that a system of local epimorphisms is a system of morphisms in the presheaf category that has the closure properties expected of epimorphisms under composition and under pullback. There is then a unique Grothendieck topology on that induces this system of local epimorphism, see Relation.