For a category, 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.
A specification of a system of local epimorphisms is equivalent to giving a Grothendieck topology and hence the structure of a site on .
Moreover the local isomorphisms among the local epimorphisms admit a calculus of fractions which equips with the structure of a category with weak equivalences. The corresponding homotopy category is the category of sheaves on the site .
Let be a category. A system of local epimorphisms on the presheaf category is a collection of morphisms satisfying the following axioms
LE1 every epimorphism in is a local epimorphism;
LE2 the composite of two local epimorphisms is a local epimorphism;
LE3 if the composite is a local epimorphism, then so is ;
LE4 a morphism is a local epimorphism precisely if for all (regarded as a representable presheaf) and morphisms , the pullback morphism is a local epimorphism.
Relation to sieves
The specification of a system of local epimorphisms is equivalent to a system of Grothendieck covering sieves.
To see this, translate between local epimorphisms to sieves as follows.
From covering sieves to local epimorphisms
Let be a category equipped with a Grothendieck topology, hence in particular with a collection of covering sieves for each object .
For a morphism in the presheaf category with and the Yoneda embedding, let be the sieve at
which assigns to all morphisms from to that factor through .
The morphism is a local epimorphism if is a covering sieve.
An arbitrary morphism in is a local epimorphism if for every and every the morphism is a local epimorphism as above.
From local epimorphisms to covering sieves
Conversely, assume a system of local epimorphisms is given.
Declare a sieve at to be a covering sieve precisely if the inclusion morphism is a local epimorphism. Then this defines a Grothendieck topology encoded by the collection of local epimorphisms.
Section 16 of