Given a site $\mathcal{S}$, then a local epimorphism is a morphism in the category of presheaves over the site which becomes an epimorphism under sheafification.
More abstractly, for $\mathcal{S}$ a small category, one says axiomatically that a system of local epimorphisms is a system of morphisms in the presheaf category $[S^{op}, Set]$ that has the closure properties expected of epimorphisms under composition and under pullback.
There is then a unique Grothendieck topology on $\mathcal{S}$ that induces this system of local epimorphism, see Relation to sieves below.
Moreover the local isomorphisms among the local epimorphisms admit a calculus of fractions which equips the category of presheaves with the structure of a category with weak equivalences. The corresponding reflective localization is the category of sheaves on the site $\mathcal{S}$.
(system of local epimorphisms)
Let $\mathcal{S}$ be a small category. A system of local epimorphisms on the presheaf category $[\mathcal{S}^{op}, Set]$ is a class of morphisms satisfying the following axioms:
LE1 every epimorphism in $[\mathcal{S}^{op}, Set]$ is a local epimorphism;
LE2 the composite of two local epimorphisms is a local epimorphism;
LE3 if the composite $A_1 \stackrel{u}{\to} A_2 \stackrel{v}{\to} A_3$ is a local epimorphism, then so is $v$;
LE4 a morphism $u \colon A \to B$ is a local epimorphism precisely if for all $U \in \mathcal{S}$ (regarded as a representable presheaf) and morphisms $y: U \to B$, the pullback morphism $A \times_B U \to U$ is a local epimorphism.
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.
Throughout, let $\mathcal{S}$ be a small category. Write $[\mathcal{S}^{op}, Set]$ for its category of presheaves and write
for the Yoneda embedding.
(local epimorphisms from Grothendieck topology)
Let the small category $\mathcal{S}$ be equipped with a Grothendieck topology.
For $U \in \mathcal{S}$ an object in the site, a morphism of presheaves into the corresponding represented presheaf
is a local epimorphism if the sieve
at $U$ which assigns to $V$ all morphisms from $V$ to $U$ that factor through $f$
is a covering sieve.
A general morphism of presheaves
is a local epimorphism if for every $U \in \mathcal{S}$ and every $y(U) \to B$ the projection morphism $y(U) \times_{B} A \overset{p_1}{\to} y(V)$ out of the pullback/fiber product
is a local epimorphism as above. By the universal property of the fiber product, this means equivalently that
is a covering sieve.
(in terms of coverages)
If instead of a Grothendieck topology we are just given a coverage, then Def. becomes:
is a local epimorphism, if for all $y(U) \longrightarrow B$ there is a covering $\{ V_i \overset{\iota_i}{\longrightarrow} U \}$ in the coverage, such that for each $i$ there exists a lift
(Grothendieck topology from local epimorphisms)
Conversely, assume a system of local epimorphisms as in Def. is given.
Declare a sieve $F$ at $U$ to be a covering sieve precisely if the inclusion morphism $F \hookrightarrow U$ is a local epimorphism. Then this defines a Grothendieck topology encoded by the collection of local epimorphisms.
(Cech nerve projection of local epimorphism is local weak equivalence)
For $\mathcal{S}$ a site, let
be a local epimorphism (Def. ). Then the projection
out of the Cech nerve simplicial presheaf
is a weak equivalence in the projective local model structure on simplicial presheaves $[\mathcal{S}^{op}, sSet_{Qu}]_{proj,loc}$.
(Dugger-Hollander-Isaksen 02, corollary A.3)
$\,$
Kashiwara-Schapira, section 16 of Categories and Sheaves
Saunders Mac Lane, Ieke Moerdijk, chapter III, section 8 of Sheaves in Geometry and Logic, Springer 1992
Daniel Dugger, Sharon Hollander, Daniel Isaksen, Hypercovers and simplicial presheaves, Mathematical Proceedings of the Cambridge Philosophical Society. Vol. 136. No. 1. Cambridge University Press, 2004 (arXiv:math/0205027)
Last revised on July 4, 2021 at 22:16:26. See the history of this page for a list of all contributions to it.