nLab
local epimorphism

Idea

For $S$ a category, a system of local epimorphisms is a system of morphisms in the presheaf category $[S^{\mathrm{op}},\mathrm{Set}]$ 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 $S$.

Moreover the local isomorphisms among the local epimorphisms admit a calculus of fractions which equips $[S^{\mathrm{op}},\mathrm{Set}]$ with the structure of a category with weak equivalences. The corresponding homotopy category is the category of sheaves on the site $S$.

Definition

Let $S$ be a category. A system of local epimorphisms on the presheaf category $[S^{\mathrm{op}},\mathrm{Set}]$ is a collection of morphisms satisfying the following axioms

LE1 every epimorphism in $[S^{\mathrm{op}},\mathrm{Set}]$ 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 $u:A\to B$ is a local epimorphism precisely if for all $U\in S$ (regarded as a representable presheaf) and morphisms $y:U\to B$, the pullback morphism $A{\times }_{B}U\to U$ is a local epimorphism.

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 $S$ be a category equipped with a Grothendieck topology, hence in particular with a collection of covering sieves for each object $U\in S$.

For a morphism $f:A\to Y(U)$ in the presheaf category $[S^{\mathrm{op}},\mathrm{Set}]$ with $U\in S$ and $Y:S\to [S^{\mathrm{op}},\mathrm{Set}]$ the Yoneda embedding, let ${\mathrm{sieve}}_A\subset Y(U)\in [S^{\mathrm{op}},\mathrm{Set}]$ be the sieve at $U$

$${\mathrm{sieve}}_f:V\mapsto \{h:V\to U\in S\mid Y(h)=Y(V)\stackrel{\exists }{\to }A\to Y(U)\}$$sieve_f : V \mapsto \{ h : V \to U \in S \;|\; Y(h) = Y(V) \stackrel{\exists}{\to} A \to Y(U)\}

which assigns to $V$ all morphisms from $V$ to $U$ that factor through $f$.

The morphism $f:A\to Y(U)$ is a local epimorphism if ${\mathrm{sieve}}_f$ is a covering sieve.

An arbitrary morphism $f:A\to B$ in $[S^{\mathrm{op}},A]$ is a local epimorphism if for every $V\in S$ and every $Y(V)\to B$ the morphism $A{\times }_{Y(U)}Y(V)\to Y(V)$ is a local epimorphism as above.

From local epimorphisms to covering sieves

Conversely, assume a system of local epimorphisms 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.

References

Section 16 of

• Kashiwara-Schapira, Categories and Sheaves