local epimorphism

For $S$ a category, 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.

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 $[S^{op}, Set]$ with the structure of a category with weak equivalences. The corresponding homotopy category is the category of sheaves on the site $S$.

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

**LE1** every epimorphism in $[S^{op}, Set]$ 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 $u : A \to B$ 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.

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.

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 $[S^{op},Set]$ with $U \in S$ and $Y : S \to [S^{op}, Set]$ the Yoneda embedding, let $sieve_A \subset Y(U) \in [S^{op}, Set]$ be the sieve at $U$

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

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 $sieve_f$ is a covering sieve.

An arbitrary morphism $f : A \to B$ in $[S^{op}, A]$ 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.

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.

Section 16 of

- Kashiwara-Schapira, Categories and Sheaves

Revised on September 10, 2011 04:19:27
by Mike Shulman
(71.136.248.27)