nLab local epimorphism

Idea

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 $\mathcal{S}$.

Definition

Definition

(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.

Properties

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.

Throughout, let $\mathcal{S}$ be a small category. Write $[\mathcal{S}^{op}, Set]$ for its category of presheaves and write

$$y \;\colon\; \mathcal{S} \longrightarrow [\mathcal{S}^{op}, Set]$$

for the Yoneda embedding.

Definition

(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

$$A \overset{f}{\longrightarrow} y(U) \;\;\; \in [\mathcal{S}^{op}, Set]$$

is a local epimorphism if the sieve

$$sieve_A \subset y(U) \in [S^{op}, Set]$$

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

$$sieve_f \;\colon\; V \;\mapsto\; \left\{ V \overset{g}{\to} U \,\in \mathcal{S} \;\Big\vert\; \array{ && A \\ & {}^{\mathllap{\exists}}\nearrow & \Big\downarrow{}^{f} \\ y(V) & \underset{y(g)}{\longrightarrow} & y(U) } \right\}$$

is a covering sieve.

A general morphism of presheaves

$$A \overset{}{\longrightarrow} B \;\;\; \in [\mathcal{S}^{op}, Set]$$

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

$$\array{ y(U)\times_{B} A &\overset{}{\longrightarrow}& A \\ {}^{\mathllap{p_1}}\Big\downarrow &{}^{(pb)}& \Big\downarrow{}^{\mathrlap{f}} \\ y(U) &\underset{}{\longrightarrow}& B }$$

is a local epimorphism as above. By the universal property of the fiber product, this means equivalently that

$$sieve_f \;\colon\; V \;\mapsto\; \left\{ V \overset{g}{\to} U \,\in \mathcal{S} \;\Big\vert\; \array{ y(V) &\overset{\exists}{\longrightarrow}& A \\ {}^{\mathllap{g}}\Big\downarrow & & \Big\downarrow{}^{f} \\ y(U) & \underset{}{\longrightarrow} & B } \right\}$$

is a covering sieve.

Remark

(in terms of coverages)

If instead of a Grothendieck topology we are just given a coverage, then Def. becomes:

$$A \overset{f}{\longrightarrow} B \;\;\; \in [\mathcal{S}^{op}, Set]$$

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

$$\array{ y(V_i) &\overset{\exists}{\longrightarrow}& A \\ {}^{\mathllap{\iota_i}}\Big\downarrow & & \Big\downarrow{}^{f} \\ y(U) & \underset{}{\longrightarrow} & B }$$

Definition

(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.

Relation to simplicial presheaves

Proposition

(Cech nerve projection of local epimorphism is local weak equivalence)

For $\mathcal{S}$ a site, let

$$A \overset{f}{\longrightarrow} B \;\colon\; [\mathcal{S}^{op}, Set]$$

be a local epimorphism (Def. ). Then the projection

$$C(f) \longrightarrow B \;\;\;\; \in [\mathcal{S}^{op}, sSet]$$

out of the Cech nerve simplicial presheaf

$$C(f)_k \;\coloneqq\; \underset{ k \; \text{factors} }{ \underbrace{ A \times_B \cdots \times_B A }}$$

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)

$\,$

References

• 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)