nLab
local epimorphism

Contents

Idea

For SS a category, a system of local epimorphisms is a system of morphisms in the presheaf category [S op,Set][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 SS.

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

Definition

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

LE1 every epimorphism in [S op,Set][S^{op}, Set] is a local epimorphism;

LE2 the composite of two local epimorphisms is a local epimorphism;

LE3 if the composite A 1uA 2vA 3A_1 \stackrel{u}{\to} A_2 \stackrel{v}{\to} A_3 is a local epimorphism, then so is vv;

LE4 a morphism u:ABu : A \to B is a local epimorphism precisely if for all USU \in S (regarded as a representable presheaf) and morphisms y:UBy: U \to B, the pullback morphism A× BUUA \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 SS be a category equipped with a Grothendieck topology, hence in particular with a collection of covering sieves for each object USU \in S.

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

sieve f:V{h:VUSY(h)=Y(V)AY(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 VV all morphisms from VV to UU that factor through ff.

The morphism f:AY(U)f : A \to Y(U) is a local epimorphism if sieve fsieve_f is a covering sieve.

An arbitrary morphism f:ABf : A \to B in [S op,A][S^{op}, A] is a local epimorphism if for every VSV \in S and every Y(V)BY(V) \to B the morphism A× Y(U)Y(V)Y(V)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 FF at UU to be a covering sieve precisely if the inclusion morphism FUF \hookrightarrow U is a local epimorphism. Then this defines a Grothendieck topology encoded by the collection of local epimorphisms.

References

Section 16 of

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