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local epimorphism

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Idea

Given a site 𝒮\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 𝒮\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][S^{op}, Set] that has the closure properties expected of epimorphisms under composition and under pullback.

There is then a unique Grothendieck topology on 𝒮\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 𝒮\mathcal{S}.

Definition

Definition

(system of local epimorphisms)

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

LE1 every epimorphism in [𝒮 op,Set][\mathcal{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 \colon 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.

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

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

for the Yoneda embedding.

Definition

(local epimorphisms from Grothendieck topology)

Let the small category 𝒞\mathcal{C} be equipped with a Grothendieck topology.

For U𝒮U \in \mathcal{S} an object in the site, a morphism of presheaves into the corresponding represented presheaf

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

is a local epimorphism if the sieve

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

at UU which assigns to VV all morphisms from VV to UU that factor through ff

sieve f:V{VgU𝒮| A f y(V) y(g) y(U)} 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

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

is a local epimorphism if for every U𝒮U \in \mathcal{S} and every y(U)By(U) \to B the projection morphism y(U)× BAp 1y(V)y(U) \times_{B} A \overset{p_1}{\to} y(V) out of the pullback/fiber product

y(U)× BA A p 1 (pb) f y(U) B \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:V{VgU𝒮|y(V) A g f y(U) B} 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:

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

is a local epimorphism, if for all y(U)By(U) \longrightarrow B there is a covering {V iι iU}\{ V_i \overset{\iota_i}{\longrightarrow} U \} in the coverage, such that for each ii there exists a lift

y(V i) A ι i f y(U) B \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 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.

Relation to simplicial presheaves

Proposition

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

For 𝒮\mathcal{S} a site, let

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

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

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

out of the Cech nerve simplicial presheaf

C(f) kA× B× BAkfactors 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 [𝒮 op,sSet Qu] proj,loc[\mathcal{S}^{op}, sSet_{Qu}]_{proj,loc}.

(Dugger-Hollander-Isaksen 02, corollary A.3)

\,

References

Last revised on July 16, 2018 at 13:03:53. See the history of this page for a list of all contributions to it.