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separated presheaf

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Separated presheaf

Idea

The condition that a presheaf be a sheaf may be seen as a condition of unique existence. A presheaf is separated if it satisfies the uniqueness part.

Definition

Let SS be a site.

Recall that a sheaf on SS is a presheaf APSh SA \in PSh_S such that for all local isomorphisms YXY \to X the induced morphism PSh S(X,A)PSh S(Y,A)PSh_S(X,A) \to PSh_S(Y,A) (under the hom-functor PSh S(,A)PSh_S(-,A)) is an isomorphism. (For an arbitrary class of morphisms VV, the corresponding condition is called being a local object.)
It is sufficient to check this on the dense monomorphisms instead of all local isomorphisms. This is equivalent to checking covering sieves.

Definition

A presheaf APSh(S)A \in PSh(S) is called separated (or a monopresheaf) if for all local isomorphisms YXY \to X the induced morphism Hom(X,A)Hom(Y,A)Hom(X,A) \to Hom(Y,A) is a monomorphism.

More generally, for a class VV of arrows in a category CC, an object ACA\in C is VV-separated if for all morphisms YXY\to X in VV, the induced morphism Hom(X,A)Hom(Y,A)Hom(X,A)\to Hom(Y,A) is a monomorphism.

Remark

As for sheaves, it is sufficient to check the separation condition on the dense monomorphisms, hence on the sieves.

For {p i:U iU}\{p_i : U_i \to U\} a covering family of an object USU \in S, the condition is that if a,bA(U)a,b \in A(U) are such that for all ii we have A(p i)(a)=A(p i)(b)A(p_i)(a) = A(p_i)(b) then already a=ba = b.

Remark

The definition generalizes to any system of local isomorphisms on any topos, such as that obtained from any Lawvere-Tierney topology, or equivalently any subtopos.

Example

Example

Let (S,J)(S,J) be a site for which every JJ-covering family is inhabited. Then for any set XX, the constant presheaf SaXS\ni a \mapsto X is separated.

See also at locally connected site.

Properties

Proposition

The full subcategory

i:Sep(S)PSh(S) i : Sep(S) \hookrightarrow PSh(S)

of separated presheaves in a presheaf category is

Being a reflective subcategory means that there is a left adjoint functor to the inclusion

(L sepi):Sep(S)L sepPSh S. (L_{sep} \dashv i) : Sep(S) \stackrel{\overset{L_{sep}}{\leftarrow}}{\hookrightarrow} PSh_S \,.
Definition

For APSh SA \in PSh_S the separafication L sepAL_{sep}A of AA is the presheaf that assigns equivalence classes

L sepA:UA(U)/ U, L_{sep}A : U \mapsto A(U)/\sim_U \,,

where U\sim_U is the equivalence relation that relates two elements aba \sim b iff there exists a covering {p i:U iU}\{p_i : U_i \to U\} such that A(p i)(a)=A(p i)(b)A(p_i)(a) = A(p_i)(b) for all ii.

This construction extends in the evident way to a functor

L sep:PSh(S)Sep(S). L_{sep} : PSh(S) \to Sep(S) \,.
Proposition

This functor L sepL_{sep} is indeed a left adjoint to the inclusion i:Sep(S)PSh(S)i : Sep(S) \hookrightarrow PSh(S).

Proof

Let APSh(S)A \in PSh(S) and BSep(S)PSh(S)B \in Sep(S) \hookrightarrow PSh(S). We need to show that morphisms f:ABf : A \to B in PSh CPSh_C are in natural bijection with morphisms L sepABL_{sep} A \to B in Sep(S)Sep(S).

For ff such a morphism and f U:A(U)B(U)f_U : A(U) \to B(U) its component over any object USU \in S, consider any covering {U iU}\{U_i \to U\}, let S(U i)US(U_i) \to U be the corresponding sieve and consider the commuting diagram

{(a iA(U i))|} {(b iF(U i))|} A(U) f U B(U) \array{ \{(a_i \in A(U_i)) | \cdots \} &\to& \{(b_i \in F(U_i)) | \cdots \} \\ \uparrow && \uparrow \\ A(U) &\stackrel{f_U}{\to}& B(U) }

obtained from the naturality of PSh S(S(U i)U,AfB)PSh_S(S(U_i) \to U, A \stackrel{f}{\to} B).

If for a,aA(U)a,a' \in A(U) two elements that are not equal their restrictions to the cover become equal in that i:a| U i=a| U i\forall i : a|_{U_i} = a'|_{U_i}, then also f(a| U i)=f(a| U i)f(a|_{U_i}) = f(a'|_{U_i}) and since the right vertical morphism is monic there is a unique bB(U)b \in B(U) mapping to the latter. The commutativity of the diagram then demands that f(a)=f(a)=bf(a) = f(a') = b.

Since this argument applies to all covers of UU, we have that f Uf_U factors uniquely through the projection map A(U)A(U)/ U=:L sep(U)A(U) \to A(U)/\sim_U =: L_{sep}(U) onto the quotient. Since this is true for every object UU we have that ff factors uniquely through AL sepAA \to L_{sep}A.

Biseparated presheaf

Idea

Often one is interested in separated presheaves with respect to one coverage that are sheaves with respect to another coverage. These are called biseparated presheaves .

This typically arises if a reflective subcategory

CSh(S) C \stackrel{\stackrel{}{\leftarrow}}{\hookrightarrow} Sh(S)

of a sheaf topos is given. This is the localization at a set WW of morphisms in Sh(S)Sh(S), with CC the full subcategory of all local objects cc: objects such that Sh (S)(w,c)Sh_(S)(w,c) is an isomorphism for all wWw \in W. A WW-separated object is then called a biseparated presheaf on SS and their collection BiSep(S)BiSep(S) factors the reflective inclusion as

CBiSep(S)Sh(S). C \stackrel{\leftarrow}{\hookrightarrow} BiSep(S) \stackrel{\leftarrow}{\hookrightarrow} Sh(S) \,.

Definition

Definition

A bisite is a small category SS equipped with two coverages: JJ and KK such that JKJ \subset K.

A presheaf APSh SA \in PSh_S is called (J,K)(J,K)-biseparated if it is

  • a sheaf with respect to JJ;

  • a separated presheaf with respect to KK.

Write

BiSep (J,K)(S)Sh J(S)PSh(S) BiSep_{(J,K)}(S) \hookrightarrow Sh_J(S) \hookrightarrow PSh(S)

for the full subcategory on biseparated presheaves.

Properties

Proposition

Biseparated presheaves form a reflective subcategory of all sheaves

BiSep (J,K)(S)L sep KSh J(S). BiSep_{(J,K)}(S) \stackrel{\stackrel{L^K_{sep}}{\leftarrow}}{\hookrightarrow} Sh_J(S) \,.

See quasitopos for the proof.

References

The general theory of biseparated presheaves and Grothendieck quasitoposes is in section C.2.2 of

A concrete description of separafication appears on page 43 of

  • Angelo Vistoli, Notes on Grothendieck topologies, fibered categories and descent theory (pdf)

category: sheaf theory

Revised on June 20, 2013 12:54:52 by Urs Schreiber (82.169.65.155)