nLab
separated presheaf

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 $S$ be a site.

Recall that a sheaf on $S$ is a presheaf $A\in {\mathrm{PSh}}_S$ such that for all local isomorphisms $Y\to X$ the induced morphism ${\mathrm{PSh}}_S(X,A)\to {\mathrm{PSh}}_S(Y,A)$ (under the hom-functor ${\mathrm{PSh}}_S(-,A)$) is an isomorphism. (For an arbitrary class of morphisms $V$, 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 $A\in \mathrm{PSh}(S)$ is called separated (or a monopresheaf) if for all local isomorphisms $Y\to X$ the induced morphism $\mathrm{Hom}(X,A)\to \mathrm{Hom}(Y,A)$ is a monomorphism.

More generally, for a class $V$ of arrows in a category $C$, an object $A\in C$ is $V$-separated if for all morphisms $Y\to X$ in $V$, the induced morphism $\mathrm{Hom}(X,A)\to \mathrm{Hom}(Y,A)$ is a monomorphism.

Remark

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

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

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.

Properties

Proposition

The full subcategory

$$i:\mathrm{Sep}(S)\hookrightarrow \mathrm{PSh}(S)$$i : Sep(S) \hookrightarrow PSh(S)

of separated presheaves in a presheaf category is

• a reflective subcategory;

• a quasitopos.

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

$$(L_{\mathrm{sep}}⊣i):\mathrm{Sep}(S)\stackrel{\stackrel{L_{\mathrm{sep}}}{\leftarrow }}{\hookrightarrow }{\mathrm{PSh}}_S.$$(L_{sep} \dashv i) : Sep(S) \stackrel{\overset{L_{sep}}{\leftarrow}}{\hookrightarrow} PSh_S \,.

Definition

For $A\in {\mathrm{PSh}}_S$ the separafication $L_{\mathrm{sep}}A$ of $A$ is the presheaf that assigns equivalence classes

$$L_{\mathrm{sep}}A:U\mapsto A(U)/{\sim }_U,$$L_{sep}A : U \mapsto A(U)/\sim_U \,,

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

This construction extends in the evident way to a functor

$$L_{\mathrm{sep}}:\mathrm{PSh}(S)\to \mathrm{Sep}(S).$$L_{sep} : PSh(S) \to Sep(S) \,.

Proposition

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

Proof

Let $A\in \mathrm{PSh}(S)$ and $B\in \mathrm{Sep}(S)\hookrightarrow \mathrm{PSh}(S)$. We need to show that morphisms $f:A\to B$ in ${\mathrm{PSh}}_C$ are in natural bijection with morphisms $L_{\mathrm{sep}}A\to B$ in $\mathrm{Sep}(S)$.

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

$$\begin{array}{ccc}\{(a_i\in A(U_i))\mid \cdots \}& \to & \{(b_i\in F(U_i))\mid \cdots \}\\ \uparrow & & \uparrow \\ A(U)& \stackrel{f_U}{\to }& B(U)\end{array}$$\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 ${\mathrm{PSh}}_S(S(U_i)\to U,A\stackrel{f}{\to }B)$.

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

Since this argument applies to all covers of $U$, we have that $f_U$ factors uniquely through the projection map $A(U)\to A(U)/{\sim }_U=:L_{\mathrm{sep}}(U)$ onto the quotient. Since this is true for every object $U$ we have that $f$ factors uniquely through $A\to L_{\mathrm{sep}}A$.

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

$$C\overleftarrow{\hookrightarrow }\mathrm{Sh}(S)$$C \stackrel{\stackrel{}{\leftarrow}}{\hookrightarrow} Sh(S)

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

$$C\overleftarrow{\hookrightarrow }\mathrm{BiSep}(S)\overleftarrow{\hookrightarrow }\mathrm{Sh}(S).$$C \stackrel{\leftarrow}{\hookrightarrow} BiSep(S) \stackrel{\leftarrow}{\hookrightarrow} Sh(S) \,.

Definition

Definition

A bisite is a small category $S$ equipped with two coverages: $J$ and $K$ such that $J\subset K$.

A presheaf $A\in {\mathrm{PSh}}_S$ is called $(J,K)$-biseparated if it is

• a sheaf with respect to $J$;

• a separated presheaf with respect to $K$.

Write

$${\mathrm{BiSep}}_{(J,K)}(S)\hookrightarrow {\mathrm{Sh}}_J(S)\hookrightarrow \mathrm{PSh}(S)$$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

$${\mathrm{BiSep}}_{(J,K)}(S)\stackrel{\stackrel{L_{\mathrm{sep}}^K}{\leftarrow }}{\hookrightarrow }{\mathrm{Sh}}_J(S).$$BiSep_{(J,K)}(S) \stackrel{\stackrel{L^K_{sep}}{\leftarrow}}{\hookrightarrow} Sh_J(S) \,.

See quasitopos for the proof.

Related concepts

• separated presheaf

• separated (2,1)-presheaf

• separated (∞,1)-presheaf

References

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

• Peter Johnstone, Sketches of an Elephant .

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