topos theory

# Contents

## 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 ${\mathrm{PSh}}_{S}\left(X,A\right)\to {\mathrm{PSh}}_{S}\left(Y,A\right)$ (under the hom-functor ${\mathrm{PSh}}_{S}\left(-,A\right)$) 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 $A\in \mathrm{PSh}\left(S\right)$ is called separated (or a monopresheaf) if for all local isomorphisms $Y\to X$ the induced morphism $\mathrm{Hom}\left(X,A\right)\to \mathrm{Hom}\left(Y,A\right)$ 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 $\mathrm{Hom}\left(X,A\right)\to \mathrm{Hom}\left(Y,A\right)$ is a monomorphism.

###### Remark

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

For $\left\{{p}_{i}:{U}_{i}\to U\right\}$ a covering family of an object $U\in S$, the condition is that if $a,b\in A\left(U\right)$ are such that for all $i$ we have $A\left({p}_{i}\right)\left(a\right)=A\left({p}_{i}\right)\left(b\right)$ 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

$i:\mathrm{Sep}\left(S\right)↪\mathrm{PSh}\left(S\right)$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

$\left({L}_{\mathrm{sep}}⊣i\right):\mathrm{Sep}\left(S\right)\stackrel{\stackrel{{L}_{\mathrm{sep}}}{←}}{↪}{\mathrm{PSh}}_{S}\phantom{\rule{thinmathspace}{0ex}}.$(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

${L}_{\mathrm{sep}}A:U↦A\left(U\right)/{\sim }_{U}\phantom{\rule{thinmathspace}{0ex}},$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 $\left\{{p}_{i}:{U}_{i}\to U\right\}$ such that $A\left({p}_{i}\right)\left(a\right)=A\left({p}_{i}\right)\left(b\right)$ for all $i$.

This construction extends in the evident way to a functor

${L}_{\mathrm{sep}}:\mathrm{PSh}\left(S\right)\to \mathrm{Sep}\left(S\right)\phantom{\rule{thinmathspace}{0ex}}.$L_{sep} : PSh(S) \to Sep(S) \,.
###### Proposition

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

###### Proof

Let $A\in \mathrm{PSh}\left(S\right)$ and $B\in \mathrm{Sep}\left(S\right)↪\mathrm{PSh}\left(S\right)$. 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 $\mathrm{Sep}\left(S\right)$.

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

$\begin{array}{ccc}\left\{\left({a}_{i}\in A\left({U}_{i}\right)\right)\mid \cdots \right\}& \to & \left\{\left({b}_{i}\in F\left({U}_{i}\right)\right)\mid \cdots \right\}\\ ↑& & ↑\\ A\left(U\right)& \stackrel{{f}_{U}}{\to }& B\left(U\right)\end{array}$\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 ${\mathrm{PSh}}_{S}\left(S\left({U}_{i}\right)\to U,A\stackrel{f}{\to }B\right)$.

If for $a,a\prime \in A\left(U\right)$ 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 $f\left(a{\mid }_{{U}_{i}}\right)=f\left(a\prime {\mid }_{{U}_{i}}\right)$ and since the right vertical morphism is monic there is a unique $b\in B\left(U\right)$ mapping to the latter. The commutativity of the diagram then demands that $f\left(a\right)=f\left(a\prime \right)=b$.

Since this argument applies to all covers of $U$, we have that ${f}_{U}$ factors uniquely through the projection map $A\left(U\right)\to A\left(U\right)/{\sim }_{U}=:{L}_{\mathrm{sep}}\left(U\right)$ 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

$C\stackrel{\stackrel{}{←}}{↪}\mathrm{Sh}\left(S\right)$C \stackrel{\stackrel{}{\leftarrow}}{\hookrightarrow} Sh(S)

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

$C\stackrel{←}{↪}\mathrm{BiSep}\left(S\right)\stackrel{←}{↪}\mathrm{Sh}\left(S\right)\phantom{\rule{thinmathspace}{0ex}}.$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 $\left(J,K\right)$-biseparated if it is

• a sheaf with respect to $J$;

• a separated presheaf with respect to $K$.

Write

${\mathrm{BiSep}}_{\left(J,K\right)}\left(S\right)↪{\mathrm{Sh}}_{J}\left(S\right)↪\mathrm{PSh}\left(S\right)$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

${\mathrm{BiSep}}_{\left(J,K\right)}\left(S\right)\stackrel{\stackrel{{L}_{\mathrm{sep}}^{K}}{←}}{↪}{\mathrm{Sh}}_{J}\left(S\right)\phantom{\rule{thinmathspace}{0ex}}.$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 March 6, 2013 19:42:33 by Zoran Škoda (161.53.130.104)