What information is lost in $X \to \mathrm{Sh}(X)$?

Question

Given a topological space or site $X$. Construct $\mathrm{Sh}(X)$ - the sheaves on $X$ with values in $\mathrm{Set}$. Is it known what information is lost in this procedure?

Thanks, Adrian.

Answer 1

Following Martin's suggestion, I will turn my comment into an answer.

If $T$ is an Grothendieck topos, then the subobjects of the terminal object form a frame. If $X_T$ is the corresponding locale, then the topos $Sh(X_T)$ of sheaves on $X_T$ is the called localic reflection of $T$. One has that $T\mapsto X_T$ is adjoint to the functor that takes a locale to its topos of sheaves. Moreover, if $X$ is any locale then $Sh(X)$ is equivalent to its localic reflection. If $X$ is a topological space, then $Sh(X)$ is equivalent to the category of sheaves on the locale corresponding to its frame $O(X)$ of open subsets. It is in this sense that you can recover the locale of open subsets of $X$.

Now associated to any locale $L$ is its space of points $Pts(L)$. Let $O(L)$ denote the frame corresponding to $L$. The functor $L$ to $Pts(L)$ is adjoint to the functor taking a space to its corresponding locale. A point of $L$ is a homomorphism of frames $p\colon O(L)\to O(pt)$ where $pt$ is the one point space. It is convenient to call the two elements of $O(pt)$ $0,1$ with $0<1$. A locale may not have any points. The topology on $O(L)$ is the usual "Zariski" topology: a basic open set is of the form $D(U)$ with $U\in O(L)$ where $D(U)$ consists of those points $p$ with $p(U)=1$.

If $X$ is a space, then to each point $x\in X$, we get a point $p_x$ given by

$p_x(U)=\begin{cases} 1 & x\in U\\ 0 & x\notin U\end{cases}$.

Notice that these points suffice to separate open sets (i.e., the locale of a space has enough points). The map $x\to p_x$ is injective iff $X$ is $T_0$ and is an isomorphism iff $X$ is a sober space (each irreducible closed subset has a unique generic point). Let $Sob(X)$ be the space of points of the locale of the space $X$. Note that $O(X)$ and $O(Sob(X))$ are isomorphic frames. The natural map $X\to Sob(X)$ is continuous and is the universal map of $X$ into a sober space. Sometimes $Sob(X)$ is called the soberification of $X$. Note that $X$ is sober iff the natural map is a homeomorphism. Therefore, the information that can be recovered about a space $X$ from its topos of sheaves is the space $Sob(X)$ (up to homeomorphism).

Answer 2

Benjamin Steinberg's answer covers the case of topological spaces, so I'll answer this question for sites.

The functor $$\mathrm{Sh}:\mathbf{Site}\rightarrow\mathbf{Topos}^{\mathrm{op}}$$ (here $\mathbf{Topos}$ is the category of Grothendieck toposes and geometric morphisms) has a right adjoint $$\mathrm{U}:\mathbf{Topos}^{\mathrm{op}}\rightarrow\mathbf{Site}$$ that takes a topos to the site formed by equiping its underlying category with the canonical topology.

As shown at the above link, the composite $\mathrm{Sh}\circ\mathrm{U}$ is the identity, so this constuction recovers from a Grothendieck topos a site that has that topos as its topos of sheaves.

The fixed points of $\mathrm{U}\circ\mathrm{Sh}$ are pricisely the sites which are toposes equipped with the canonical topology. So one could refer to these as the "sober sites" by analogy to sober topological spaces being the ones that can be recovered from their locales.

The analogy isn't particularly close though, because the relationship between topological spaces and locales isn't especially analogous to that between sites and toposes. A better analogy is to say that locales are to topological spaces as toposes are to ionads. Each topos produces a canonical ionad on its set of points (geometric morphisms from $\mathbf{Set}$). So the functor from toposes to ionads is $\mathrm{Hom}(\mathbf{Set},-)$. I don't know a nice classification of the "sober" ionads though.

Answer 3

About sites, if I remember correctly, we can say the following. Given a category $\mathcal C$, there is a bijection

$$\{\textrm{Grothendieck topologies}\; \mathcal J\; \textrm{on}\; \mathcal C\}\leftrightarrow\{\textrm{reflective subcategories}\;\mathcal E \subseteq\mathrm{Psh}(\mathcal C)\}$$

given by $\;\mathcal J\mapsto \mathrm{Sh}(\mathcal C,\mathcal J)$. So, in some sense, nothing is lost in passing to the category of sheaves, if we're dealing with a fixed category $\mathcal C$.