Essential geometric morphism seen topologically

Question

I know that any geometric morphism between toposes of sheaves on spaces $f^*\colon Sh(X)\leftrightarrows Sh(Y)\colon f_*$ comes from a continuous map $f\colon X\to Y$. But what does it mean for $f$ the fact that $(f^*,f_*)$ is essential? I can't find anything more than the page of nlab, where the condition is linked to "connectedness" of the spaces. http://ncatlab.org/nlab/show/essential+geometric+morphism

An interesting topic on MO shows that - If I consider an etale morphism on the etale site of schemes, then it induces an essential geometric morphism; - I think the same holds in the case of the etale site of topological spaces.

So, is there a pattern?

Answer 1

This is Zhen Lin's comment turned into an answer. The following can be found in the article Representing topoi by topological groupoids by Carsten Butz and Ieke Moerdijk, see Section 3.

Definition: A continuous map $f\colon X \to Y$ of topological spaces is called locally connected if it is an open map, and $X$ can be covered by open subspaces $\{U_\alpha\}$ such that the fibers $f^{-1}(y) \cap U_{\alpha}$ of the restriction $f\vert_{U_{\alpha}}\colon U_{\alpha} \to Y$ are either empty or connected.

Proposition: If $f\colon X \to Y$ is locally connected, then the induced map of topoi $f^*\colon Sh(X)\leftrightarrows Sh(Y)\colon f_*$ is locally connected, and thus in particular essential.

Answer 2

To collect some more references into a partial answer:

• For $Y = \operatorname{pt}$ and in the setting of $\infty$-topoi, Lurie shows [HA, A.1.8] that this is equivalent to $X$ being locally of constant shape, i.e. for every $\mathscr F \in \mathbf{Sh}(X)$, the terminal morphism of $\infty$-topoi $\pi \colon \mathbf{Sh}(X)_{/\mathscr F} \to \mathscr S$ has the property that $\pi_*\pi^*$ preserves (small) limits (roughly speaking, cohomology of constant abelian sheaves commutes with limits on every open $U \subseteq X$). Paracompact CW complexes are given as an example.
• Conversely, if $X = \{y\}$ is a point, then $f \colon \{y\} \to Y$ gives an essential geometric morphism if and only if there is a smallest neighbourhood containing $y$ [SGA 4.1, Exp. IV, §7.6]. For closed points this means that $y$ is isolated, but also any point in a finite $T_0$ space has this property (the smallest open is the cosieve $Y_{\geq y}$ of points specialising to $y$).