Let $(P,J)$, $(Q,K)$ be sites. A geometric morphism from $Sh(P,J)$ to $Sh(Q,K)$ is (following Moerdijk-MacLane- Sheaves in Geometry and Logic, in the sequel MM) a pair of adjoint functors $f^*:Sh(Q,K)\to Sh(P,J)$,$f_*:Sh(P,J)\to Sh(Q,K)$ such that $f^*$ is left adjoint to $f_*$ (Def 1, Pag. 348). An essential Geometric morphism is then defined to be a geometric morphism such that $f^*$ has a left adjoint $f_!$ (Pag. 360) finally open geometric morphisms are defined on pag. 495 (Def. 2) by asserting that locally $f^*$ has a left adjoint (with certain compatibility condition between these local left adjoints).
It appears that essential geometric morphisms shold be open (even though this is never stated in MM for my exploration of the book). Is this correct? (at least is it so in the case one deals with sites on posets or on complete heyting algebras)? I do not see why the restriction of the global left adjoint given by an essential geometric morphism should not provide the local geometric morphisms whose existence is required by the notion of open geoemtric morphism.
As for the converse: why the local left adjoints provided by an open geometric morphisms may not be glued in a global left adjoint? An answer covering just the case for sheaves of sites on posets would be also extremely helpful.
Thanks in advance for any clarification.
Not every open geometric morphism is essential. For example, for any topological space $X$ whatsoever, the unique geometric morphism $\textbf{Sh} (X) \to \textbf{Set}$ is open. (After all, the unique continuous map $X \to 1$ is open too.) But $\Delta : \textbf{Set} \to \textbf{Sh} (X)$ may fail to preserve limits, so $\textbf{Sh} (X) \to \textbf{Set}$ may fail to be essential. Indeed, this is so when $X$ is the Cantor set.
(Recall that the Cantor set is homeomorphic to $2^\mathbb{N}$ with the product topology. To avoid confusion, I will keep writing $X$ for this topological space and reserve $2^\mathbb{N}$ for the discrete set. The canonical comparison morphism $\Delta (2^\mathbb{N}) \to \Delta (2)^\mathbb{N}$ is not an isomorphism in $\textbf{Sh} (X)$: it is a monomorphism but fails to be an epimorphism.
Indeed, the sheaf $\Delta (2)^\mathbb{N}$ can be identified with the functor $\textbf{Top} (-, X)$. On the other hand, the sheaf $\Delta (2^\mathbb{N})$ can be identified with the functor $\textbf{Top} (-, 2^\mathbb{N})$. Under these identifications, the canonical comparison morphism $\Delta (2^\mathbb{N}) \to \Delta (2)^\mathbb{N}$ is identified with the morphism induced by the identity-on-points continuous map $2^\mathbb{N} \to X$. In particular, $\Delta (2^\mathbb{N}) \to \Delta (2)^\mathbb{N}$ is a monomorphism. But since $\textrm{id}_X$ does not factor through this map, we have a global section of $\Delta (2)^\mathbb{N}$ not present in $\Delta (2^\mathbb{N})$, and therefore $\Delta (2^\mathbb{N}) \to \Delta (2)^\mathbb{N}$ is not an epimorphism.)
Also, not every essential geometric morphism is open. For example, given any small category $\mathcal{C}$ and any object $C$ in $\mathcal{C}$, we have an essential geometric morphism $\textbf{Set} \to \textbf{Psh} (\mathcal{C})$ whose inverse image functor is evaluation at $C$, i.e. the functor $F \mapsto F (C)$. The direct image functor is fully faithful, so this is actually a geometric inclusion of toposes. When $\mathcal{C}$ is a preorder then this is a geometric inclusion between localic toposes, so openness of the geometric inclusion is equivalent to openness of the locale inclusion. A locale inclusion is open if and only if it is (isomorphic to) the embedding of an open sublocale. But, for example, if $\mathcal{C} = \{ B \to C \to D \}$, then $\textbf{Psh} (\mathcal{C})$ is equivalent to sheaves on $\{ 0, 1, 2 \}$ with the Alexandrov topology and the geometric morphism $\textbf{Set} \to \textbf{Psh} (\mathcal{C})$ corresponds to the inclusion of the subspace $\{ 1 \}$, which is not open.