Can you phrase more in detail the question? I see what is a "level" from nLab, though some more context would help potential readers.

I would say you have an adjunction $(L\dashv i) : \text{Sh}(B) \leftrightarrows \text{PSh}(B)$ (the direction of the arrow is that of $i$); now $L$ has an additional left adjoint $L'$.

Is this interpretation correct? Is it correct to argue that such an essential localisation comes from an open map whose domain (codomain?) is a Stonean locale?

Another observation, don't take it as well-informed: the poset of essential localisations is usually not as well-behaved as one would expect; probably in the case of a presheaf topos it's easier, but have you tried to classify at least the locally connected geometric morphisms (those for which $f_!$ is an indexed left adjoint to $f^*$, and not just an adjoint)? Look into Elephant C.1.5.9 : is it true that there are no nontrivial such geometric morphisms (motivated by that theorem)?

I would suggest start with Kelly, Lawvere 1989. A locally connected inclusion is an open inclusion, whence a subobject of 1.

Before you asked this question, my intuition was that essential subtoposes of a presheaf topos $\mathbf{PSh}(\mathcal{C})$ were all of the form $\mathbf{PSh}(\mathcal{D})$ for some full subcategory $\mathcal{D} \subseteq \mathcal{C}$, under the assumption that $\mathcal{C}$ is idempotent-complete.

Thanks to the paper by Kelly and Lawvere, I now have the following counterexample:

Take the real interval $[1,+\infty)$ under multiplication, as a monoid. Let $\mathcal{C}$ be the corresponding 1-object category. There are no nontrivial idempotents, so $\mathcal{C}$ is idempotent-complete.

Let $\mathcal{I}$ be the two-sided idempotent ideal given by the interval $(1,+\infty)$. If I understand correctly, then by Theorem 4.4 from the Lawvere–Kelly paper this defines a non-trivial essential subtopos.

The Grothendieck topologies on such that the corresponding inclusion is essential are precisely those such that every object of admits a smallest covering sieve. Note that, as shown at page 9 of these slides, such topologies are in bijective correspondence with the two-sided idempotent ideals on .

This class of topologies notably includes all the rigid ones, which, as shown in section 8.2.2 of my book *Theories, Sites, Toposes*, correspond exactly to the essential subtoposes induced by full subcategories of , if is Cauchy-complete.

Thanks to everyone for your answers! Sorry to have let so much time pass since I posted this: it's kind of a side-project and I didn't return on it until now.

Your reply, Olivia, is very intriguing.

1. When you say 'smallest covering sieve', does it mean that every other covering sieve factors through this one?

2. When you say rigid topologies correspond to the essential subtopoi induced by full subcategories of C (when it is Cauchy complete, which is my case since Boolean algebras are posets), do you mean that every essential subtopos of Psh C is in the form Psh D for D full subcategory of C? What's the link with rigidity then?

Concerning 1, I mean that every object $c$ has a $J$-covering sieve $S_{c}$ such that for any $J$-covering sieve $S$ on $c$, $S\supseteq S_{c}$.

Concerning 2, not every essential subtopos of $[{\cal C}^{\textrm{op}}, \textbf{Set}]$ (for a Cauchty-complete $\cal C$) comes from a rigid topology on $\cal C$, since there are topologies with smallest covering sieves on each object which are not rigid.

I can post a note here explaining all the details, if that helps.

Ok thanks for the clarification.

Let me see if I got it right: you're saying, for Cauchy complete , rigid topologies correspond exactly to essential subtopoi induced (I assume by taking presheaves) by full subcategories, but not every essential subtopos is of this form, so some essential subtopoi *are not* induced by full subcategories hence do not come from rigid topologies?

Then these latter subtopoi necessarily come from sheaves for non-rigid topologies with the property of having a smallest covering sieve.

If the above checks out: do you have a conceptual intuition regarding the 'smallest covering' property? I mean, when is it sensible to expect it?

Yes, you have understood correctly. Any two-sided idempotent ideal $I$ on $\cal C$ gives rise to a Grothendieck topology $J_{I}$ on $\cal C$ inducing an essential subtopos, and in general not every such ideal is associated with a full subcategory of $\cal C$. For any object $c$ of $\cal C$, the smallest $J_{I}$-covering sieve on $c$ consists precisely of the arrows in $I$ with codomain $c$.

It is instructive to look at what happens in the particular case you are interested in, namely that of a complete Boolean algebra $B$ endowed with the canonical topology $J^{\textrm{can}}_{B}$. If the algebra is atomic then every object has a smallest covering sieve (generated by the atoms covering it); in this case, the associated topology is rigid and by the Comparison Lemma $\textbf{Sh}(B, J^{\textrm{can}}_{B})\simeq [At(B), \textbf{Set}]$, where $At(B)$ is the set of atoms (regarded as a discrete category). If $B$ is not atomic then there could in principle be objects that do not admit a smallest $J^{\textrm{can}}_{B}$-covering sieve; so, for a general $B$, the subtopos $\textbf{Sh}(B, J^{\textrm{can}}_{B})\hookrightarrow [B^{\textrm{op}}, \textbf{Set}]$ is not necessarily essential.

Thanks, it's been really helpful! Unfortunately my algebras are non-atomic, so the mystery remains. I believe then sheafification is not going to produce an essential subtopos, but I have to prove it.

Actually, I thought I understood the distinction between rigid and 'admitting smallest coverings' for posets, but now I don't anymore. If I spell out the definition of 'two sided idempotent ideal' for posets I get 'upward and downard closed subset'. But then this is a full subcategory of the initial poset, and not every full subcategory is of this kind, hence it seems that the implication 'rigid => smallest coverings' is inverted. What did I get wrong?

EDIT: also downward and upward closed sets are just the improper subcategories!

I agree that sheafification will not give an essential subtopos if the boolean algebra is not atomic. A proof is as follows. If the subtopos is essential then on every object there is a minimal covering sieve. Now let $U$ be an element of the boolean algebra, and suppose that the minimal covering sieve is given by a family of open subsets $U_i \subseteq U$ , $i \in I$. Then the claim is that each $U_i$ is an atom. To see this, take a nontrivial open subset $V \subset U$. Then $V$ and $U_i-V$ together give an open covering of $U_i$. But then replacing $U_i$ by $V$ and $U_i-V$ gives a refinement of the minimal covering on $U$, so it isn't minimal after all. So each $U_i$ must be an atom and as a result the boolean algebra is atomic.

Actually, I thought I understood the distinction between rigid and 'admitting smallest coverings' for posets, but now I don't anymore. If I spell out the definition of 'two sided idempotent ideal' for posets I get 'upward and downard closed subset'. But then this is a full subcategory of the initial poset, and not every full subcategory is of this kind, hence it seems that the implication 'rigid => smallest coverings' is inverted. What did I get wrong?

EDIT: also downward and upward closed sets are just the improper subcategories!

Maybe the confusion is due to the non-standard use of the term 'ideal'? By a two-sided ideal on $\cal C$ I meant a collection of arrows (rather than objects) of $\cal C$ which is closed under composition with arbitrary arrows both on the right and on the left.