Questions about coherent topology For each coherent category $C$, let $J_C$ be the topology on $C$ in which a sieve $\{f_i\colon U_i\to X\}_{i\in I}$ is covering if and only if there exists a finite set $I_0\subseteq I$ such that $\bigcup_{i\in I_0} \operatorname{im}(f_i)=X$ as subobjects of $X$. (This is a Grothendieck topology by Proposition 12 of Lecture 8 (Grothendieck topologies) of Lurie's "Categorical logic" notes.)

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*Is $J_C$ equivalent to what other people call "the coherent topology on $C$"?

*If $C$ is a coherent category which is Boolean, is the topos $\operatorname{Sh}(C, J_C)$ Boolean too?

*Can one find for each Boolean coherent topos $\mathcal E$ a Boolean coherent category $C$ such that both


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*$\mathcal E\simeq \operatorname{Sh}(C, J_C)$ and

*there exists an object $X\in C$ such that every object of $C$ is a subobject of $X^n$?



*Is the étale topos of the spectrum of any field coherent?

 A: Edit :  I should clarify that I've interpreted "Etale topos" to mean the petit/small étale topos everywhere. What I've said about Grothendieck-Galois duality only apply to the petit étale topos. If you are talking about the Gros topos, then these part no longer holds. I actually don't know if the Gros étale topos of a fields has a boolean category of coherent object or not.

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*Yes. I can't give you a proof because as far as I'm concerned this is the definition of the coherent topology. If you see a different definition, maybe edit your question!


*Essentially no. Take for example a Boolean algebra $B$. It can be seen as a coherent category (I see $B$ as a poset, and every poset as a category in the usual way). Then the associated topos is the topos of sheaves over the Stone spectrum of $B$, and unless $B$ is finite it has plenty of open that are not also closed (in fact the open that are complemented corresponds exactly to the element of $B$). The general case looks like this though: a Boolean coherent category will gives a topos that "looks like" a Stone spectrum.


*The answer is yes for the first half, no for the second, but only because there are very few coherent boolean topos. I would say, a coherent topos is essentially never Boolean (the only exception being the framework of Galois theory): A coherent topos has always enough points, and it can be proved that a boolean topos with enough points is "atomic", that is a disjoint union of topos of the form $BG_i$ where the $G_i$ are localic group. (Here $BG$ is the topos of sets endowed with a continuous action of the localic group $G$.) Adding back the fact that we want this topos to be coherent, we get that the Boolean coherent toposes are exactly the toposes that are finite coproducts of $BG_i$ where the $G_i$ are profinite groups.
The second condition you ask for doesn't hold if some of the $G_i$ are non-discrete though. If $G$ is a profinite group (take $G = \mathbb{Z}_p$ for example), then a coherent $G$-set $X$ is a finite $G$-set, so there is going to be an open normal subgroup of $G$ that stabilise all the points of $X$, and the things you get as subobjects of $X^n$ will all be stabilised by the same subgroup.


*Yes. The étale topos of any affine scheme is coherent. (For a general scheme it is locally coherent. I'll let an algebraic geometer give you the precise condition under which we get coherence.) In fact, thanks to Grothendieck Galois duality, the étale topos of a field is one of the rare examples of Boolean coherent toposes: it is $BG$ where $G$ is the absolute Galois group of the field, with its profinite topology.
Answering some of the follow up question in the comments.

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*I would recomand to double checking what I'm going to say here if you plan on using it - I haven't looked at it in enough to details, but I think toposes associated to boolean coherent categories can be characterized as the coherent toposes in which coherent subobject of coherent object have complement. A topos satisfying these condition is clearly the topos of coherent sheaf on a booleancoherent category (by taking all coherent objects) but the converse also seems true. The other condition you have (every object of $C$ is a subobject a power of some fixed object $X$) should corresponds to have a coherent "pre-bound".


*The class descriebd you in (1) (including both conditions) are the topos that classyfies single sorted "boolean" first order theory. This additional condition of having a coherent prebound is not automatic at all as the exemple of the topos $BG$ for $G$ a non discrete profinite group shows.


*Yes. étale topos of fields are Boolean topos (from their explicit descrition given by Galois duality) so in particular coherent subobject have complements.
