Topos properties from coverage conditions

Question

For any category $C$ and coverage $J$ on it, let $\mathcal{E}:=\mathsf{Shv}(C,J)$ denote topos of sheaves on the site $(C,J)$. What sorts of results are known about the relationship between properties of the topos $\mathcal{E}$, and conditions on the 'diagrammatic shape' of the covering families in $J$?

My interest includes any sort of property of $\mathcal{E}$, for example logical properties. To that end, let $\mathtt{Prop}$ denote the subobject classifier in the internal language of $\mathcal{E}$.

Here are some first examples of the kinds of relationships I'm talking about:

1. if all covering families in $J$ are empty, then $\mathcal{E}$ is terminal.
2. if all covering families in $J$ consist of the identity, then $\mathcal{E}$ is of presheaf type.
3. if all covering families in $J$ are filtered, then constant objects satisfy a dual Frobenius property.

By #3, I mean the following. Say that a sheaf $K\in\mathcal{E}$ satisfies the dual Frobenius property if, for any proposition $P:\mathtt{Prop}$, and predicate $P':K\to\mathtt{Prop}$, the following logical statement is sound in $\mathcal{E}$: $$\big(\forall(k:K)\ldotp P\vee P'(k)\big)\implies\big(P\vee\forall(k:K)\ldotp P'(k)\big).$$ It is relatively straightforward—though seems to require classical reasoning—to prove that constant sheaves satisfy this property when every covering family in $J$ is filtered.

Questions:

• Are there more examples of such relationships? Obviously the three above are not very thorough (they don't have much 'coverage,' so to speak).
• Is there a name for toposes of type 3 above?

Answer 1

This is a very broad question, we have a huge numbers of such characterization.

But part C of "Sketches of an elephant" contains most of those I know (especially C3). Basically all the notion introduced have such a "site characterization", i.e. a properties of a topos or a morphism is characterized by the existence of a Site description having some properties.

And I would like to add that Part C is in my opinion the better written and easiest to read part of the elephant.

From memory, you can find there at the very least conditions for:

• Atomic toposes, as sheaves for a topology where all non-empty sieve are covering (called an atomic topology).
• Coherent toposes by combining the fact that every covering is finitely generated with existence of finite limits in the site. ( it is a bit more subtle than that, I refer you to the elephant for the precise statement)
• A characterization of proper geometric morphisms. (but maybe not the simplest possible ? )

and a lots of other examples (open geometric morphisms, locally connected morphisms and so one...) but I do not remember all of them.

This dual frobenius property that you sate seem related to properness (maybe be not equivalent, but that is what it makes me think about...) you should have a look to the section of the elephant on properness.