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:

- if all covering families in $J$ are empty, then $\mathcal{E}$ is terminal.
- if all covering families in $J$ consist of the identity, then $\mathcal{E}$ is of presheaf type.
- 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?