In the category of schemes the objects of the form $Spec(K)$ with $K$ a field can be characterized as follows. They are precisely the schemes which have no proper subobjects in $Sch$.

Consider now the category $Ring^{op}$ with its various Grothendieck topologies (Zariski, etale, fpqc, etc.). From those we get various big sheaf topoi which contain the category of schemes as a full subcategory. As a functor, $Spec(K)$ is the representable presheaf $Hom_{Ring}(K,-)$.

Even though $Spec(K)$ has no proper subschemes when $K$ is a field, it can have non-trivial sub-pre-sheaves. Can it have non-trivial Zariski sub-sheaves? If yes, can we switch to one of the finer topologies to prevent this? Is one of the topologies on $Ring^{op}$ fine enough, so that the sheaves of the form $Spec(K)$ with $K$ a field are precisely the objects with a trivial subobject lattice in $Sh(Ring^{op},sth)$?