I thought I heard something like this theorem somewhere, but I may be wrong:
Theorem: (Informally stated) Sheaves in the category $[\text{Ring}^{op}, \text{Set}]$ the same as filtered colimits of representable functors.
If this is true, I'd like to know a general result along these lines for Grothendieck topologies.
Thanks so much!
Here is some more of what I was thinking.
Taking $F : I^{op} \rightarrow \text{Set}$, $F$ is the colimit of $\text{el}(F) \rightarrow I \rightarrow [I^{op}, \text{Set}]$, where $\text{el}(F)$ is the category of elements of $F$, whose objects are pairs $(i, x)$ where $i \in I$ and $x \in F(i)$, and morphisms $\phi : (i, x) \rightarrow (j, y)$ are maps $\phi : i \rightarrow j$ such that $\phi(y) = x$. This construction shows that each functor $F$ in $[I^{op}, \text{Set}]$ is canonically the colimit of representable functors. It can also be developed to show that $[I^{op}, \text{Set}]$ is a sort of free colimit completion of $I$, though there are notable size issues involved in that consideration.
My question is then, noticing two major classes of maps (filtered colimits of presheaves and sheaves given a grothendieck topology) we might wonder which grothendieck topologies on $I$ would make these two classes coincide.
Now, if the category $\text{el}(F)$ is filtered, then we are done. So what kind of covers in a Grothendieck topology would make $\text{el}(F)$ filtered? It seems like jointly surjective maps would do nicely in many examples.
Example: Let $I$ be the category of compact hausdorff spaces (a full subcategory of the category of topological spaces). I happen to be interested in presheaves on this category at the moment, but it makes a nice example. Put a Grothendieck topology on $I$ where covers are jointly surjective collections of maps into a given object.
The claim is that the sheaves here are filtered colimits of representables. Take two objects $(i, x)$ and $(j, y)$ in the category of elements of a given sheaf $F$ on $I$ (note that $F : I^{op} \rightarrow \text{Set}$). Take $(i \amalg j, (x, y) \in F(i) \prod F(j))$, noting that $F$ must preserve products as $I, J \rightarrow I \amalg J$ is a cover.
... I haven't finished this.
This is not an answer but a collection of likely relevant facts, references, and concerns.
Categories of sheaves are locally presentable. This means that every object of such a category is a $\kappa$-filtered colimit of $\kappa$-compact objects. Allegedly, all representables in a sheaf topos are $\kappa$-compact objects. This doesn't say all $\kappa$-compact objects are representables, though, or that the representables suffice.
Also, sites are required to be small usually which $\mathbf{Ring}$ is not. This causes issues for this referenced proposition because you need $\kappa$ to be a regular cardinal of strictly greater cardinality than $\mathsf{Mor}(\mathbf{Ring})$ which won't normally exist.
$\mathbf{Ring}$ itself being a category of models for a Lawvere theory is locally finitely presentable which may or may not give you enough of a handle on size issues to adapt the above arguments.