Let $\mathcal{C}$ be a category and $\mathcal{J}$ be a Grothendieck topology on it (i.e., $(\mathcal{C},\mathcal{J})$ is a site). Then what is a good notion of locality in it?

I came up with the following definition. Suppose $P$ is a property of topological spaces. Then I say it holds locally for $(\mathcal{C},\mathcal{J})$ if, for every $A\in\textrm{ob }\mathcal{C}$ we have a morphism $f$ with codomain $A$ and a topological space $(X,\mathcal{O}(X))$ such that $\mathcal{J}(\textrm{dom }f)\cong X$ and $X$ has the property $P$.

Here, I say $\mathcal{J}(B)\cong X$ if there is an isomorphism between the following two posets:

- Sieves on $B$: Set of sieves on $B$ ordered by inclusion
- $Z(X)$: Elements of this are sieves on $X$. That is, collection of open subsets which are closed under taking open subsets. I.e., if $U\subseteq V\in z\in Z(X)$ then $U\in z$. Order this by inlcusion.

This isomorphism must take families of open sets that cover $X$ to elements of $\mathcal{J}(B)$ (covering sieves) and vice versa.

Edit: Cross-posted from MSE (https://math.stackexchange.com/questions/2684050/locality-in-grothendieck-topologies)