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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:

  1. Sieves on $B$: Set of sieves on $B$ ordered by inclusion
  2. $Z(X)$: 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)

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    $\begingroup$ In topos theory it is said that a property holds locally if there exists an effective epimorphism $U\to *$ (a "covering") such that the property holds in the slice topos $\mathcal{X}_{/U}$. You definition seems to work only if the topos is locally 0-localic ("locally a topological space") which is a rather rare condition. $\endgroup$
    – Denis Nardin
    Commented Mar 12, 2018 at 8:18
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    $\begingroup$ @ChetanVuppulury: The main (maybe only?) use of Grothendieck topologies is for presenting sheaf toposes — I’ve never heard of them being used/studied for any other purpose. So when trying to find the “right” definition of a concept for Grothendieck topologies, it is at the very least important to think about the relationship with that concept in toposes! $\endgroup$
    – Peter LeFanu Lumsdaine
    Commented Mar 12, 2018 at 9:04
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    $\begingroup$ @PeterLeFanuLumsdaine, the reason I was thinking about this was to generalise the definition of a scheme, where we need the space to 'locally' look like $\textrm{Spec }R$ for some $R$. Hence my interest in in locality in Grothendieck topologies in particular. I obviously agree that this any notion of locality in Grothendieck topologies must be 'compatible' with that of sheaf toposes. $\endgroup$
    – Chetan Vuppulury
    Commented Mar 12, 2018 at 9:10
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    $\begingroup$ @ChetanVuppulury Are you familiar with the description of Deligne-Mumford stacks as locally ringed topoi locally equivalent to $\mathrm{Spét}R$? If not, I think that would be a good starting point. Grothendieck topologies are a "generators and relations" description of topoi, and so tend to be unsuited to describing what "locally equivalent" means. $\endgroup$
    – Denis Nardin
    Commented Mar 12, 2018 at 10:29
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    $\begingroup$ The link I put in the comment above is a good start. I like the presentation of chapter 1.2 of Lurie's Spectral Algebraic Geometry for more details, but it might be a bit aggressive if you're not confortable with topoi. $\endgroup$
    – Denis Nardin
    Commented Mar 12, 2018 at 10:39

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