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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.

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  • $\begingroup$ I'm voting to close this question as off-topic because it is more appropriate on mathoverflow, where it has just been crossposted $\endgroup$ – Hurkyl Mar 12 '18 at 9:55
  • $\begingroup$ How do you close a question? I just cross-posted to mathoverflow myself. $\endgroup$ – Chetan Vuppulury Mar 12 '18 at 9:57
  • $\begingroup$ As the author, you should be able to delete the question. Or if not, you could flag the question to have a diamond-moderator do something. $\endgroup$ – Hurkyl Mar 12 '18 at 11:29

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