I find the definition for locales and sheaves on locales to be straightforward, but I'm stumbling over the idea of a Grothendieck topology. Is there a nice way to see roughly how the latter generalises the former? And is there a nice way to interpret the idea of a Grothendieck topos in a similar way to the sheaf topos on some locale? If it's easier to replace "locale" with "topological space" then that works just as well.

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    $\begingroup$ The standard way to make a locale a site is that an object is covered by a collection of subobjects iff that object is the join of the (objects representing the) subobjects. Also, have you yet seen Lawvere-Tierney topology? $\endgroup$
    – user13113
    Sep 27, 2015 at 22:38
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    $\begingroup$ The other basic example of a Grothendieck topos is the category of $G$-sets for a group $G$, which I intuit as being a way that Grothendieck toposes can vary that is completely orthogonal to how locales vary. And those two options essentially cover all possibilities: every Grothendieck topos is (equivalent to) the category of sheaves for a localic groupoid. $\endgroup$
    – user13113
    Sep 27, 2015 at 22:41
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    $\begingroup$ The correct analogy is this: (Grothendieck sites) : (Grothendieck topos) :: (presentation of a frame) : (locale). $\endgroup$
    – Zhen Lin
    Sep 28, 2015 at 0:24


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