Given a complex manifold or complex analytic space, one has the standard notion of open set. There are two different Grothendieck topologies that one can define using this notion, one where covers must have finite subcovers and one without this restriction. There are other variants such as asking for subcovers which are locally finite. Is there anything known about the relation between these topologies, or various categories of sheaves with respect to them? Of course I am looking for more than just the obvious morphisms. Are there certain classes of spaces or sheaves where one has an equivalence?
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$\begingroup$ Excellent question. Have you found out any news since 2015? $\endgroup$– Ingo BlechschmidtDec 21, 2018 at 19:22
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$\begingroup$ no, nothing yet $\endgroup$– Oren Ben-BassatAug 7, 2019 at 14:32
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