Is there a standard term for a morphism of sites $(C,J)\to(C',J')$ which induces an equivalence on sheaf categories $\operatorname{Sh}(C',J')\xrightarrow\sim\operatorname{Sh}(C,J)$? The nLab entry suggests the term Morita equivalence, however I have not yet found any other sources which use this term in this way.
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asked Oct 28, 2019 at 2:02
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$\begingroup$ Sometimes people describe it as 'satisfying the comparison theorem' or words to that effect, citing something in SGA4. The precise hypotheses or definitions that go into that theorem may not be what you have in mind, but I think it's a reasonable claim. $\endgroup$– David Roberts ♦Commented Oct 28, 2019 at 3:41
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$\begingroup$ The comparison lemma of SGA 4 (Expose 3, 4.1) is specifically about fully faithful functors, so that would be a very inaccurate reference in the generality of the question. Morita equivalence is indeed the term used in topos theoretic references (e.g. Caramello uses that term a lot). I don’t think I’ve seen it come up in algebraic geometry literature before. $\endgroup$– user147129Commented Oct 28, 2019 at 8:10
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$\begingroup$ The correct reference for the most general version of the comparison lemma is Moerdjik & Kock "Presentation of etendues" numdam.org/article/CTGDC_1991__32_2_145_0.pdf see the end of section 2. It is not claimed in the paper, but if I remember correctly the conditions they give are necessary and sufficient. $\endgroup$– Simon HenryCommented Oct 28, 2019 at 15:20
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Johnstone's Sketches of an Elephant (2 volumes) is a standard reference which uses "Morita equivalence" in this way. In fact, Jonstone systematically uses "Morita equivalence" in a similar way across many types of categorical logic. Relevant here is the case of geometric logic.
Although some of the terminology in the Elephant is a bit idiosyncratic, I think this use of "Morita equivalence" is pretty common in the categorical logic literature. And the analogy to ring theory which it brings to mind is spot-on, I think.
answered Oct 28, 2019 at 14:45