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Since Grothendieck introduced them in the 1960s, topoi have found many applications in algebraic geometry, category theory, and logic. For instance, they appear in the development of étale and crystalline cohomology, and feature prominently in SGA4.

However, it appears (to me, at least) that they are not as widely used today in algebraic geometry (that is, not counting developments in logic, higher category theory, and derived algebraic geometry, with a concrete example being Lurie's work on spectrally ringed $\infty$-topoi)).

So, how are topoi used in algebraic geometry today?

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    $\begingroup$ A good place to start is the papers by Ingo Blechschmidt: ingo-blechschmidt.eu $\endgroup$
    – Dmitri Pavlov
    Commented Sep 10, 2019 at 1:50
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    $\begingroup$ Faltings topos in p-adic Simpson correspondence, the oriented product of topos for studying nearby cycle over general bases... $\endgroup$
    – Zhiyu
    Commented Sep 11, 2019 at 3:22
  • $\begingroup$ J Giraud ICM 1970 pdf, UTILISATION DES CATÉGORIES EN GÉOMÉTRIE ALGÉBRIQUE $\endgroup$
    – user366545
    Commented Sep 13, 2021 at 1:01
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    $\begingroup$ Of course étale and crystalline cohomology are still widely used! Then there is the new pro-étale topos of Bhatt and Scholze, which both clarifies conceptually what is going on, but also has technical advantages (for instance to the construction of filtered derived categories). $\endgroup$
    – R. van Dobben de Bruyn
    Commented Apr 18, 2023 at 9:37
  • $\begingroup$ @R.vanDobbendeBruyn I guess that, for crystalline cohomology, the point of view of topoi is not that important (one usually studies quasicoherent sheaves there). $\endgroup$
    – Z. M
    Commented Apr 18, 2023 at 14:23

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