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The question is about how did the person who invented Grothendieck topoi (presumably Grothendieck) arrive at the necessity of a such a notion. I do not know much about the history of the subject. What I do know is that functoriality for crystalline cohomology is most elegantly done via the introduction of topoi and can not be always be done if you work at the level of sites.

So was this person solely motivated by the crystalline cohomology? Were the any other geometric, topological or number-theoretic (not set-theoretic or logical) problems that "forced" the notion of topos (as opposed to site) upon you? I am not asking for overly hypothetical justifications (for example, by notions whose existence was only discovered well after Grothendieck topoi were introduced) but the justifications by problems which could reasonably be of interest to people (that is, already were around and had not yet received a completely satisfactory treatment at that moment) in 1950s and 1960s. If the justifications you propose were explicitly stated as motivation in the research literature of the era, that is even better.

The question is somewhat speculation-ish but not overly so. It still is answerable, just give relevant examples of problems and how Grothendieck topoi come up when you try to solve them.

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  • $\begingroup$ I'm not sure I understand the question, in particular the "(as opposed to site)" part. My impression is as follows: The reason sites were invented was to be able to define etale cohomology as the derived functor of global sections on the category of etale sheaves. Therefore the category of etale sheaves (the topos) was in play from the very start. I assume your comment about functoriality for crystalline cohomology refers to the existence of morphisms of topoi which don't come from morphisms of sites, which I see as a separate phenomenon... $\endgroup$
    – dhy
    Commented Jun 3, 2019 at 8:44
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    $\begingroup$ You can also, in my opinion, view the theory of Grothendieck toposes as a precursor to the theory of locally presentable categories. So if you believe the latter is important in category theory, then you could view part of the historical interest in toposes as because they were partially fulfilling that need. $\endgroup$
    – user13113
    Commented Jun 3, 2019 at 8:50
  • $\begingroup$ @dhy I think the "as opposed to site" part is pretty clear. With etale cohomology you don't need to think of topos as the essential object, you can think of the site as the essential object, but with crystalline cohomology, in at least some of the parts of the theory, you do (the distinction is somewhat similar to the distinction between model categories and $\infty$-categories, I think). So I am confused by your confusion, sort of. $\endgroup$
    – user141414
    Commented Jun 3, 2019 at 8:54
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    $\begingroup$ @dhy really? I thought topoi came at least a little bit later than sites (I have heard some category theorists, for example, emphasize the fundamental conceptual difference between a topos and a site generating it, so I thought it requred some more effort). Probably my ignorance is showing, I retract my statement then. $\endgroup$
    – user141414
    Commented Jun 3, 2019 at 9:00
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    $\begingroup$ @Cutthewood Seeing the topos as the fundamental object rather than the site is certainly a conceptual advance, and I can't say whether or not Grothendieck had this idea in mind from the very start (though if he didn't, he developed it soon after.) At least in SGA4, the emphasis is already on topoi and not just sites. I suppose if you interpret the question as "What led Grothendieck to consider the notion of topos as more fundamental than the notion of site?" this is an interesting question... $\endgroup$
    – dhy
    Commented Jun 3, 2019 at 9:16

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