Timeline for Historically, how were Grothendieck topoi motivated?
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9 events
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| Jun 3, 2019 at 10:46 | comment | added | Denis Nardin | I have no reason to really believe this, but I always assumed that one of the important features is that there are morphisms of topoi that do not come from morphisms of the "usual" sites (basically because the pullback of a representable does not need to be representable in general). | |
| Jun 3, 2019 at 9:16 | comment | added | dhy | @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... | |
| Jun 3, 2019 at 9:00 | comment | added | user141414 | @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. | |
| Jun 3, 2019 at 8:57 | comment | added | dhy | @Cutthewood But the question is "how were they invented" - and the answer is "they were invented at the same time as sites, because it is hard to talk about sites without also inventing topoi." | |
| Jun 3, 2019 at 8:54 | comment | added | user141414 | @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. | |
| Jun 3, 2019 at 8:50 | comment | added | user13113 | 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. | |
| Jun 3, 2019 at 8:44 | comment | added | dhy | 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... | |
| Jun 3, 2019 at 8:05 | review | First posts | |||
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| Jun 3, 2019 at 8:02 | history | asked | user141410 | CC BY-SA 4.0 |