# Is there a concrete application of topos theory?

The notion of topos was originally formulated in SGA 4 in the context of attacking the Weil conjectures. This formalism turned out to be unnecessary for the purposes of proving those conjectures. But the Weil conjectures do give a good example of a theorem where other abstract things, such as Grothendieck topologies and etale cohomology, are used in an essential way.

Is there any example of a concrete result in which the usage of topos theory is essential?

• A concrete result in which field? Algebraic geometry? Differential geometry? Any field that's not category theory? (Genuinely curious, since you haven't used a top-level tag) Commented Jan 16, 2021 at 13:42
• I'm confused by how you consider that topos theory is not used in the proof of the Weil conjecture. You seem to be saying that there use can be replaced by Grothendieck topologies and étale cohomology (which I agree with) but topos theory as introduce in SGA 4 is really nothing more than the theory of Grothendieck topologies. So I wouldn't consider this as getting ride of topos theory. I'm pointing this out, because the same thing apply to all application of Grothendieck toposes : you can always write everything in terms of Sites and Grothendieck topologies. Commented Jan 16, 2021 at 14:58
• Yes "Grothendieck topos theory" and "site theory" are two different point of view on exactly the same thing. Anything you can do with one you can also do it with the other. Of course there are indeed a lot of technical and conceptual advantages of working with toposes rather than sites, but at the end of the day you can always translate everything in terms of sites. Commented Jan 16, 2021 at 15:13
• Since many different sites can give the same topos of sheaves, I'm inclined to view a topos as capturing the "important" aspects of a site and discarding irrelevant details. I could describe this as an analogy "topos : site :: group : group-presentation. In principle, group theory could be developed entirely in terms of presentations, never mentioning the groups themselves, and in some situations that's useful, but in most situations it just makes things less clear. Commented Jan 16, 2021 at 15:50
• @Kim I think you can't lift the functoriality of the crystalline topos to sites so that might be an example. Commented Jan 16, 2021 at 16:12