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I split off locally representable structured (infinity,1)-topos from generalized scheme .
This is about Lurie’s $\mathcal{G}$-schemes, but I decided to change the title. For one to avoid the continuous conflict of notions of “generalized scheme” that made generalized scheme a mess, but also because it seems quite reasonable terminology to. Would’t you agree?
In locally representable structured (infinity,1)-topos I changed two “Proj” into “Pro”. Objections?
I don’t know what $Proj$ is supposed to stand for in the first place (is it supposed to evoke $Proj$ as in spectrum of a graded algebra?), but I have a feeling that $Pro$ would rub me the wrong way: it would remind me of “profunctor” or “profinite completion” which I strongly suspect are wrong associations.
The entry in question draws from DAG V and I am rather sure that pro-objects are intended.
Okay, thanks! I stand corrected. To further expose my ignorance: is DAG V some work by Lurie on Derived Algebraic Geometry? (I cannot find it on his home page, but maybe I didn’t look hard enough.)
Should that reference be in the reference section? Also, why was $Proj$ there in the first place; was it a typo?
Yes, I referred to
It is in the reference section, but not under its full title.
I guess the $Proj$ was a typo. However there are other deviations in notation compared to Lurie’s text (e.g. an op-ing from ind-objects to pro-objects) hence I was not sure in first place.
Todd, “DAG” refers to a series of 14 articles by Jacob Lurie, which lay out the foundations of higher geometry. However, I like to suppress the “algebraic geometry” when citing these articles, because I find it misleads people. The development in the articles is about geometry quite generally. I kep running into differential geometers who would keep saying things like “in algebraic geometry there are these higher categories”, not realizing that in differential geometry, topology, supergeometry, etc. there are just as well, and I don’t want to participate in further prolonging this decade-old confusion.
Concerning the pro-objects: I think the choice of notation convention does not matter much if only it is made clear what is meant. I have now added a pointer to pro-object in an (infinity,1)-category to the entry. This had been there originally, but got stripped off when the entry was split off from structured (infinity,1)-topos.
Concerning whether this is a typo: it feels a bit like saying that stating Pythagoras’ theorem like “$x^2 + y^2 = z^2$” is a typo, because it ought to be “$a^2 + b^2 = c^2$”. But I agree that changing it to “Pro” is probably better.
Thanks for your information and help, Urs. It’s of course fine with me either way how you guys want to refer to Lurie’s work; I just wasn’t sure what Stephan was referring to in #4 and was confirming. Having a pointer to the pro-object article is also useful.
I don’t agree with the analogy in your last sentence, since I had guessed $Proj$ referred to something totally different from pro-object. I would never think that about an inessential change of variables. I’m glad it’s been sorted out.
Here is further confusion of notation: In St. Sp. (:-) in definition 2.1.2 in the composit diagram the relative spectrum functor goes in the wrong direction and perhaps the Ind-objects are included in the wrong category. In the nlab here it is syntactically correct but if one has a transformation of geometries which is not the identity (which it is in this case), the relative spectrum functor goes in the wrong direction.
Hi Stephan,
I just got off the plane after a transatlantic flight. Don’t quite feel like digging through this variance-issue right now. If you feel you understand what’s going on and feel sure the direction of the arrow in the entry is reversed, then please correct it!
Hi Urs, welcome back.
I made the change I consider right: $p:\mathcal{G}_0\to \mathcal{G}$ (not the other way round) since this shall map admissible morphisms to such and $\mathcal{G}_0$ denotes the discrete geometry underlying $\mathcal{G}$. In St Sp is the same typo.
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