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created a “category: reference”-page The Stacks Project
I have only now had a closer look at this and am impressed by the scope this has. Currently a total of 2288 pages. It starts with all the basics, category theory, commutative algebra and works its way through all the details to arrive at algebraic stacks.
So besides my usual complaint (Why behave as if there are not sites besides the usual suspects on $CRing^{op}$ and either give a general account or call this The Algebraic Stacks Project ? ) I am enjoying seeing this. We should have lots of occasion to link to this. Too bad that this did not start out as a wiki.
One thing they deal with is foundations for algebraic stacks: they work up from showing that there are no set-theoretical problems with talking about sheaves on such and such a site, and that various categories are locally small, essentially small and so on as necessary, which I find admirable. See e.g. lemma 3.8.9.
I agree, this could all be worked out in general. But! all this is application of the general theory, which shows that the whole thing is far from vacuous.
Yes, it is our taste to have it written in more general way, with little expense. On the other hand, there is lots of material there which does not readily generalize and is a specific toolbox for algebraic stacks.
What do you think than of the publicly grown book by Ravi Vakil on schemes then ? Each single lecture is beautfully written and full of expert insight which is far deeper than I could ever speak about commutative algebraic geometry; on the other hand so little of general perspective and view in wider context, like if it were written in 1970-s…
Notice that I wrote:
either give a general account or call this The Algebraic Stacks Project
(New emphasis added.)
Not that I am loosing sleep over this, it was just a remark. But a remark that I keep feeling triggered to make. Also for instance the authors of the Wikipedia page topos behave as if the notion were somehow intrinsic to algebraic geoemtry.
I just think that this is a habit worth discarding. Eventually. I think it is helpful to realize the vast generality of notions such as “stack” and “topos” and be conciously aware of what it means to look at them over special sites. Which of course is something perfectly fine to do!
I agree with you.
On the other hand, the fact that so many people contributed to stacks project (and some others) hwile not contributing to more flexible $n$Lab project, could also be discussed. I had some feedback from some mathematicians, who usually say that they are not category theorists and like more concrete problems. I tell them that we are interested in those (people mentioned things like vertex algebras, motives and so on, which are all so interesting to our scope!). The thing which I share with them is sometimes impression that we go too often right away with say, infinity-categorical picture; the gentle way in my opinion is to have idea section more traditional and the hypermodern version somewhere below; unless the difference can be expressed without scaring people.
I agree, it would be good to add such material. But I only have time to add the material that I need myself. And even that time I don’t really have! :-)
I spent lots of time writing just entries to make things more feasible to the outside world. I hope it will pay back once by attracting new contributors. These days I added few pages about applied mathematics in this manner…
BTW Does anybody have a file of Taylor’s 1999 book on Practical foundations which is not corrupted ? I mean I have some file which is everywhere available on the web and which is taken sometime from a web page prerelease where unfortunately LaTeX has not been properly displayed in many web formulas, so many many pages have unreadable formulas.
I spent lots of time writing just entries to make things more feasible
I’d dare say I have my share of trying to make things nicer, too. When I scroll down the “Recently Revised” page I don’t get the impression that it is necessarily me who is neglecting work on prettifying the nLab. ;-)
@David #2: It’s odd that they use the phrase ’ “big” category ’ for what everyone else I have ever heard of uses the term large category. But I agree that it’s nice that they take set-theoretic issues seriously and also don’t retreat to universes.
I’d dare say I have my share of trying to make things nicer, too.
Surely you did, more than any of us (I do not understand what provoked your defensive statement). I am talking about going outside of strict infinity stuff in content where I feel more isolated (typical entry: integrable systems), rather than making more accessible the esoteric stuff. Hence I take this is a more important resposibility for a mortal like me and leave myself less time to interact in infinity issues where I am even less competent and lagging behind.
(I do not understand what provoked your defensive statement)
I regretted saying it the moment I did.
But speaking of all this: in the other thread we found that we need on the $n$Lab a clear discussion of how algebraic stacks relate to groupoid objects internal to algebraic spaces. That’s an urgent isse of content away from $\infty$-stuff that needs to be added to the $n$Lab. The entry “algebraic stack” badly needs improvement. The chapter on this in The Stacks Project looks very good. Somebody should read it and then extract the missing definitions and properties to the $n$Lab page algebraic stack.
I am glad that you explicitly called here for an action as far as improving algebraic stacks, as the issue is not that difficult for others like me to contribute, while there is more than one reason to do it soon. In differentiable stacks there is also a distinction between DM and Artin.
Ah please, let’s have a good discussion on algebraic stacks vs groupoid schemes. One thing I find frustrating (only because of my own lack of experience with alg geom) is that conditions for such and such a stack to be a “blah” stack are stated in terms of separation properties of the diagonal and only there. I know this translates through to something about what the groupoid scheme presenting the stack looks like, and that people want presentation-independent conditions on a presheaf of groupoids for it to be a “blah” stack, but at some point one has to look at groupoid (at least, I do). My anafunctors paper is crying out for algebraic examples, but my oblique MO question that tried to drum up examples (something about categories of groupoid schemes fibred over Sch, if you care) got no bites.
So here is what I know, and would like to know:
1) I know that the sort of cover $X \to \mathcal{X}$ of a stack one uses becomes the sort of maps the source and target maps $s,t:X\times_\mathcal{X} X \rightrightarrows X$ are.
2) I’m fairly sure that the sort of map the diagonal $\mathcal{X} \to \mathcal{X}\times \mathcal{X}$ is, becomes the sort of map that $(s,t):X\times_\mathcal{X} X \to X$ is.
Now I have no problem with the cover, because I know how pretopologies behave. But the sort of conditions that people put on the diagonal I have no idea how they behave: like, a separated open immersion (or something - this is just an example, noone need answer that).
Am I on the right track?
The condition on the cover $X \to \mathcal{X}$ is a way of saying what the space of objects is like.
The condition on the diagonal is a way of specifying what the space of morphisms is like.
The first statement should be clear. Let’s look at the second. Suppose we have a homotopy pullback diagram
$\array{ V = f^* \mathcal{X} &\to& \mathcal{X} \\ \downarrow &\swArrow_{\eta}& \downarrow \\ U &\stackrel{f}{\to} & \mathcal{X} \times \mathcal{X} } \,.$By the defining property of homotopy pullbacks, the points (generalized points) of $V$ are to be thought of as triples consisting of
a point $u \in U$ with two images $f_1(u) := p_1 (f(u))$ and $f_2(u) := p_2 (f(u))$
a point $x \in \mathcal{X}$
two composable morphisms $f_1(u) \stackrel{}{\to} x \stackrel{}{\to} f_2(u)$ (this is the component of the natural transformation $\eta$ in $\mathcal{X} \times \mathcal{X}$, using that all morphisms are invertible, so that the direction we disply does not matter).
In other words, such a triple is just a morphism in $\mathcal{X}$. Hence $V$ in the above is the space of morphisms in $\mathcal{X}$ that start inside the image $f_1(U)$ and end inside the image $f_2(U)$.
Hence if we demand that for all $f$ the $V$ here is such-and-such, this says that the space of morphisms of $\mathcal{X}$ is such-and-such.
@Zoran I agree with your sentiments at no. 11. Some of the entries in the Lab plunge into n-stuff a bit quickly and the ideas section is sometimes too abstract too quickly. A motivation page with the non-infinity cat version explained is then needed. I have tried to put in a few, but seem to have not enough time to do all that I would feel competent to do, let alone alll those that I would like to do. (Happy New Year!)
15 Urs - The homotopy pullback in the case of sheaves should be just ordinary pullback. No ? I thought that representability for ordinary spaces (that is sheaves on Aff in some subcanonial pretopology) does not have any 2-cells.
Yes. But I was replying to David in #14, who asked how to relate the condition on the diagonal of a stack to the properties of the corresponding groupoid.
Oh, this remark is helpful (I mean the goal statement, not to whom is addressed).
There is some discussion about the set theoretical foundations of the stacks project on their blog. Is there an informed opinion about what would be the most convenient formalism to develop stacks. SEAR, homotopy type theory, …
There is problem with your link. This here should work: http://math.columbia.edu/~dejong/wordpress/?p=530
Or that’s at least what I gather you are pointing us to. But notice that this discussion is from June 2010, over seven years ago, and the comment section is closed.
Also, I would be surprised to learn that the StacksProject authors would care to consider anything but ZF+X. But who knows.
The insistence on ZFC, as opposed to a structural set theory, surprised me a little for the work by Grothendieck. However, it seems that they are interested in showing that they are using the “standard” axioms.
Also, the aim is to show that one can use small sites for any given situation, so that all categories are locally small etc
I came across this comment
when nlab cites the @stacks_project, it doesn’t use tags shudders but instead gives (unstable) chapter numbers, eg in the definition of an algebraic stack.
Does that make sense to anyone?
Also, it shouldn’t be a citation to deJong, but to the project itself; people don’t cite the nLab and write (Schreiber 20xx). I agree that we should only cite tags, not numbers (there are many more chapters now than when I first learned of the project).
So to be concrete
is to replaced by
Then we’d need to find the tags for those 4 results.
Re #27: Considering that Johan de Jong wrote 703685 out of 709534 lines in the Stacks Project, i.e., 99.18%, Urs’s attribution was correct and appropriate. The Stacks Project is not a collaborative project like the nLab, and should be cited with an author.
To put things in perspective (with the same proportions), if a group of people writes 2 pages of text for a 243-page book, while the other 241 pages are written by a single author, the resulting book only has 1 author.
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