Not signed in (Sign In)

Not signed in

Want to take part in these discussions? Sign in if you have an account, or apply for one below

  • Sign in using OpenID

Site Tag Cloud

2-category 2-category-theory abelian-categories adjoint algebra algebraic algebraic-geometry algebraic-topology analysis analytic-geometry arithmetic arithmetic-geometry book bundles calculus categorical categories category category-theory chern-weil-theory cohesion cohesive-homotopy-type-theory cohomology colimits combinatorics comma complex complex-geometry computable-mathematics computer-science constructive cosmology deformation-theory descent diagrams differential differential-cohomology differential-equations differential-geometry digraphs duality elliptic-cohomology enriched fibration finite foundation foundations functional-analysis functor gauge-theory gebra geometric-quantization geometry graph graphs gravity grothendieck group group-theory harmonic-analysis higher higher-algebra higher-category-theory higher-differential-geometry higher-geometry higher-lie-theory higher-topos-theory homological homological-algebra homotopy homotopy-theory homotopy-type-theory index-theory integration integration-theory k-theory lie-theory limits linear linear-algebra locale localization logic mathematics measure-theory modal modal-logic model model-category-theory monad monads monoidal monoidal-category-theory morphism motives motivic-cohomology nlab noncommutative noncommutative-geometry number-theory of operads operator operator-algebra order-theory pages pasting philosophy physics pro-object probability probability-theory quantization quantum quantum-field quantum-field-theory quantum-mechanics quantum-physics quantum-theory question representation representation-theory riemannian-geometry scheme schemes set set-theory sheaf simplicial space spin-geometry stable-homotopy-theory stack string string-theory superalgebra supergeometry svg symplectic-geometry synthetic-differential-geometry terminology theory topology topos topos-theory tqft type type-theory universal variational-calculus

Vanilla 1.1.10 is a product of Lussumo. More Information: Documentation, Community Support.

Welcome to nForum
If you want to take part in these discussions either sign in now (if you have an account), apply for one now (if you don't).
    • CommentRowNumber1.
    • CommentAuthorUrs
    • CommentTimeDec 29th 2010
    • (edited Dec 29th 2010)

    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 opCRing^{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.

    • CommentRowNumber2.
    • CommentAuthorDavidRoberts
    • CommentTimeDec 29th 2010

    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.

    • CommentRowNumber3.
    • CommentAuthorzskoda
    • CommentTimeDec 29th 2010

    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…

    • CommentRowNumber4.
    • CommentAuthorUrs
    • CommentTimeDec 29th 2010
    • (edited Dec 29th 2010)

    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!

    • CommentRowNumber5.
    • CommentAuthorzskoda
    • CommentTimeDec 29th 2010

    I agree with you.

    • CommentRowNumber6.
    • CommentAuthorzskoda
    • CommentTimeDec 29th 2010

    On the other hand, the fact that so many people contributed to stacks project (and some others) hwile not contributing to more flexible nnLab 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.

    • CommentRowNumber7.
    • CommentAuthorUrs
    • CommentTimeDec 29th 2010

    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! :-)

    • CommentRowNumber8.
    • CommentAuthorzskoda
    • CommentTimeDec 29th 2010
    • (edited Dec 29th 2010)

    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.

    • CommentRowNumber9.
    • CommentAuthorUrs
    • CommentTimeDec 29th 2010

    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. ;-)

    • CommentRowNumber10.
    • CommentAuthorMike Shulman
    • CommentTimeDec 29th 2010

    @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.

    • CommentRowNumber11.
    • CommentAuthorzskoda
    • CommentTimeDec 30th 2010

    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.

    • CommentRowNumber12.
    • CommentAuthorUrs
    • CommentTimeDec 30th 2010

    (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 nnLab 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 nnLab. 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 nnLab page algebraic stack.

    • CommentRowNumber13.
    • CommentAuthorzskoda
    • CommentTimeDec 30th 2010

    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.

    • CommentRowNumber14.
    • CommentAuthorDavidRoberts
    • CommentTimeDec 30th 2010

    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 X𝒳X \to \mathcal{X} of a stack one uses becomes the sort of maps the source and target maps s,t:X× 𝒳XXs,t:X\times_\mathcal{X} X \rightrightarrows X are.

    2) I’m fairly sure that the sort of map the diagonal 𝒳𝒳×𝒳\mathcal{X} \to \mathcal{X}\times \mathcal{X} is, becomes the sort of map that (s,t):X× 𝒳XX(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?

    • CommentRowNumber15.
    • CommentAuthorUrs
    • CommentTimeDec 31st 2010
    • (edited Dec 31st 2010)

    The condition on the cover X𝒳X \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

    V=f *𝒳 𝒳 η U f 𝒳×𝒳. \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 VV are to be thought of as triples consisting of

    1. a point uUu \in U with two images f 1(u):=p 1(f(u))f_1(u) := p_1 (f(u)) and f 2(u):=p 2(f(u))f_2(u) := p_2 (f(u))

    2. a point x𝒳x \in \mathcal{X}

    3. two composable morphisms f 1(u)xf 2(u)f_1(u) \stackrel{}{\to} x \stackrel{}{\to} f_2(u) (this is the component of the natural transformation η\eta in 𝒳×𝒳\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 𝒳\mathcal{X}. Hence VV in the above is the space of morphisms in 𝒳\mathcal{X} that start inside the image f 1(U)f_1(U) and end inside the image f 2(U)f_2(U).

    Hence if we demand that for all ff the VV here is such-and-such, this says that the space of morphisms of 𝒳\mathcal{X} is such-and-such.

    • CommentRowNumber16.
    • CommentAuthorTim_Porter
    • CommentTimeDec 31st 2010

    @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!)

    • CommentRowNumber17.
    • CommentAuthorzskoda
    • CommentTimeDec 31st 2010
    • (edited Dec 31st 2010)

    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.

    • CommentRowNumber18.
    • CommentAuthorUrs
    • CommentTimeDec 31st 2010

    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.

    • CommentRowNumber19.
    • CommentAuthorzskoda
    • CommentTimeJan 3rd 2011

    Oh, this remark is helpful (I mean the goal statement, not to whom is addressed).

    • CommentRowNumber20.
    • CommentAuthorspitters
    • CommentTimeJul 4th 2017

    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, …

    • CommentRowNumber21.
    • CommentAuthorUrs
    • CommentTimeJul 4th 2017

    There is problem with your link. This here should work: http://math.columbia.edu/~dejong/wordpress/?p=530

    • CommentRowNumber22.
    • CommentAuthorUrs
    • CommentTimeJul 4th 2017
    • (edited Jul 4th 2017)

    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.

    • CommentRowNumber23.
    • CommentAuthorspitters
    • CommentTimeJul 4th 2017

    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.

    • CommentRowNumber24.
    • CommentAuthorDavidRoberts
    • CommentTimeJul 5th 2017

    Also, the aim is to show that one can use small sites for any given situation, so that all categories are locally small etc

    • CommentRowNumber25.
    • CommentAuthorDavid_Corfield
    • CommentTimeAug 6th 2020

    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?

    • CommentRowNumber26.
    • CommentAuthorUrs
    • CommentTimeAug 6th 2020

    I am often citing their tags (or should I say “their tags shudders”?), for instance here or here or here, as others have (e.g here).

    I have never seen “the nlab” cite anyone, would have to check with our admin whether automation is that advanced already.

    • CommentRowNumber27.
    • CommentAuthorDavidRoberts
    • CommentTimeAug 6th 2020

    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).

    • CommentRowNumber28.
    • CommentAuthorDavid_Corfield
    • CommentTimeAug 6th 2020
    • (edited Aug 6th 2020)

    So to be concrete

    is to replaced by

    Then we’d need to find the tags for those 4 results.

    • CommentRowNumber29.
    • CommentAuthorDmitri Pavlov
    • CommentTimeAug 6th 2020
    • (edited Aug 6th 2020)

    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.

    • CommentRowNumber30.
    • CommentAuthorDavidRoberts
    • CommentTimeApr 11th 2023

    Updated to point to current website and pdf link. Also updated page estimate from more than 6000 to more than 7500.

    diff, v6, current