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 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 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 internal-categories k-theory lie-theory limits linear linear-algebra locale localization logic mathematics measure 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 28th 2010
    • CommentRowNumber2.
    • CommentAuthorzskoda
    • CommentTimeDec 28th 2010

    never heard :)

    • CommentRowNumber3.
    • CommentAuthorUrs
    • CommentTimeDec 28th 2010

    It’s kind of important in the context of the discusison of “2-affine” geometry that we are having on the nnCafe: a projective variety X/𝔾 mX/\mathbb{G}_m is not affine, but if we think of it as a weak quotient X//𝔾 mX // \mathbb{G}_m instead thus realizing it as a geometric stack, we see that it is “2-affine” (in the sense of Tannaka duality of geometric stacks.)

    • CommentRowNumber4.
    • CommentAuthorUrs
    • CommentTimeDec 28th 2010

    never heard :)

    by the way: if you ask Google, you’ll see that the term is used by a bunch of autors.

    • CommentRowNumber5.
    • CommentAuthorzskoda
    • CommentTimeDec 29th 2010

    It’s kind of important in the context of the discusison of “2-affine” geometry that we are having on the nnCafe: a projective variety X/𝔾 mX/\mathbb{G}_m is not affine, but if we think of it as a weak quotient X//𝔾 mX // \mathbb{G}_m instead thus realizing it as a geometric stack, we see that it is “2-affine”

    Right, but one can not think of one object as something else. Most of algebraic geometers find many substantial advantages and applications of coarse moduli spaces compared with stack solutions like fine moduli spaces. Resolutions are nicer than the objects themselves, but the object of the study (say in algebraic geometry), most of the time is the object itself. I understand that it is easier to take the trivial replacement (stack) and this is what I do most of the time in my work in noncommutative geometry, and that is one of the reasons why most of algebraic geometers are not interested in my work.

    by the way: if you ask Google, you’ll see that the term is used by a bunch of autors

    Has nothing to do with me.

    • CommentRowNumber6.
    • CommentAuthorzskoda
    • CommentTimeDec 29th 2010

    One of the areas where the difficulty of coarse moduli spaces is very apparent is GIT (geometric invariant theory). It is a well defined area of mathematics, with important 19-th century problems making its core. While generalizing stacks to noncommutative setup is not difficult, I would be very happy if I would know anything about how to generalize GIT to noncommutative setup to use it in noncommutative invariant theory which is in its infancy.

    • CommentRowNumber7.
    • CommentAuthorUrs
    • CommentTimeDec 29th 2010
    • (edited Dec 29th 2010)

    Maybe help me: isn’t the action of 𝔾 m\mathbb{G}_m on 𝔸 n+1{0}\mathbb{A}^{n+1}-\{0\} free (and transitive)? Doesn’t that make the weak quotient equivalent to the strict quotient?

    • CommentRowNumber8.
    • CommentAuthorzskoda
    • CommentTimeDec 29th 2010

    Yes, but already the simplest and most widely used generalizations, like the weighted projective spaces do not share this feature.

    • CommentRowNumber9.
    • CommentAuthorUrs
    • CommentTimeDec 29th 2010

    Yes, but that’s not the case that I mentioned in the paragraoh that you quoted and replied to.

    • CommentRowNumber10.
    • CommentAuthorUrs
    • CommentTimeDec 29th 2010

    Oh, I see, it is. I said “projective variety”. I should have said “projective space”. Sorry.

    • CommentRowNumber11.
    • CommentAuthorzskoda
    • CommentTimeDec 29th 2010

    The matter is just the relation between the emphasis on two kinds of content; one very specific and another aimed at the simplest gadgets expressing quotients. Both ideas aree very powerful, I just object to either side if one claims that one of the two ideas replaces another. Grothendieck was master of both. Not only he devised stacks but also solved the question of representability of a number of interesting functors, the questions which were open for about a century, what lead to solutions as Hilbert schemes, Quot schemes etc.