Not signed in (Sign In)

Start a new discussion

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 bundles calculus categorical categories category category-theory chern-weil-theory cohesion cohesive-homotopy-theory 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 education 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-theory modal modal-logic model model-category-theory monad monads monoidal monoidal-category-theory morphism motives motivic-cohomology nforum nlab noncommutative noncommutative-geometry number 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 string string-theory superalgebra supergeometry svg symplectic-geometry synthetic-differential-geometry terminology theory topology topos topos-theory 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
    • CommentTimeJan 18th 2010
    • (edited Jan 18th 2010)

    I started quasicoherent infinity-stack. Currently all this contains is a summary of some central definitions and propositions in Toen/Vezzosi's work. I tried to list lots of direct pointers to page and verse, as their two articles tend to be a bit baroque as far as notation and terminology is concerned.

    This goes parallel with the blog discussion here.

    In the process I also created stubs for SSet-site and model site. These are terms by Toen/Vezzosi, but I think these are obvious enough concepts that deserve an entry of their own. Eventually we should also have one titled "(oo,1)-site", probably, that points to these as special models.

    • CommentRowNumber2.
    • CommentAuthorTobyBartels
    • CommentTimeJan 18th 2010
    • (edited Jan 18th 2010)

    Do you realise that you've created both quasicoherent infinity-stack and quasicoherent ?-stack?

    (Well, the forum software just destroyed one of those links, but you can probably guess what it should be.)

    • CommentRowNumber3.
    • CommentAuthorUrs
    • CommentTimeJan 18th 2010
    • (edited Jan 18th 2010)

    Hm, strange. I started with "quasicoherent oo-stack" and then hit "change entry name". But something went wrong.

    Sorry for that, no sure what happened. Now I renamed "quasicoherent oo-stack" to "quasicoherent oo-stack > history". But I suppose the cache bug also still has a word to say...

    • CommentRowNumber4.
    • CommentAuthorUrs
    • CommentTimeJan 18th 2010
    • (edited Jan 18th 2010)

    Here is a remark that I would like to bounce off in particular Zoran:

    If we use, for definiteness, the projective model structure on complexes, then the cofibant complexes are those that are degreewise projective (and every object is fibrant). That means (doesn't it?) that in the model category of quasicoherent modules, the fibrant-cofibrant objects are in fact the projective chain complexes. But aren't this the complexes of vector bundles?

    In other words, I am wondering if in the oo-setting the oo-category of derived quasicoherent modules isn't much closer to being the oo-category of oo-vector bundles than it may seem.

    • CommentRowNumber5.
    • CommentAuthorzskoda
    • CommentTimeJan 18th 2010
    • (edited Jan 18th 2010)

    I assume you are talking about bounded complexes.

    If you ask weather the algebraic vector bundle is the same as a projective 0-module, then the answer is yes and no. Literally the Serre's theorem (one half of Serre-Swan) is about qcoh sheaves of O-modules on an affine scheme (and even Noetherianess is assumed in Serre's paper what is less essential). For complex analytic manifolds, the Serre's GAGA is extending this projectivity philosophy with really talking about complexes of vector bundles and more generally coherent sheaves in the statements. If you sheafify the condition, you come to local/affine situation, namely under mild assumptions on a scheme, locally free qcoh sheaves and locally projective qcoh sheaves are the same. But algebraic vector bundles are finite-dimensional by definition, what has many reasons. Vector bundles should properly generalize to the sub-infinity-category of perfect complexes (roughly: of compact objects) in infinity setup and this is exactly what Ben Zvi and others are really careful about: to state fine conditions on the subcategory of perfect complexes when relevant. Now it is essential that for the smooth schemes the infinity subcategory of perfect complexes tells you all, while the nonsmooth part (singularities) really need the rest of the information. Look for Orlov's paper on "triangulated categories of singularities" to get convinced into that.

    I just created a stub Serre-Swan theorem.

    • CommentRowNumber6.
    • CommentAuthorUrs
    • CommentTimeJan 18th 2010

    Thanks, Zoran. I need to have a closer look at all this.

    Now that you mention it: what does the homotopy category of modules over a simplicial ring actually give: the derived category of bounded or of unbounded complexes? (I should look at Toen's lectures...)

    • CommentRowNumber7.
    • CommentAuthorUrs
    • CommentTimeJan 18th 2010

    I just created a stub Serre-Swan theorem.

    Thanks! I added some links.

    we should write algebraic vector bundle at some point...

    • CommentRowNumber8.
    • CommentAuthorzskoda
    • CommentTimeJan 18th 2010

    I think bounded from one side only. I just created entry Max Karoubi.

    • CommentRowNumber9.
    • CommentAuthorTobyBartels
    • CommentTimeJan 18th 2010

    @ Urs #3

    Caches cleared.

    • CommentRowNumber10.
    • CommentAuthorUrs
    • CommentTimeJan 18th 2010

    Thanks, Toby!

Add your comments
  • Please log in or leave your comment as a "guest post". If commenting as a "guest", please include your name in the message as a courtesy. Note: only certain categories allow guest posts.
  • To produce a hyperlink to an nLab entry, simply put double square brackets around its name, e.g. [[category]]. To use (La)TeX mathematics in your post, make sure Markdown+Itex is selected below and put your mathematics between dollar signs as usual. Only a subset of the usual TeX math commands are accepted: see here for a list.

  • (Help)