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).

nLab > Latest Changes: plus construction on presheaves

Bottom of Page

1 to 16 of 16

  1.  
    • CommentRowNumber1.
    • CommentAuthorUrs
    • CommentTimeApr 30th 2012

    Stephan Spahn added stuff to plus construction on presheaves

    • CommentRowNumber2.
    • CommentAuthorUrs
    • CommentTimeApr 30th 2012

    Thanks, Stephan!

    you ask

    To address the question of compact objects in Sh ∞(SmoothMfd) there should be an (∞,1)-plus construction, too. Is in this case where the (∞,1)-site is just a 1-site somehow clear how this works?

    For nn-truncated objects it is in principle clear: one has to apply the plus-construction (n+2)-times in a row!

    See for instance section 6.5.3 of HTT.

    • CommentRowNumber3.
    • CommentAuthorUrs
    • CommentTimeApr 30th 2012

    I have added a brief remark on this in the Idea section. But am out of time now.

    • CommentRowNumber4.
    • CommentAuthorzskoda
    • CommentTimeApr 30th 2012
    • (edited Apr 30th 2012)

    I have added the reference to the famous Heller-Rowe article where the plus construction in the case of abelian sheaves is studied. In the abelian context one usually says Heller-Rowe functor. Interestingly, the theory of Q-categories of Rosenberg has been written out in 1988 to exhibit the common and nontrivial generalization of the functor H ℱH_{\mathcal{F}} in the theory of Gabriel localization and Heller/Rowe functor, as the instances of the same construction. Gabriel localization is also H ℱ 2H^2_{\mathcal{F}}, just like sheafification. Here ℱ\mathcal{F} is a Gabriel filter.

    • CommentRowNumber5.
    • CommentAuthorZhen Lin
    • CommentTimeApr 30th 2012

    I always thought that the fact that we needed to do (−) +(-)^+ twice had something to do with the fact that the equaliser diagram has two stages, but I never did find a good technical explanation of this point. What is clear, though, is that doing it once is a generalisation of computing Hˇ 0\check{H}^0. I imagine these two facts are related via nn-categories…

  6. Added to plus construction on presheaves that applying the plus construction once is not left adjoint to the inclusion of separated presheaves in all presheaves.

    • CommentRowNumber7.
    • CommentAuthorMike Shulman
    • CommentTimeApr 26th 2021

    I just realized that the description of the plus-construction in 6.2.2.9 of HTT is exactly an instance of (the ∞\infty-version of) the description of a p.r.a. functor in terms of a polynomial of discrete fibrations here.

    That’s interesting.

    • CommentRowNumber8.
    • CommentAuthorMike Shulman
    • CommentTimeApr 26th 2021

    Well, not quite. The last functor ρ\rho is not a discrete fibration. But it’s closely related.

    • CommentRowNumber9.
    • CommentAuthorMike Shulman
    • CommentTimeMay 23rd 2021

    The link to descent morphism didn’t match the usage of that term on that page.

    diff, v12, current

    • CommentRowNumber10.
    • CommentAuthorMike Shulman
    • CommentTimeMay 25th 2021

    The first and third definitions were really the same, so I collapsed them, and added in the internal description to the same list for good measure.

    diff, v13, current

    • CommentRowNumber11.
    • CommentAuthorMike Shulman
    • CommentTimeMay 25th 2021

    Added a definition in terms of Kan extensions, inspired by Lurie’s. Also remarked on which definitions work for ∞\infty-presheaves.

    diff, v13, current

    • CommentRowNumber12.
    • CommentAuthorMike Shulman
    • CommentTimeMay 25th 2021

    Sketched a proof that the plus-construction is a well-pointed endofunctor whose fixedpoints are the sheaves, using an observation that there is a situation of cohesion involved.

    diff, v13, current

    • CommentRowNumber13.
    • CommentAuthorMike Shulman
    • CommentTimeMay 25th 2021

    Sketched a proof that the plus-construction is a well-pointed endofunctor whose fixedpoints are the sheaves, using an observation that there is a situation of cohesion involved.

    diff, v13, current

    • CommentRowNumber14.
    • CommentAuthorMike Shulman
    • CommentTimeMay 25th 2021

    Completed a proof, inspired by Anel and Leena Subramaniam, that the plus-construction for nn-separated presheaves converges after n+2n+2 steps.

    diff, v13, current

    • CommentRowNumber15.
    • CommentAuthorMike Shulman
    • CommentTimeMay 28th 2021

    Added link to the Coq formalization.

    diff, v16, current

    • CommentRowNumber16.
    • CommentAuthorMike Shulman
    • CommentTimeMay 28th 2021

    Added remarks about generalization to lex modulators and non-presheaf toposes.

    diff, v16, current

1 to 16 of 16

  1.