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 bundle bundles calculus 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 elliptic-cohomology enriched fibration foundations functional-analysis functor galois-theory gauge-theory gebra geometric-quantization geometry graph graphs gravity 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 itex k-theory lie-theory limit limits linear linear-algebra locale localization logic manifolds mathematics measure measure-theory modal-logic model model-category-theory monads monoidal monoidal-category-theory morphism motives motivic-cohomology multicategories 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 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.
    • CommentAuthorMarc Hoyois
    • CommentTimeNov 23rd 2013

    Fixed a couple incorrect statements at hypercomplete (infinity,1)-topos:

    • Remark 1 claimed that having enough points in the 1-topos sense implies having enough points in the ∞-sense. I replaced it with a counterexample from HTT.
    • In Proposition 1 I replaced “finite homotopy dimension” by “locally of homotopy dimension n\leq n”.
    • CommentRowNumber2.
    • CommentAuthorMarc Hoyois
    • CommentTimeNov 23rd 2013

    Here’s a simple counterexample to “finite homotopy dimension implies hypercompleteness”:

    Let H\mathbf{H} be the big (∞,1)-topos on the site TopTop of topological spaces. Then H\mathbf{H} has homotopy dimension 00 but is not hypercomplete because it contains non-hypercomplete (∞,1)-topoi as essential retracts of its slices. More precisely, if XX is a topological space, the embedding i:Open(X)Top /Xi\colon Open(X)\to Top_{/X} is continuous and cocontinuous, so that we have an adjunction i !i *i *i_!\vdash i^*\vdash i_* with i !i_! fully faithful. In that situation, i *i^* preserves nn-truncated morphisms, hence i !i_! preserves \infty-connective morphisms, hence i *i^* preserves hypercomplete objects, hence i !i_! preserves non-hypercomplete objects. Thus, for appropriate XX, H /X\mathbf{H}_{/X} and hence H\mathbf{H} are not hypercomplete.

    • CommentRowNumber3.
    • CommentAuthorUrs
    • CommentTimeNov 24th 2013
    • (edited Nov 24th 2013)

    Remark 1 claimed that having enough points in the 1-topos sense implies having enough points in the ∞-sense. I replaced it with a counterexample from HTT.

    Thanks for looking into this. But wait, the argument that you removed had just another “hypercomplete” implicit/missing.

    The argument said that in the Jardine model structure it is true that when the underlying 1-topos has enough points, then the Jardine weak equivalences are equivalently the stalkwise weak equivalences. Since the Jardine model structure presents the hypercomplete \infty-topos over the site, it does follow that this has enough points when the 1-topos does.

    • CommentRowNumber4.
    • CommentAuthorMarc Hoyois
    • CommentTimeNov 24th 2013

    I added the remark that the hypercompletion has enough points.

    • CommentRowNumber5.
    • CommentAuthorUrs
    • CommentTimeNov 24th 2013

    Okay, thanks.

    (yeah, that darn hypercompletion subtlety… ;-)

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)