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-type-theory cohomology colimits combinatorics 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 foundations functional-analysis functor galois-theory 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 lie-theory limit limits linear linear-algebra locale localization logic manifolds mathematics measure-theory modal modal-logic model model-category-theory 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 20th 2011
    • (edited Jan 20th 2011)

    expanded and polished the entry model structure on simplicial sheaves (to be distinguished from the one of simplicial pre-sheaves!)

    Made explicit the little corollary that for DCD \to C a dense sub-site, the corresponding hypercompleted \infty-sheaf \infty-toposes are equivalent.

    • CommentRowNumber2.
    • CommentAuthorMike Shulman
    • CommentTimeJan 20th 2011

    Is that corollary not true without hypercompletion? That would be surprising to me.

    • CommentRowNumber3.
    • CommentAuthorUrs
    • CommentTimeJan 20th 2011
    • (edited Jan 20th 2011)

    That would be surprising to me.

    True. I just went the way of least resistance.

    So let’s see. Let f:DCf : D \to C be a dense sub-site. Then by standard arguments ( I have now typed that out here) we have that restriction and right Kan extension constitute a simplicial Quillen adjunction between the Cech-local injective model structures on simplicial presheaves

    (f *f *):[D op,sSet] inj,loc[C op,sSet] inj,loc (f^* \dashv f_*) : [D^{op}, sSet]_{inj,loc} \stackrel{\leftarrow}{\to} [C^{op}, sSet]_{inj,loc}

    Is this a Quillen equivalence?

    Since the (f *f *)(f^* \dashv f_*)-counit is the identity, it would be sufficient to check that for all fibrant A[C op,sSet] inj,locA \in [C^{op}, sSet]_{inj,loc} the unit

    Af *f *A A \to f_* f^* A

    is a weak equivalence (we don’t even need to throw in an extra fibrant replacement, since the left adjoint f *f^* preserves fibrant objects in our case). That’s the analog statement of the last paragraph on p. 547 of the Elephant .

    Now, that looks like it ought to be easy, but I may nevertheless be stuck.

    By Dugger-Hollander-isaksen it would be sufficient to show that for every XCX \in C and morphism Xf *f *AX \to f_* f^* A there is a cover {U iX}\{U_i \to X\} and local lifts

    U i A X f *f *A. \array{ U_i &\to& A \\ \downarrow && \downarrow \\ X &\to& f_* f^* A } \,.

    How do I deduce this? I will need to use that by fibrancy of AA we have that the morphisms

    [C op,sSet](X,f *f *A)[C op,sSet](S({U i}),f *f *A) [C^{op}, sSet](X, f_*f^* A) \to [C^{op}, sSet](S(\{U_i\}), f_*f^* A)

    are acyclic fibrations, for S({U i})XS(\{U_i\}) \to X the sieve generated by the {U iX}\{U_i \to X\}.


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)