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 beauty 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 education elliptic-cohomology enriched fibration 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 k-theory lie 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 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 science set set-theory sheaf simplicial space spin-geometry stable-homotopy-theory string string-theory subobject 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.
    • CommentAuthorTobyBartels
    • CommentTimeJan 15th 2011

    I added a definition to locally contractible space, but is it correct?

    • CommentRowNumber2.
    • CommentAuthorUrs
    • CommentTimeJan 15th 2011
    • (edited Jan 15th 2011)

    Thanks, Toby. Discussion of this and many related aspects has been on my to-do list for quite a while, but I have not really gotten around to it.

    Let’s first clarify the case of locally contractible topological spaces. I have now added a discussion of this at locally infinity-connected (infinity,1)-site.

    So the general statement is that (hypercompleted) \infty-sheaves on a 1-site CC form a locally \infty-connected (,1)(\infty,1)-topos if all constant \infty-presheaves are already sheaves, which is the case if there is for each object UCU \in C a split hypercover YUY \to U.

    This is a sufficient, not a necessary condition. For XX a locally contractible topological space, the condition is not satisfied for the category of open subsets Op(X)Op(X). Becuse even though every open subset is covered by contractibles, it need not even be connected, and hence constant presheaves are not even sheaves.

    But there is another site of definition for Sh (1,1)(X)Sh_{(1,1)}(X) and Sh^ (,1)(X)\hat Sh_{(\infty,1)}(X): the full subcategrory

    cOp(X)Op(X) cOp(X) \hookrightarrow Op(X)

    on the contractible open subsets: since by assumption every UOp(X)U \in Op(X) has a cover by contractbles, two sheaves are equivalent already when they are equivalent on contractibles.

    And the site cOp(X)cOp(X) does have the property that all constant (n,1)(n,1)-presheaves on it are already (n,1)(n,1)-sheaves, for all nn.

    So if XX is locally contractible then Sh (,1)(X)Sh_{(\infty,1)}(X) is locally \infty-connected.

    For the converse statement, let me think…

    • CommentRowNumber3.
    • CommentAuthorUrs
    • CommentTimeJan 15th 2011
    • (edited Jan 15th 2011)

    I wrote:

    For the converse statement, let me think…

    I went back to the 1-categorical statement that the sheaf topos over a topological space is a locally connected topos precisely if the topological space is so. The key step is to prove that if we do have an essential geometric mophism (Π 0LConstΓ):Set(\Pi_0 \dashv L Const \dashv \Gamma) : \mathcal{E} \to Set, then every object is the coproduct of connected objects.

    In the Elephant, lemma 3.3.6. this is a 6-line argument. But I am not sure how to generalize that argument to \infty-sheaves. (Maybe it’s easy, but I am not sure.)

    So what I did instead now is that I re-proved this in a fashion that ought to have a straightforward, almost verbatim, \infty-categorical analog.

    What i came up with is now the proof here at locally connected topos.

    I’ll think a bit more now to see if from this I can really deduce that Sh (,1)(X)Sh_{(\infty,1)}(X) is locally \infty-connected precisely if XX is locally contractible…

    • CommentRowNumber4.
    • CommentAuthorUrs
    • CommentTimeJan 15th 2011
    • (edited Jan 15th 2011)

    I wrote:

    I’ll think a bit more now to see if from this I can really deduce that Sh (,1)(X)Sh_{(\infty,1)}(X) is locally \infty-connected precisely if XX is locally contractible…

    It seems that the argument that is needed is that which we already discussed once at Whitehead tower in an (infinity,1)-topos. I try to go through this now carefully to see if it serves to prove the above, but I will likely be interrupted before I am done.

    But let me just briefly notice the relation to Whitehead towers:

    Along the lines of the proof for n=1n = 1 we want to say that for (Π nΔΓ):HnGrpd(\Pi_n \dashv \Delta \dashv \Gamma) : \mathbf{H} \to n Grpd a locally n-connected (n,1)-topos and XX any object, that the nn-connected objects U iU_i covering AA are the pullbacks

    U i * X ΔΠ nX, \array{ U_i &\to& * \\ \downarrow && \downarrow \\ X &\to& \Delta \Pi_n X } \,,

    where the bottom morphism is the (ΠΔ)(\Pi \dashv \Delta)-unit. It is clear that this collection of objects does cover:

    lim iΠXU iX {\lim_\to}_{i \in \Pi X} U_i \stackrel{\simeq}{\to} X

    this follows from universal colimits. The question is if the U iU_i are really all nn-connected in that Π n(U i)*\Pi_n(U_i) \simeq *.

    But this is exactly the implicit claim at Whitehead tower in an (infinity,1)-topos. Notice that

    Π nX:=ΔΠ nX \mathbf{\Pi}_n X := \Delta \Pi_n X

    is what I call the intrinsic path n-groupoid .

    In suitable situations where both notions exist, this does coincide with the nn-truncation of the standard fundamental infinity-groupoid of a topological space. As observed at universal cover, we have that the universal 1-connected cover of a connected topological space is the \infty-pullback

    X^ * X Π 1(X). \array{ \hat X &\to& * \\ \downarrow && \downarrow \\ X &\to& \mathbf{\Pi}_1(X) } \,.

    Back then this was the starting point for the idea of defining the whole Whitehead tower of any object in any locally \infty-connected (,1)(\infty,1)-topos by the obvious iteration of this construction.

    Now, at Whitehead tower in an (ininity,1)-topos I had once typed the argument for how the internal homotopy groups do behave as expected. If this goes through as one might expect, it would serve to yield a proof that XX is locally contracible precisely if Sh (infty,1)(X)Sh_{(infty,1)}(X) is locally \infty-connected.

    [edit: hm, some nns are off by ±1\pm 1, as usual…]

    • CommentRowNumber5.
    • CommentAuthorTobyBartels
    • CommentTimeJan 15th 2011

    Thanks, that sounds reasonable. I’ll read through the proofs that you have so far and see if they make sense to me.

    There does seem to be some discussion of locally contractible locales in the literature already, although I can’t read any of it online right now. Ieki Moerdijk in particulary has mentioned them, and presumably he has a definition in mind.

    • CommentRowNumber6.
    • CommentAuthorUrs
    • CommentTimeJan 15th 2011
    • (edited Jan 15th 2011)

    I’ll read through the proofs that you have so far and see if they make sense to me.

    Please do. I’d be grateful if you’d tried hard to poke holes into everything.

    Meanwhile, I still don’t have a general proof that the U iU_i in comment #4 are nn-connected in the \infty-topos over a locally nn-connected topological space, except for n=0n = 0.

    I do have something different, but related: in the cohesive \infty-topos ETop ∞Grpd I think I can show (see there) the analogous statement for the case that XX is a paracompact space: that all homotopy fibers of XΔΠ(X)X \to \Delta \Pi(X) are geometrically contractible. In fact, in that case I think I can show that the homotopy fibers of XΔτ nΠ(X)X \to \Delta \tau_{\leq n} \Pi(X) for all nn \in \mathbb{N} map under Π\Pi to the whole standard Whitehead tower of XX.

    But the proof of this is not purely general abstract, so I am not sure right now how to generalize it. Maybe I am just being dense. Or maybe there is more to it.

    There does seem to be some discussion of locally contractible locales in the literature already, although I can’t read any of it online right now. Ieke Moerdijk in particulary has mentioned them, and presumably he has a definition in mind.

    I’ll see what i can find out next week.

    • CommentRowNumber7.
    • CommentAuthorMike Shulman
    • CommentTimeJan 18th 2011

    Maybe I missed the mention of something like this, but won’t you need some sort of q- or m-cofibrancy? Since isn’t toposic (local) connectiveness more about trivial homotopy (pro)groups?

    • CommentRowNumber8.
    • CommentAuthorUrs
    • CommentTimeJan 18th 2011
    • (edited Jan 18th 2011)

    but won’t you need some sort of q- or m-cofibrancy? Since isn’t toposic (local) connectiveness more about trivial homotopy (pro)groups?

    Hm, let’s see, my argument is as follows:

    1. by Artin-Mazur we have that if XX is a locally contractible topological space, then the simplicial set

      lim (YX)Y contr {\lim_\to}_{(Y \to X)} Y_{contr}

      (where the colimit ranges over all hypercovers and Y contrY_{contr} is the simplicial set obtained by contracting degreewise each connected summand to a point) has the same homotopy groups as XX.

    2. Observe that when switching site of definition Sh(Op(C))Sh(cOp(X))Sh(Op(C)) \simeq Sh(cOp(X)) (right?) we have that this procedure computes the left derived functor of lim :[cOp(X) op,sSet]sSet\lim_\to : [cOp(X)^op, sSet] \to sSet;

    3. By Dugger-Hollander-Isaksen this means that the colimit in the above localizes on any split hypercover, degreewise a coproduct of contractibles.

    • CommentRowNumber9.
    • CommentAuthorMike Shulman
    • CommentTimeJan 18th 2011

    I was talking about the converse: if Sh(X) is locally \infty-connected then XX is locally contractible.

    • CommentRowNumber10.
    • CommentAuthorUrs
    • CommentTimeJan 19th 2011

    Hi Mike,

    sorry, I had missed that you had already replied here. (That nnPublication-discussion is drawing a lot of energy and attention…)

    First, just for the record, concerning item 2 in #8: I have recorded the standard facts and the example of locally contractible spaces at dense sub-site.

    Then finally concerning your comment:

    so you are suggesting that for an \infty-topos over a topological space XX being toposophically \infty-connected to imply that XX is locally contractible it is necessary that XX is an m-cofibrant space?

    Not sure. Maybe I need to think more about this.

    By the way, I asked about the definition of “locally contractibe locale” as given somewhere by Ieke: presumeably it was by saying there is a cover such that the patches are contractible as seens by homming the locales Δ n\Delta^n into them.

    • CommentRowNumber11.
    • CommentAuthorMike Shulman
    • CommentTimeJan 19th 2011

    so you are suggesting that for an
    ∞-topos over a topological space X being toposophically [locally] ∞-connected to imply that X is locally contractible it is necessary that X is an m-cofibrant space?

    Yes, I’m suggesting that something of the sort at least might be necessary (perhaps even “local m-cofibrancy”). I would be a bit surprised if no conditions at all were required. Of course the toposophic homotopy groups are more refined than the classical ones, so my intuition could be off.

    • CommentRowNumber12.
    • CommentAuthorDavidRoberts
    • CommentTimeJan 19th 2011

    I edited the example of good open covers at dense sub-site to read as follows:

    every paracompact manifold has a good open cover by open balls diffeomorphic to a Cartesian space.

    I changed homeomorphic to diffeomorphic. Just checking this is correct (not quite sure about exotic smoothness, that’s all)

    • CommentRowNumber13.
    • CommentAuthorUrs
    • CommentTimeJan 19th 2011
    • (edited Jan 19th 2011)

    Just checking this is correct

    Yes, it is. A while back I spelled out the full detailed proof for this at good open cover. This statement has a curious status in the literature: in textbooks one finds both the statement that this is a difficult open problem and the statement that this is obvious, but never a statement of the proof.

    I’d prefer, though, if we kept topological and smooth manifolds as two distinct cases at dense sub-site. I think I’ll do that now.

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)