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 comma 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 finite 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 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 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).
    • CommentRowNumber1.
    • CommentAuthorHarry Gindi
    • CommentTimeDec 8th 2010

    I stated this for presheaf categories, but I’m pretty sure that it carries over for any Grothendieck topos.

    Check it out: lawvere interval

    • CommentRowNumber2.
    • CommentAuthorUrs
    • CommentTimeDec 8th 2010

    Thanks.

    So is this the result of your quest to find a natural orighin of the model structure for Cartesian fibrations: you are claiming that its some construction by Cisisnki applied to the Lawvere cylinder in the preseaf topos of simplicial sets?

    I can’t quite parse yet the details of your example at the end. I am sure I could if I now went to Cisinski’s book and looked things up. But since I need to be doing something else, maybe you could expand a bit around where you say “we get exactly”. By doing what exactly?

    When you go to the entry you’ll see that I did some editorial editing. Feel free to change if you don’t approve of what I did.

    • CommentRowNumber3.
    • CommentAuthorHarry Gindi
    • CommentTimeDec 8th 2010
    • (edited Dec 8th 2010)

    @Urs: I plan on finishing the article later. Basically, Cisinski has a construction that given a cylinder, a set of monomorphisms, and a set of generating cofibrations, we can complete the set of cofibrations to a class of anodyne morphisms for the cylinder. Then, by applying the machinery in chapter 1.3, we get a model structure. What I wrote up at the bottom there is from private correspondence with Cisinski (I asked him how to get Joyal’s right anodyne morphisms using his formalism. He said that the way to do it is to generate the anodyne maps by taking Lawvere’s cylinder and generating a class of anodyne morphisms using his construction). This is kinda easy to see, in some sense, since in particular, it generates the class of morphisms Δ n×Δ 1 Δ n×{1}Δ n×{1}Δ n×Δ 1\partial\Delta^n\times \Delta^1\coprod_{\partial \Delta^n\times \{1\}} \Delta^n\times \{1\}\to \Delta^n\times \Delta^1, so it obviously contains the class of right anondyne morphisms, but the class of right anodyne morphisms basically has to contain all of the “extra” stuff lying around, so this is identical to the class of right anondyne morphisms.

    (Note that I’m claiming this is the contravariant model structure, not the cartesian model structure)

    Regarding the cartesian fibrations thing, I’m not trying it over simplicial sets, but over a different presheaf category where things are easier to encode (either that category of relative simplices or the category Δ +\Delta^+).

    • CommentRowNumber4.
    • CommentAuthorUrs
    • CommentTimeDec 8th 2010

    Okay, I’d be quite interested in this, but can’t look into it right now. But maybe later when you have found time to add mode details.

    (Note that I’m claiming this is the contravariant model structure, not the cartesian model structure)

    Right, I changed the link to point to model structure for right fibrations. Let’s use that term. Using terms like “contravariant model structure” may work in a book such as Lurie’s where it does not collide with anything else, but here on the Lab we should use a more descriptive term.

    • CommentRowNumber5.
    • CommentAuthorHarry Gindi
    • CommentTimeDec 8th 2010
    • (edited Dec 8th 2010)

    By the way, something very interesting about Cisinski’s construction that I didn’t realize before: The “data” for his construction can be sliced in a way so they generate the model structure for right fibrations over any base, but if you slice the model structure, you get the wrong answer (for instance, slicing the model structure for right fibrations based at the terminal object, you get the relative Kan model structure, but if you slice the generator data over an object and complete to a model structure, you get the model structure for right fibrations over the slice.

    Also, the “wrong” slice is a left bousfield localization of the “correct” slice.

    • CommentRowNumber6.
    • CommentAuthorjonsterling
    • CommentTimeMay 27th 2023
    • (edited May 27th 2023)

    I noticed that this page claims that in an arbitrary presheaf topos, the terminal object has exactly two subobjects and that the subobject classifier has exactly two global points. This is obviously false (unless I am misreading?), but I wanted to understand what was intended here before I fixed it…

    • CommentRowNumber7.
    • CommentAuthorTodd_Trimble
    • CommentTimeMay 27th 2023

    Doesn’t look like you’re misreading. I’m not sure what was intended. It was written a long time ago, but I think the author Harry Gindi is still active as a mathematician (Edinburgh, last I heard).

    • CommentRowNumber8.
    • CommentAuthorHurkyl
    • CommentTimeMay 27th 2023

    Since the original author was thinking about simplicial sets, maybe they had in mind a two-valued topos.

    Taking the bottom and top as endpoints is the correct thing to define the Lawvere interval; it’s just the uniqueness statement that was in error.

    • CommentRowNumber9.
    • CommentAuthorjonsterling
    • CommentTimeMay 29th 2023

    Thanks! I’ve fixed it now.