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 k-theory lie-theory limit 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 subobject 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
    • CommentTimeJun 18th 2010
    • (edited Jun 18th 2010)

    It is a theorem of Joyal that the class of all anodyne maps in the Joyal model structure is generated by (equivalently):

    Inner horn inclusions


    Δ n×Λ 1 2Δ n×Δ 2Δ 2×Δ n\Delta^n\times \Lambda^2_1 \cup \partial\Delta^n \times \Delta^2\to \Delta^2 \times \Delta^n


    S×Λ 1 2S×Δ 2Δ 2×SS' \times \Lambda^2_1 \cup S \times \Delta^2\to \Delta^2\times S'

    for all inclusions SSS\hookrightarrow S'.

    Now, if we use Cisinski’s formalism of classes of anodyne maps, the equivalence of the second two become obvious (by paragraph 1.3.12 and remark 1.3.15) if we can figure out what the cylinder functor for the Joyal model structure is. Now, for sure we know that the actual cylinder functor is taking the product with Δ 2\Delta^2, but we need slightly more information regarding the actual structure of the cylinder. Obviously, the retraction of the two maps XΔ 2×XX\to \Delta^2\times X will be the projection on the second factor, but I can’t figure out for the life of me what the two maps XΔ 2×XX\to \Delta^2\times X actually are. It seems to be related to the inclusion of the inner 2-horn, but I can’t get any further.

    Maybe you guys can figure out what the situation is? I’m inclined to believe that the maps are the inclusions of the legs of the inner 2-horn, where the inner 2-horn takes the role of XX=I×XX\coprod X = \partial I \times X.

    • CommentRowNumber2.
    • CommentAuthorMike Shulman
    • CommentTimeJun 19th 2010

    Do you mean inner anodyne maps? I didn’t think the class of all acyclic cofibrations in the Joyal model structure had any explicit generating set.

    • CommentRowNumber3.
    • CommentAuthorHarry Gindi
    • CommentTimeJun 19th 2010

    I meant what I said, although Cisinski’s terminology conflicts with the terminology of Lurie and Joyal. A class of anodyne maps for a cylinder means something rather specific.

    What I was asking for was finding a cylinder functor generating the class of weak cofibrations in the Joyal model structure (i.e. inner anodyne maps) by means of the empty generating set for the anodyne structure for that cylinder object. I don’t have time at the moment, but I will translate all of the relevant material if you don’t feel like reading it yourself.