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.
    • CommentAuthorUrs
    • CommentTimeMar 24th 2010
    • CommentRowNumber2.
    • CommentAuthorUrs
    • CommentTimeMar 26th 2010

    further expanded the list of central propositions at model structure for Cartesian fibrations

    • CommentRowNumber3.
    • CommentAuthorUrs
    • CommentTimeMar 30th 2010
    • (edited Mar 30th 2010)

    I expanded on the discussion of how marked simplicial sets are cartesian closed. This is now a new section called "Cartesian closure".

    In particular I write out what I think is a detailed proof how the Cartesian closure works. The strategy chosen is to use presheaves on a category \Delta^+ that is like the simplex category, but with one more object  [1^+] added, that will parameterize the marked edges. Then there is the standard cartesian closed structure on presheaves on  \Delta^+ and one checks that this restricts to one on the full subcategory which is marked simplicial sets.

    Possibly overkill, but I struggled a bit to find a really clean argument.

    • CommentRowNumber4.
    • CommentAuthorUrs
    • CommentTimeApr 7th 2010

    expanded the section on Marked anodyne morphisms

    • CommentRowNumber5.
    • CommentAuthorTim Campion
    • CommentTimeFeb 28th 2020

    Lurie only claims that the covariant / contravariant model structures are left proper. In fact, they are not right proper, even over a point: the usual counterexample to right properness of the Joyal model structure works here too. Namely, the inclusion of the 1-face Δ[1]d 1Δ[2]\Delta[1] \xrightarrow {d_1} \Delta[2] into the 2-simplex is an isofibration of nerves of gaunt categories, and hence a fibration in the model structure on sSet +sSet^+. The inclusion Λ 1[2]Δ[2]\Lambda^1[2] \to \Delta[2] of the 1-horn into the 2-simplex is a weak equivalence. But the pullback of the latter along the former is not a weak equivalence.

    diff, v27, current

    • CommentRowNumber6.
    • CommentAuthorHurkyl
    • CommentTimeJan 14th 2021

    Noted that Map (X,Y)Map^\flat(X,Y) is a full simplicial subset of the simplicial internal hom spanned by the mark-preserving maps.

    diff, v28, current