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    • 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 27th 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

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