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1. • CommentRowNumber1.
   • CommentAuthorUrs
   • CommentTimeMar 24th 2010
   • PermaLink
   Author: Urs
   Format: Markdown* renamed [[model structure on marked simplicial oversets]] to [[model structure for Cartesian fibrations]] * merged the material that was at [[marked simplicial set]] (now [[marked simplicial set > history]]) into this entry * expanded the entry a bit, but still working on it...

   • renamed model structure on marked simplicial oversets to model structure for Cartesian fibrations

   • merged the material that was at marked simplicial set (now marked simplicial set > history) into this entry

   • expanded the entry a bit, but still working on it...

2. • CommentRowNumber2.
   • CommentAuthorUrs
   • CommentTimeMar 26th 2010
   • PermaLink
   Author: Urs
   Format: Markdownfurther expanded the list of central propositions at [[model structure for Cartesian fibrations]]

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

3. • CommentRowNumber3.
   • CommentAuthorUrs
   • CommentTimeMar 30th 2010
   • (edited Mar 30th 2010)
   • PermaLink
   Author: Urs
   Format: MarkdownI 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 <latex>\Delta^+</latex> that is like the simplex category, but with one more object <latex> [1^+]</latex> added, that will parameterize the marked edges. Then there is the standard cartesian closed structure on presheaves on <latex> \Delta^+ </latex> 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.

   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 that is like the simplex category, but with one more object added, that will parameterize the marked edges. Then there is the standard cartesian closed structure on presheaves on 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.

4. • CommentRowNumber4.
   • CommentAuthorUrs
   • CommentTimeApr 7th 2010
   • PermaLink
   Author: Urs
   Format: Markdownexpanded the section on <a href="http://ncatlab.org/nlab/show/model+structure+for+Cartesian+fibrations#MarkedAnodyne">Marked anodyne morphisms</a>

   expanded the section on Marked anodyne morphisms

5. • CommentRowNumber5.
   • CommentAuthorTim Campion
   • CommentTimeFeb 28th 2020
   • PermaLink
   Author: Tim Campion
   Format: MarkdownItexLurie 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 $\Delta[1] \xrightarrow {d_1} \Delta[2]$ into the 2-simplex is an isofibration of nerves of gaunt categories, and hence a fibration in the model structure on $sSet^+$. The inclusion $\Lambda^1[2] \to \Delta[2]$ 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. <a href="https://ncatlab.org/nlab/revision/diff/model+structure+for+Cartesian+fibrations/27">diff</a>, <a href="https://ncatlab.org/nlab/revision/model+structure+for+Cartesian+fibrations/27">v27</a>, <a href="https://ncatlab.org/nlab/show/model+structure+for+Cartesian+fibrations">current</a>

   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 $\Delta[1] \xrightarrow {d_1} \Delta[2]$ into the 2-simplex is an isofibration of nerves of gaunt categories, and hence a fibration in the model structure on $sSet^+$. The inclusion $\Lambda^1[2] \to \Delta[2]$ 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

6. • CommentRowNumber6.
   • CommentAuthorHurkyl
   • CommentTimeJan 14th 2021
   • PermaLink
   Author: Hurkyl
   Format: MarkdownItexNoted that $Map^\flat(X,Y)$ is a full simplicial subset of the simplicial internal hom spanned by the mark-preserving maps. <a href="https://ncatlab.org/nlab/revision/diff/model+structure+for+Cartesian+fibrations/28">diff</a>, <a href="https://ncatlab.org/nlab/revision/model+structure+for+Cartesian+fibrations/28">v28</a>, <a href="https://ncatlab.org/nlab/show/model+structure+for+Cartesian+fibrations">current</a>

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

   diff, v28, current