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

]]>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.

expanded the section on Marked anodyne morphisms

]]>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.

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

]]>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...