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Jacob Lurie defined a model structure on the category of marked simplicial sets sliced over a fixed simplicial set $S$ called the cartesian model structure. (For a definition, see here or HTT Ch.3.1). How does this model structure behave with respect to slicing over objects? That is, is the natural model structure on the slice over a simplicial set $S$ the same as the cartesian model structure on the slice over $S$?

Do any of the other flexible models of higher categories (specifically complete segal spaces and segal categories) have relative forms? If so, do these model categories give the right results under slicing? Whether they do or not, are there appropriate analogues of the straightening and unstraightening constructions? If so, do they give a powerful enough theory of fibered and cofibered categories (which are extremely important in ordinary 2-category theory (these give the "right" slice bicategories)).

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up vote 11 down vote accepted

It seems that the first question only makes sense for marked simplicial sets $X$ over $S$ where every edge of $X$ is marked (otherwise, the slice category is not equivalent to marked simplicial sets over $X$). Under this assumption, the answer is yes at least if $X$ is fibrant (so that the underlying map of simplicial sets $X \rightarrow S$ is a right fibration).

If you take any model category ${\mathbf A}$ for higher category theory and take the slice category ${\mathbf A}_{/X}$ for some fibrant object $X$, it will be a model for higher categories $Y$ with an arbitrary functor $p: Y \rightarrow X$. If you want to enforce the requirement that $p$ should be a Grothendieck fibration, you need to modify the definitions somehow. In the quasicategory model, this is what the markings are for.

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@Jacob: With respect to the first part, do you know of any counterexamples to the stronger claim (when they're not right fibrations, that is)? –  Harry Gindi Dec 2 '10 at 7:00
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If S is a point, then the Cartesian and coCartesian model structures coincide, so induce the same model structure after slicing at X. So the claim can't be true in general, since the Cartesian and coCartesian model structures on marked simplicial sets over X are generally different (they agree when X is a Kan complex). All of these model categories are closely related to the Joyal structure on simplicial sets, which is not right proper. It's generally a bad idea to slice them over non-fibrant objects. –  Jacob Lurie Dec 2 '10 at 13:43
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