TITLE: Presheaves on a complete Segal space
QUESTION [6 upvotes]: Let C be an $(\infty,1)$-category, incarnated as a complete Segal space, hence in particular a bisimplicial set.  Is there a model structure on the slice category of bisimplicial sets over C which presents the $(\infty,1)$-presheaf category of C?  Ideally, such a model structure would be Quillen equivalent to the contravariant model structure over a quasicategory incarnation of C, and to the projective model structure for simplicial presheaves on a simplicial-category incarnation of C.

REPLY [5 votes]: Yes.  Let $W$ be a complete Segal space, thought of as a simplicial "space" $(W_q)$.  The fibrant objects of your model category will be the fibrations $f:X\to W$ such that for each simplicial operator $\delta:[q]\to [p]$ with $\delta(q)=p$, the evident map from $X_p$ to the pullback of 
$$X_q \xrightarrow{f} W_q \xleftarrow{\delta} W_p$$
is a weak equivalence of spaces.  (Edit: in fact, it suffices to require the evident map to the pullback to be a weak equivalence only for $\delta:[0]\to[p]$ with $\delta(0)=p$.)
I worked out some of this years ago, but never finished it; somebody should do this (or perhaps someone has already?).  Lurie has done pretty much exactly the same thing in the context of quasi-categories, in HTT.