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.
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asked Dec 5, 2011 at 8:36
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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.
answered Dec 5, 2011 at 15:57
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$\begingroup$ Thanks! Two questions. (1) By "fibrations $f\colon X\to W$", do you mean Reedy fibrations? Or fibrations in the CSS model structure? (2) Is the "evident map" necessarily itself a fibration (say, if $f$ is a fibration according to your answer to (1))? $\endgroup$ Commented Dec 6, 2011 at 0:45
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$\begingroup$ Also, (3) I presume that this model structure is constructed as a localization of some over-model-structure for simplicial spaces over W? And (4, I guess) does that imply that the weak equivalences between fibrant objects in this model structure are still just the levelwise equivalences? $\endgroup$ Commented Dec 6, 2011 at 0:47
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$\begingroup$ (1) I mean Reedy fibration. It turns out that if W is a CSS, and f is a Reedy fibration with the above property, then X is also a CSS, and thus f (being a Reedy fibration between CSS's) is also a fibration in the CSS model category. (2) I believe it is a fibration if $\delta$ is injective, but not generally. (In the condition describing fibrant objects, it actually suffices to require a weak equivalence only in the case $\delta:[0]\to [p]$.) (3) Yup. (4) Yup. $\endgroup$ Commented Dec 6, 2011 at 16:22
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$\begingroup$ Beautiful! That's all I could have hoped for. Someone should really work all of this out and write it up. $\endgroup$ Commented Dec 7, 2011 at 0:25