@JonBeardsley actually, Emily's code is already good enough for me, I don't need those packages she was having problems with. Thanks! I wasn't successful installing mbboard...
Let me say more about the question that @NatStapleton , Toby and myself have: 1) Is the Barwick-Kan Model Structure Right proper? 2) Is there a nice description for path objects in Barwick-Kan Model Structure
The Barwick-Kan model structure is probably not right proper. At it is transfered from the Rezk model structure on simplicial spaces, it would be if the Rezk model structure was right proper. But according to mathoverflow.net/questions/40938/… it is not.
Depends on what you mean by path object. If you demand for you factorization X -> PX -> X x X only that the first is a weak equivalence, then you are fine. If you want that the second is a fibration, then you are screwed in general.
I think, it does not even work for C = \hat{1}. (have just discussed this with Viktoriya)
and this is actually fibrant in the Barwick-Kan model structure.
You might have better chances if you look at C^{\xi(\hat{1}}, but this is also not clear. The N_xi functor has quite bad properties with respect to products and internal homs
I should mention that there is a standard trick for getting around the failure of right properness: I think if you can write down what Cartesian or coCartesian fibrations are in your favorite model category C for infty-categories, then pulling back along those should always preserve categorical equivalences.
It's possible a proof of this could be patched together from things that exist in the literature.
Also, if you use quasi-categories instead of relative categories, then Lurie studies a particular example of when you can pullback categorical equivalences. These are a generalization of (co)cartesian fibrations called "flat inner fibrations"
You might be able to use Lennart's work to eke out a condition for pulling back categorical equivalences along the particular map you're interested in?
The relevant Lurie reference is B.3 of Higher Algebra... e.g. B.3.15
Dunno if any of that was relevant but there it is.
