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I am interested in comparison of homotopy pullback squares in the category of simplicial sets with respect to Joyal' model structure and Quillen's one.

Suppose we are given a (strict) pullback square

$\begin{array}{ccc}W&\to&X\\ \downarrow && \downarrow\rlap{p}\\ S&\to&T\end{array}$

of quasi-categories with $p$ a categorical fibration; i.e. a fibration in the Joyal model structure, so it is a homotopy pullback in the Joyal model structure.

Question

Are there any criteria for the square to be a homotopy pullback also in Quillen model structure?

I also appreciate criteria with additional assumption on $p$; e.g. being a left / right fibration.

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    $\begingroup$ Left fibration and right fibration are insufficient. In fact the only right or left fibrations satisfying this condition (for all square of quasi categories) are the Kan fibrations. Though there are some $p$ that satisfies this without being a Kan fibrations. (for example, unless I making a mistake, all Joyal fibrations to the terminal simplicial sets...). I'm not sure what it the appropriate condition. $\endgroup$ – Simon Henry Sep 27 '18 at 12:31
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    $\begingroup$ Just my opinion, but I think the question title is too broad. Would be better if it said something about homotopy pullbacks in the Joyal vs Quillen model structures on sSet. $\endgroup$ – David White Sep 27 '18 at 14:23
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    $\begingroup$ Quillen's theorem B gives sufficient conditions for this to hold. One can extend it to infinity-categories. See for instance Prop. 4.6.11 in my book Higher Categories and Homotopical Algebra (available here: mathematik.uni-regensburg.de/cisinski/publikationen.html). Prop. 4.6.2 in loc. cit. might be relevant as well. $\endgroup$ – Denis-Charles Cisinski Sep 27 '18 at 14:46
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    $\begingroup$ This is a pervasive question in homotopy theory. I would really like to see a comprehensive survey of the different sufficient conditions available. One source I like is Rezk's note. $\endgroup$ – Tim Campion Sep 27 '18 at 22:10
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    $\begingroup$ @Jun Yoshida: asserting that (ii) implies (iii) in 4.6.11 precisely means that 4.6.12 holds for quasi-categories. $\endgroup$ – Denis-Charles Cisinski Sep 29 '18 at 13:39

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