Julie Bergner defined the homotopy pullback of a diagram of model categories, in Homotopy fiber products of homotopy theories, and Homotopy limits of model categories and more general homotopy theories. In the case of homotopy pullbacks, the construction appeared a little bit earlier in Bertrand Toen's Derived Hall Algebras. Given a diagram of model categories and left Quillen functors

\begin{align} M_1 \stackrel{F_1}{\to} M_3 \stackrel{F_2}{\gets} M_2 \end{align}

Bergner defines the homotopy pullback to be a category $M$ whose objects are 5-tuples $(x_1,x_2,x_3,u,v)$ with $x_i \in M_i$ and $F(x_1) \stackrel{u}{\to} x_3 \stackrel{v}{\gets} F(x_2)$. Morphisms are the obvious thing. This category $M$ can be given a model structure where the weak equivalences and cofibrations are levelwise (on each $x_i$), and that model structure can be localized if desired to force $u$ and $v$ to be weak equivalences in the local objects of $M$. Bergner then proves $M$ has the correct homotopy type, meaning that, upon passage to complete Segal spaces, it becomes the actual homotopy pullback of the diagram. It appears that there is nothing special about $F_i$ being left Quillen, and you could make an analogous construction of the homotopy pullback of a diagram of right Quillen functors.

Here's my question:

(1) In a homotopy pullback diagram of right Quillen functors, if $F_2$ is a right Quillen equivalence, is the same true for the induced functor from the homotopy pullback $M$ to $M_1$?

Dually, consider a homotopy pushout of left Quillen functors

$$ \begin{array}{ccc} M_3 & \xrightarrow{F_1} & M_1 \newline \downarrow & & \downarrow \newline M_2 & \xrightarrow[F_1']{} & P \end{array} $$

$P$ is defined in Homotopy Colimits of Model Categories, where objects are 5-tuples with $F_1(x_3) \to x_1$ in $M_1$ and $F_2(x_3)\to x_2$ in $M_2$, with weak equivalences objectwise. Then:

(2) If $F_1$ is a left Quillen equivalence, will $F_1'$ give an equivalence of homotopy theories?

  • $\begingroup$ If you have a homotopy pushout/pullback along a weak equivalence in relative categories, it is a weak equivalence again. Quillen equivalences should define weak equivalences of the underlying relative categories; by Barwick-Kan 'A characterization of simplicial localization functors and a discussion of DK equivalences' we just have to show it induces an equivalences on hammock localizations (i.e. on mapping spaces), which should be classical. As the Rezk classifying diagram from relative categories to complete Segal spaces is an equivalence of homotopy theories, it detects homotopy.... $\endgroup$ – Lennart Meier Apr 27 '18 at 8:58
  • $\begingroup$ ...pushouts and pullbacks. Bergner and the more general staff.science.uu.nl/~meier007/HomotopyColimits2.pdf thus show these constructions to be homotopy pushouts/pullbacks in relative categories. Btw: Bergner does not really give a general construction of the localized model structure on $M$; do you have one? $\endgroup$ – Lennart Meier Apr 27 '18 at 9:07
  • $\begingroup$ @LennartMeier, thanks for your insightful response. I proceeded along the same lines, but got stuck at the part you said "should be classical." Anyway, I learned today that, even if this is true, it doesn't help, so I'm going to stop thinking about it. I still believe this MO question is interesting, and I hope it gets a full answer at some point. As for Bergner and localization, Thm 3.1 in her "homotopy fiber products" paper seems best-possible to me. Combinatorial is probably necessary. $\endgroup$ – David White Apr 30 '18 at 9:38

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