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Assume given a pullback square of simplicial categories

$$\begin{array}[c]{ccc} A&{\rightarrow}&B\\ \downarrow&&\downarrow\\ C&{\rightarrow}&D. \end{array}$$

Suppose further that one of the induced arrows $Ho (B) \to Ho(D)$ or $Ho(C) \to Ho(D)$ is an isofibration, and for each couple of objects $x,y \in A$, the induced pullback square of simplicial mapping spaces (I abuse the notation by writing $x,y$ instead of their images in $B,C,D$) $$\begin{array}[c]{ccc} A(x,y)&{\rightarrow}&B(x,y)\\ \downarrow&&\downarrow\\ C(x,y)&{\rightarrow}&D(x,y) \end{array}$$

is a homotopy pullback.

Does this imply that the original square is in fact a homotopy pullback of simplicial categories?

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Yes. In fact such a square can be replaced with a weakly equivalent Reedy fibrant pullback square without changing the object set of any of the simplicial categories. For a proof see, e.g., Lemma 3.1.11 in this paper.

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  • $\begingroup$ I assume in the case of simplicial categories, the mentioned lemma works with only one arrow being a Ho-isofibration? $\endgroup$
    – Edouard
    Dec 3 '17 at 22:01
  • $\begingroup$ @Edouard, Yes, that should be the case. $\endgroup$ Dec 5 '17 at 16:20

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