It is true that in the category of spaces there exists a characterization of homotopy pullbacks in terms of homotopy fibers ([Proposition 4.1](https://ncatlab.org/nlab/show/homotopy+pullback)).


I want to know a category (or $\infty$-category) where I can find a square diagram where there is an equivalence in all the homotopy fibers but the diagram is not an homotopy pullback. 



Homotopy pullbacks and fibers
For the record, the statement in spaces is this. A diagram
$$
\array{
 A &\stackrel{f}{\longrightarrow}& B
 \\
 \downarrow && \downarrow^{p}
 \\
 C &\stackrel{g}{\longrightarrow}& D
 }
$$
is a homotopy pullback if and only if, for every point $b: \ast \to B$, the map of homotopy fibers $hofib\_b(f) \to hofib\_{p(b)}(g)$ is an equivalence.


We could try a direct translation of this into a general $\infty$-category. Suppose that we have an $\infty$-category that has (homotopy) pullbacks and a (homotopy) final object $\ast$. Then, given a homotopy pullback diagram as above, for any map $b: \ast \to B$, the map of homotopy fibers $A \times\_B \ast \to C \times\_{D} \ast$ is an equivalence. Does this have a converse?


Here is an example where this fails: the *opposite* category of the category of spaces. In the opposite category, homotopy pullbacks translate into homotopy pushouts and the final object translates into the initial object $\emptyset$. In these terms, the question translates into the following: is a square diagram as above a pushout diagram if and only if, for every map $\epsilon: C \to \emptyset$, the map $B \coprod\_A \emptyset \to D \coprod\_C \emptyset$ is an equivalence? This criterion does not work, because there are no maps $C \to \emptyset$ unless $C$ itself is empty -- every diagram where $C$ is nonempty satisfies this criterion.


(Another example that works is the category of *pointed* spaces, because the map $\ast \to B$ is forced.)




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I believe that the real key behind the original criterion is that it is not about pullback diagrams. It is about the fact that $\ast$ generates the ($\infty$-)category of spaces: all spaces are built from $\ast$ under homotopy colimits. The same is not true of $\emptyset$ in the opposite category, or $\ast$ in the category of pointed spaces; it *is* true of $S^0$ in the category of pointed spaces. You can detect whether a square diagram is a pullback by testing it against maps from a collection of generators.


