# Techniques for computing homotopy pullbacks

I am in the context of simplicial sets. I have a square diagram that I want to show to be homotopy pullback. What I can grasp is that it is a pullback up to some equivalences in the middle of my reasoning, and I would like to promote this to a real statement.

So let $$p:X \to Z, q:Y \to Z$$ map of ssets, and $$f:P \to X, g:P\to Y$$ other two maps from the supposed-to-be homotopy pullback. Suppose that for every $$x \in X$$, the induced map on homotopy fibers $$P \to Y$$ is an equivalence. Furthermore, you can additionally suppose that $$p,q$$ are Kan fibrations. Is it true that $$P$$ is the homotopy pullback of the diagram?

Th other point is the following: when the homotopy divers are equivalent to the fibers? Maybe when maximal connected groupoids at the base are contractible?

Related question: Can homotopy pullbacks of spaces be checked on fibers?

The difference is that we are not dealing with spaces (=Kan complexes) but with general simplicial sets. Moreover, I'd like to use actual and not homotopical fibers.

Proposition 3.3.18. Thanks!

• Let $X$ be any non-empty fibrant simplicial set with a single vertex and let $P = Y = Z = \Delta^0$. Then fibers of $P$ and $Y$ are always equal to $\Delta^0$, but the square is a homotopy pullback only if $X$ is contractible. – Valery Isaev Jun 14 at 1:43
• Ok, thanks. I will edit the question so that it includes homotopy fibers (seems to solve jour counterexample). – Andrea Marino Jun 14 at 9:40