Let $\dots\to X_n\to X_{n+1}\to \dots$ and $\dots\to Y_n\to Y_{n+1}\to \dots$ be a sequences of (fibrant) simplicial sets and $f_n:X_n\to Y_n$ are fibrations that commute with maps in sequences. Let $X=\mathrm{hocolim} X_n$ and $Y=\mathrm{hocolim} Y_n$ be a corresponding homotopy colimits and $f: X\to Y$ be an induced map. Is it true that homotopy fiber of $f$ is a homotopy colimit of fibers of $f_n$ ? (so, does homotopy colimits commute with homotopy fibers). All that I know, is that in case when $X_n$ and $Y_n$ are simplicial abelian groups, we can write down a condition of $f$ being a fibration as a condition for a certain square to be a pushout square. So at least $f$ should be a fibration in this case. Not sure about the fiber though


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