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Suppose that we have a (combinatorial if necessary) model category $M$, and let $F:\Delta^{op}\rightarrow M$ a simplicial object in $M$, such that for any natural number $n$, $F([n])$ is a fibrant object in $M$. We define a new object $X= colim_{n} F([n]) $. Is it true that $X$ is a fibrant object ?

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No, it is not.

If what you mean by $\rm colim_n$ is the actual colimit of $F$ as a diagram of shape $\Delta$, then this colimit is isomorphic to the coequalizer of the two maps $F([1]) \rightrightarrows F([0])$. Coequalizers rarely preserve fibrancy, and with a little thought we can think of a counterexample that extends to a simplicial object. Let $M$ be a model category in which coproducts of fibrant objects are fibrant (e.g. simplicial sets), and let $G:D\to M$ be any diagram of fibrant objects whose colimit is not fibrant (such as a cospan $X \leftarrow \Delta[0] \to Y$ in simplicial sets for almost any Kan complexes $X$ and $Y$). Let $F$ be the simplicial bar construction of $G$, with $F([n]) = \coprod_{d_0 \to \cdots \to d_n} G(d_0)$. Then each $F([n])$ is fibrant by our assumptions on $M$ and $G$, but the colimit of $F$ is the coequalizer of $\coprod_{d_0\to d_1} G(d_0) \rightrightarrows \coprod_d G(d)$, which is just the colimit of $G$ (it is the usual computation of colimits in terms of coproducts and coequalizers).

If instead what you mean by $\rm colim_n$ is actually the geometric realization, then there is an even easier counterexample. Let $M$ be simplicial sets, let $X$ be a simplicial set that is not a Kan complex, and let $F([n]) = X_n$ regarded as a discrete simplicial set. Discrete simplicial sets are Kan complexes, so each $F([n])$ is fibrant, but the geometric realization of $F$ is just $X$ itself, which is not fibrant.

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  • $\begingroup$ I see, do you think it is still false if we impose that the two maps $F([1]) \rightrightarrows F([0])$ are fibrations ? $\endgroup$ – Paris Feb 21 at 20:36
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    $\begingroup$ @Paris Yes, probably. Colimits rarely preserve fibrant objects and fibrations. $\endgroup$ – Mike Shulman Feb 22 at 0:00

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