Let $C$ be a stable $\infty$-category (presentable, if you like) and let $map(-,-)$ denote the simplicial mapping space. If $X \to Y \to Z$ is a fiber sequence, and $W$ is an object, when is $map(W,X) \to map(W,Y) \to map(W,Z)$ a fiber sequence?


I suspect that this does not come for free. I'm more willing to believe that an internal hom object would have this property. I'm sure this is somewhere in Higher Algebra (probably in the first 300 pages), but I can't find it.



Simplicial mapping spaces, stable $\infty$-categories, and triangles
This is always true, even without the hypothesis of stability. In an ∞-category a fiber sequence $X\to Y\to Z$ is a pullback square
$$\require{AMScd}
\begin{CD}
X @>>> Y\\
@VVV @VVV \\
\* @>>> Z
\end{CD}\,.$$


So you are pretty much asking whether the functor $\mathrm{Map}(W,-)$ preserves pullback squares. In fact it preserves all limits. In fact we can write it as the composition of the functors
$$ C\to P(C)\xrightarrow{ev\_W} \mathcal{S}$$
where the first arrow is the Yoneda embedding and the second is evaluation at $W$.


Proposition 5.1.3.2 of Higher topos theory says that the Yoneda embedding preserves limits, while proposition 5.1.2.3 says that for any functor category (like $P(C)=\mathrm{Fun}(C^{\mathrm{op}},\mathcal{S})$) evaluation preserves (and in fact detects) (co)limits.


