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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.

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    $\begingroup$ Wait what am I talking about it's way simpler: map(W,-) always takes limits to limits. What were we worried about? $\endgroup$
    – Dylan Wilson
    Commented Jun 3, 2017 at 14:04
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    $\begingroup$ This is always true since fiber sequences are defined by pullback squares and the covariant hom is a continuous functor. (whoops, didn't see Dylan's comment above.) $\endgroup$
    – skd
    Commented Jun 3, 2017 at 14:06
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    $\begingroup$ This has nothing to do with enrichment- it comes from the universal property of a limit in an infty category. I'm on my phone but I'm sure skd could write the reference from HTT. (It'll probably be near the Yoneda embedding stuff?) $\endgroup$
    – Dylan Wilson
    Commented Jun 3, 2017 at 14:10
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    $\begingroup$ (On a side note, thinking about infty categories as simplicial or topologically enriched categories is good for intuition but misleading once things get more technical, especially if you're trained in various aspects of enriched category theory. For example, a limit in an infinity category has an entirely different universal property than the usual notions of 'enriched limit' since various homeomorphisms are replaced by weak equivalences). $\endgroup$
    – Dylan Wilson
    Commented Jun 3, 2017 at 14:19
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    $\begingroup$ Hi Dylan. That last comment is very helpful. $\endgroup$
    – David White
    Commented Jun 3, 2017 at 14:28

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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.

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  • $\begingroup$ Thanks! So, I guess dually, Map(-,W) preserves colimits? So for a stable infinity category either of the two Map functions will preserve fiber sequences? $\endgroup$
    – David White
    Commented Jun 3, 2017 at 15:34
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    $\begingroup$ @DavidWhite Not quite. Map(-,W):C^{op}→S preserves limits too, that is it turns colimits in C to limits in spaces. So it will send cofiber sequences to fiber sequences. What you said for stable∞-categories is true though, since there fiber and cofiber sequences coincide. $\endgroup$
    – Denis Nardin
    Commented Jun 3, 2017 at 15:40

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