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I believe it is well-known that for a based map $f:X\to Y$ of simplicial sets (possibly with some extra hypotheses on $X$ and $Y$), there is a long exact sequence $$ \ldots \to \pi_n(F)\to \pi_n(X) \to \pi_n(Y) \to \pi_{n-1}(F) \to \ldots, $$ where $F$ is the homotopy fibre of $f$. However, I cannot actually find a precise statement of this result. The closest I've come is this nLab page, which states essentially the same result for $\infty$-groupoids. I know that the classical model structure on simplicial sets is a presentation for the $(\infty,1)$-category of $\infty$-groupoids, so I suspect that maybe the result holds specifically for fibrant-cofibrant simplicial sets, but I'm not sure.

Ideally, I would like a reference that states the result explicitly for simplicial sets. If no such reference exists, I'd also be grateful for an explanation of the result.

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  • $\begingroup$ See Lemma I.7.3 in Goerss-Jardine. $\endgroup$
    – JHF
    Commented Feb 22, 2021 at 15:03
  • $\begingroup$ @JHF that lemma refers to a strict fibre, rather than a homotopy fibre though. Is there some reason why it amounts to the same thing? $\endgroup$ Commented Feb 22, 2021 at 15:10
  • $\begingroup$ To compute the homotopy fibre, take a fibrant replacement of your map and compute the strict fibre. The replacement is weak homotopy equivalent, so its homotopy groups are (functorially) isomorphic. $\endgroup$
    – JHF
    Commented Feb 22, 2021 at 15:23
  • $\begingroup$ @JHF, thank you, I see now! $\endgroup$ Commented Feb 22, 2021 at 16:40

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