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In Quillen's paper, he states localization, as:

Let $B$ be a Serre subcategory of $A$, let $A/B$ be the quotient abelian category, and let $e: B \to A$ and $s: A \to A/B$ denote the canonical functors. Then there is a long exact sequence $$ \cdots \to^{s*} K_1(A/B) \to K_0B \to^{e_*} K_0A \to^{s_*} K_0(A/B) \to 0$$

My question is, does this come from a homotopy fibration $QB \to QA \to Q(A/B)$? From the proof, he applies Quillen's Theorem B to the functor $QA \xrightarrow{Qs} A/B $, so, if I understand correctly, there is a homotopy fibration. But then why doesn't he also include it in the theorem, because surely its a stronger statement?

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    $\begingroup$ I'm not sure why Quillen does not include it in his paper but today localization sequences are commonly described as fiber sequences of spectra. See for example theorem 1.8.2 in Thomason-Trobaugh. $\endgroup$ May 1, 2017 at 1:17
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    $\begingroup$ Quillen was an excellent expositor and chose a formulation whose meaning was immediately clear to contemporary readers. $\endgroup$ May 1, 2017 at 7:56

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