Proposition 20.1.14 ([Nikolaus and Scholze (2018), Proposition A.14]). Let \(p\colon C \to S\) be a cocartesian fibration which is left derivable with respect to classes of morphisms \(\{W_s\}_{s \in S}\). Let \(W\) denote the collection of morphisms in \(C\) given by the union of all \(W_s\), and let \(q\colon C[W^{-1}] \to S\) denote the unique factorization of \(p\) through the localization functor \(\gamma \colon C \to C[W^{-1}]\). Then the following statements hold true:
- (1)
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The functor \(q\) is a cocartesian fibration;
- (2)
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For every object \(s \in S\), the map on fibers \[ \gamma _s\colon C_s \to C[W^{-1}]_s \] induces an equivalence \(C_s[W_s^{-1}] \iso C[W^{-1}]_s\);
- (3)
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Consider a morphism \(f\colon s \to t\) in \(S\). Under the equivalences from part (2), the transport functor \[ C[W^{-1}]_s \to C[W^{-1}]_t \] identifies with the total derived functor \(\bL f_!\) of \(f_!\colon C_s \to C_t\).
Proof sketch. We refer to [Nikolaus and Scholze (2018), Proposition A.14] for the proof and only indicate its structure. By Example 20.1.13, one may first restrict to the bases \([0]\) and \([1]\). The result is immediate over \([0]\). Over \([1]\), the cocartesian fibration is classified by a functor \(F\colon C_0\to C_1\). Its absolute left derived functor unstraightens to a cocartesian fibration with the desired fibers. The substantive point is that absoluteness identifies the mapping simplex of this derived functor with the localization of the mapping simplex of \(F\), by the universal property of right Kan extension and the description in Example 23.2.6.
Finally, the total category of a cocartesian fibration commutes with colimits in the base, and localization commutes with colimits because it is a left adjoint on relative \(\infty \)-categories. The class of bases for which the proposition holds is therefore closed under colimits. Since \(\Cat _{\infty }\) is generated under colimits by \([0]\) and \([1]\) by Proposition 24.1.14, this proves the result for arbitrary \(S\). □
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