Proposition 6.6.3 (Cisinski (2019), 7.5.25). Let \((C,W,I)\) be an \(\infty \)-category with weak equivalences and cofibrations, let \(i\colon C_c \hookrightarrow C\) denote the full subcategory of cofibrant objects, and let \(F\colon C \to D\) be a functor such that the restriction \(F\vert _{C_c}\colon C_c \to D\) inverts the weak equivalences between cofibrant objects. Then \(F\) admits a total left derived functor \[ \bL F\colon C[W^{-1}] \to D. \] It is uniquely characterized by the existence of an isomorphism \(\bL F \circ \gamma \circ i \simeq F \circ i\). In particular, if \(q\colon Q \to X\) is any weak equivalence with \(Q\) cofibrant, then \[ \bL F(\gamma X) \simeq F(Q). \] Moreover, the structure transformation \(\bL F\circ \gamma \Rightarrow F\) exhibits \(\bL F\) as an absolute right Kan extension: for every functor \(H\colon D\to E\), the composite \(H\circ \bL F\) is a total left derived functor of \(H\circ F\).
Proof. We give the construction of \(\bL F\), and refer to [Cisinski (2019), 7.5.25] for the verification that it has the required universal property. (The reference treats the dual situation of an \(\infty \)-category with weak equivalences and fibrations, and constructs the total right derived functor of a functor which inverts the weak equivalences between fibrant objects.)
Write \(W_c\) for the weak equivalences between cofibrant objects. The restriction \(F\vert _{C_c}\) descends to a functor \(F_c\colon C_c[W_c^{-1}] \to D\). By part (1) of Corollary 2.2.9, the inclusion induces an equivalence \(\overline i\colon C_c[W_c^{-1}] \xrightarrow {\simeq } C[W^{-1}]\), and we may thus define \[ \bL F \quad := \quad F_c \circ \overline i^{\,-1}\colon \; C[W^{-1}] \to D \] by choosing an inverse of \(\overline i\). By construction, the restriction of \(\bL F\) to the cofibrant objects is isomorphic to \(F\vert _{C_c}\), that is, \(\bL F \circ \gamma \circ i \simeq F \circ i\). Since \(\overline i\) is an equivalence, a functor out of \(C[W^{-1}]\) is determined by its restriction along \(\gamma \circ i\), which gives the asserted uniqueness. Finally, if \(q\colon Q \to X\) is a weak equivalence with \(Q\) cofibrant, then \(\gamma (q)\) is an isomorphism, and hence \[ \bL F(\gamma X) \simeq \bL F(\gamma Q) \simeq F(Q). \] The construction of \(\bL F\) is also compatible with postcomposition: for every \(H\colon D\to E\), the composite \(H\circ \bL F\) is obtained by applying the same construction to \(H\circ F\). Its structure transformation is therefore again a right Kan extension by [Cisinski (2019), (7.5.25.3)], proving absoluteness. □
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