Lemma 6.6.7 (Products of cofibration structures). Let \((C_i,W_i,I_i)_{i\in K}\) be a finite collection of \(\infty \)-categories with weak equivalences and cofibrations. Then \(\prod _{i\in K}C_i\), equipped with the componentwise weak equivalences and cofibrations, is again an \(\infty \)-category with weak equivalences and cofibrations. Its cofibrant objects are precisely the tuples of cofibrant objects. Consequently, a functor \(F\colon \prod _{i\in K}C_i\to D\) which inverts weak equivalences between cofibrant objects admits a total left derived functor \[ \bL F\colon \prod _{i\in K}C_i[W_i^{-1}]\longrightarrow D. \]
Proof. The axioms for weak equivalences and cofibrations, including the factorization and pushout axioms, hold componentwise in the product, and the description of the cofibrant objects follows from the componentwise initial object. By currying and applying the universal property of localization successively in each variable, the canonical functor identifies the localization of the product with \(\prod _{i\in K}C_i[W_i^{-1}]\). The final assertion now follows from Proposition 6.6.3. □
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