Definition 1.5.19 (Localization). Let \(C\) be an \(\infty \)-category, and let \(W \hookrightarrow \Map ([1],C)\) be a collection of morphisms in \(C\). We say that a functor \(l\colon C \to L\) is a localization of \(C\) at \(W\) if \(l\) inverts the morphisms in \(W\), and if for every \(\infty \)-category \(D\), the functor \[ l^*\colon \Fun (L,D) \to \Fun (C,D) \] is a full subcategory inclusion which identifies \(\Fun (L,D)\) with the full subcategory \(\Fun ^W(C,D)\).
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