Functors inverting morphisms

Definition 1.5.17. 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 \(F\colon C \to D\) inverts the morphisms in \(W\) if for every morphism \(f\colon x \to y\) in \(C\) contained in \(W \subseteq \Map ([1],C)\), the image \(F(f)\colon F(x) \to F(y)\) is invertible in \(D\). We let \(\Fun ^{W}(C,D)\) be the full subcategory of \(\Fun (C,D)\) spanned by those functors that invert the morphisms in \(W\).