Collections of morphisms

Definition 1.5.8. Let \(C\) be an \(\infty \)-category. A collection of morphisms in \(C\) is a monomorphism \(M \hookrightarrow \Map ([1],C)\). We say it is closed under composition if the following two conditions are satisfied:

• For a morphism \(f\colon x \to y\) in \(C\), if \(f \in M\), then also \(\id _x \in M\) and \(\id _y \in M\);
• For morphisms \(f\colon x \to y\) and \(g\colon y \to z\) in \(C\), if \(g,f \in M\), then also \(g \circ f \in M\).