Definition 1.5.10. A collection of objects of an \(\infty \)-category \(C\) is a monomorphism \(\Gamma \hookrightarrow C^{\simeq }\). The endpoint functors on \(\Ar (C)\) restrict along the groupoid cores to functors \(s,t\colon \Map ([1],C)\to C^{\simeq }\), for which we use the same notation. Given a collection of objects \(\Gamma \), we define \(M_{\Gamma }\) as the following pullback:
The collection \(M_{\Gamma }\) is closed under composition, since identities and composites have endpoints in \(\Gamma \) whenever the original morphisms do. Hence Axiom H.2 determines a monomorphism \(\lra {M_{\Gamma }}_C \hookrightarrow C\). We refer to the subcategory \(\lra {M_{\Gamma }}_C\) of \(C\) as the full subcategory spanned by the objects in \(\Gamma \).
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