Left fibrations

Definition 23.1.17. A functor \(p\colon E \to C\) is called a left fibration if it is a cocartesian fibration satisfying the equivalent conditions (1), (2) and (3) from the proposition. This gives rise to a full subcategory \[ \LFib (C) \subseteq \Cocart (C). \] Note that \(\LFib (C)\) is also a full subcategory of \((\Cat _{\infty })_{/C}\) (in contrast to \(\Cocart (C)\)).

Dually, \(p\) is called a right fibration if it is a cartesian fibration which satisfies the equivalent conditions (1), (2) and (3’): Every morphism in \(E\) is cartesian. We write \(\RFib (C) \subseteq (\Cat _{\infty })_{/C}\) for the resulting full subcategory.