Class of fibrations

Definition 2.2.3 (Class of fibrations). Let \(C\) be an \(\infty \)-category with a terminal object \(*\). A collection of morphisms \(F \subseteq \Map ([1],C)\) is called a class of fibrations if the following conditions are satisfied:

(1)

It contains all the isomorphisms;

(2)

It is closed under composition;

(3)

It is closed under base change: for all morphisms \(f\colon X \to Y\) in \(F\) and \(v \colon Y' \to Y\) in \(C\), the pullback \(Y' \times _Y X\) exists, and the projection \(Y' \times _Y X \to Y'\) is again in \(F\).

A morphism in \(F\) is called a fibration. An object \(X\) in \(C\) is called fibrant if the unique map \(X \to *\) is a fibration. Dually, a collection of morphisms \(I\) is called a class of cofibrations if the corresponding class \(I\catop \) in \(C\catop \) defines a class of fibrations in \(C\catop \). A morphism in \(I\) is a cofibration, and an object \(X\) is cofibrant if \(\emptyset \to X\) is a cofibration.