Fiber and cofiber sequences

Definition 4.1.5 (Fiber and cofiber sequences). Let \(C\) be a pointed \(\infty \)-category. A nullsequence in \(C\) is a sequence of morphisms \[ X \xrightarrow {f} Y \xrightarrow {g} Z \] equipped with a specified nullhomotopy \(g \circ f \cong 0\). Equivalently, a nullsequence is a commutative square in \(C\) of the form

If this square is a pullback square in \(C\), we say it is a fiber sequence, and refer to \(X\) as the fiber of \(g\), written \(X \simeq \fib (g)\). If the square is a pushout square, we say it is a cofiber sequence, and refer to \(Z\) as the cofiber of \(f\), written \(Z \simeq \cofib (f)\).

We say that \(C\) admits fibers if such a fiber sequence exists for every morphism \(g\colon Y \to Z\). Dually, we say that \(C\) admits cofibers if such a cofiber sequence exists for every morphism \(f\colon X \to Y\).