Fiber and cofiber sequences of pointed animae

Definition 2.4.4 (Fiber and cofiber sequences in \(\An _*\)). A nullsequence in \(\An _*\) is a sequence of morphisms \[ X \xrightarrow {f} Y \xrightarrow {g} Z \] equipped with a specified nullhomotopy \(g \circ f \cong 0\). Equivalently, it is a commutative square in \(\An _*\) of the form

Such a nullsequence is called:

• a fiber sequence if this square is a pullback square in \(\An _*\) (or, equivalently, in \(\An \)). In this case we call \(X\) the fiber of \(g\) and write \(X \simeq \fib (g)\);
• a cofiber sequence if the same square is a pushout square in \(\An _*\) (or, equivalently, in \(\An \)). In this case we call \(Z\) the cofiber of \(f\) and write \(Z \simeq \cofib (f)\).