Cofiber of spectra ($\mathrm{cofib}(u)$)

The homotopy cofiber of spectra of $u:A\to B$ is the derived pushout $B{\cup }_{A}CA$. It fits into a cofiber sequence of spectra.

A cofiber sequence of spectra $A\to B\to C\to \Sigma A$ induces exact sequences after either mapping into or out of a fixed topological spectrum, and on stable homotopy groups. In particular the second map is zero precisely when the first map admits a right inverse in the stable homotopy category.