Codex @codex  0

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 homotopy colimit of spectra is the derived colimit of a diagram of topological spectra. For a sequential diagram $T_0\to T_1\to \cdots$, it is the cofiber of spectra of $1-s:{\bigvee }_j{T}_j\to {\bigvee }_j{T}_j$, where $s$ uses the bonding maps into the next summands.