Holmstrom Cofiber sequence

Let $C$ be a Pointed model category. A cofiber sequence in $Ho \ C$ is a diagram $X \to Y \to Z$ in $Ho \ C$ together with a right coaction of $\Sigma X$ on $Z$ that is IMic in $Ho \ C$ to a diagram of the form $A \displaystyle_{\longrightarrow}^{\ f} B \displaystyle_{\longrightarrow}^{\ g} C$ where $f$ is a cofibration of cofibrant objects in $C$ with cofiber $g$ and where $C$ has the standard right coaction by $\Sigma A$.

To a cofiber sequence one can associate a boundary map $\partial: Z \to \Sigma X$. See Hovey p. 156.

Some further properties: Cofiber sequences are replete in $Ho \ C$. The sequence $* \to X \to X$ is a cofiber sequence. Any map fits as the first map in a cofiber sequence, and as the last in a fiber sequence. If $[X, Y, Z]$ is a cofiber sequence, then $[Y, Z, \Sigma X]$ is too (must specify coaction here). Applying this last statement again and again gives a long exact sequence, called the Puppe sequence. If the first two maps are given between two cofiber sequences, can find a third map making everything commute. Also another fill-in statement. Verdier’s octahedral axiom holds for a cofiber sequence. A left Quillen functor preserves cofiber sequences, and a right Quillen functor preserves fiber sequences. Both kind of sequences are preserved by the closed $Ho \ Sset_*$-module structure induced by the framing (i.e. “smashing” and “RHom”).

The above properties are summarized in Hovey’s def of pre-triangulated category, see page 170.

nLab page on Cofiber sequence