Holmstrom Pointed model category

The homotopy category of a pointed MC is a closed $Ho \ Sset_*$-module. However, it also has additional structure. This is the subject of Hovey, chapter 6. In particular:

• Can define suspension functor $\Sigma : Ho \ C \to Ho \ C$
• and its right adjoint, the loop functor $\Omega$

The iterated suspension functor $\Sigma^t$ lifts to a functor to the category of (abelian) cogroups in $Ho \ C$ if $t \geq 1$ ($t \geq 2$). Same for $Omega^t$ and the category of groups. (Recall that a group object represents a contravariant functor to Grp, and a cogroup object represents a covariant such functor.)

Section 6.2: There is a natural coaction in $Ho \ C$ of the cogroup $\Sigma A$ on the cofiber of a cofibration of cofibrant objects $A \to B$. Also something about an action of the group object $\Omega B$ on some fiber. Cofiber sequences and fiber sequences.

nLab page on Pointed model category