In their text *Foundations of Stable Homotopy Theory*, Barnes and Roitzheim define the suspension of a cofibrant object X of a pointed model category to be the pushout of the diagram $\*\leftarrow X\coprod X\to Cyl(X)$, where the second map is the structure map of the cylinder object. By contrast, there is a more manifestly homotopy-invariant definition of suspension given in e.g. [Dwyer and Spalinski](http://folk.uio.no/paularne/SUPh05/DS.pdf), which is the homotopy pushout of the diagram $\*\leftarrow X\to \*$. It is not clear to me why these definitions agree; if we don't assume properness, I don't even see why the first is homotopy-invariant! (If we assume the model category is proper, then the first diagram's pushout is equal to its homotopy pushout.) There is a similar issue with the cofiber, which they define for a cofibration of cofibrant objects $f:A\to X$ as the pushout of $\*\leftarrow A\to X$: again, it is not clear why this is homotopy-invariant (with respect to maps between such $f$ in the comma category) unless the model category is proper. Can we drop the properness assumption and still get homotopy colimits or at least homotopy invariance? Even if so, why are the definitions of suspension equivalent?



Why does this construction give a (homotopy-invariant) suspension (resp. homotopy cofiber) in an arbitrary pointed model category?

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> if we don't assume properness, I don't even see why the first is homotopy-invariant!
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The pushout of a diagram A←B→C in which all objects are cofibrant and one of the maps is a cofibration
is always its homotopy pushout in any model category,
see Proposition A.2.4.4 in Lurie's Higher Topos Theory.


This is the case for both of your examples, since the initial object is cofibrant.


