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 homotopyinvariant definition of suspension given in e.g. Dwyer and Spalinski, 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 homotopyinvariant! (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 homotopyinvariant (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?

1$\begingroup$ Shouldn't the first of these diagrams be $\ast \amalg \ast \leftarrow X \amalg X \to \text{Cyl}(X)$? $\endgroup$– John KleinJun 16, 2020 at 2:46

2$\begingroup$ @JohnKlein In a category with a zero object, we have $*\coprod *\cong *$, so the diagrams are naturally isomorphic. $\endgroup$– Doron GrossmanNaplesJun 16, 2020 at 5:03

2$\begingroup$ Ah...I didn't notice that you were assuming a pointed model structure. I wrote my comment and answer without that assumption. $\endgroup$– John KleinJun 16, 2020 at 12:12

1$\begingroup$ I changed my answer to reflect that we are in the pointed case. $\endgroup$– John KleinJun 16, 2020 at 14:04
3 Answers
if we don't assume properness, I don't even see why the first is homotopyinvariant!
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.
An argument showing that the two models of suspension are equivalent will probably be based on something like the following:
Assertion: Suppose we are given a commutative diagram of the form $\require{AMScd}$ \begin{CD} \ast @<<< C @= C \\ @VVV @VVV @VV V \\ Y @<<< A @>g>> X \\ @ @VVV @VVV\\ Y @<<< A/C @>>h > X/C \end{CD} in which the vertical directions form cofibration sequences (when I write $A/C$, I mean $A \amalg_C \ast$, where $\ast$ is the zero object), and the maps $g$ and $h$ are cofibrations.
Then the map of pushouts $$ Y \cup_A X \to Y \cup_{A/C} X/C $$ is a weak equivalence, or better still, it is an isomorphism.
It seems to me that this is true by the assumption of properness, since we have a cofibration sequence given by the pushouts $$ \ast\cup_C C \to Y \cup_A X \to Y \cup_{A/C} X/C $$ in which the first term is isomorphic to $\ast$.
Let's call the first suspension $SX$ and the second one $\Sigma X$.
Given the assertion, we can show that the two models for suspension are weakly equivalent as follows:
Apply the assertion to the diagram \begin{CD} \ast @<<< \ast\amalg X @= X \\ @VVV @VVV @VVV \\ \ast @<<< X\amalg X @>g >> \text{Cyl}(X) \\ @ @VVV @VVV\\ \ast @<<< X @>>h > CX \end{CD} (where $CX = \text{Cyl}(X)/X$) to get that the map $$ SX\to \Sigma X $$ is a weak equivalence.
If you're looking to learn more about homotopy colimits, I strongly recommend:
 Dugger's Primer on Homotopy colimits
 Shulman's Homotopy limits and colimits and enriched homotopy theory
 Rehmeyer's 1997 master's thesis (under Mike Hopkins), "Homotopy Colimits"
 Homotopy Limit Functors on Model Categories and Homotopical Categories by Dwyer, Hirschhorn, Kan, Smith
 Riehl's book Categorical Homotopy Theory
I note that the first four predate Lurie's books, and the fifth works out many examples. The fact that the pushout and homotopy pushout agree for a span diagram when all objects are cofibrant and one leg is a cofibration (even without left properness) is 13.10 in Dugger's manuscript. A detailed treatment of the cofiber is in Rehmeyer's thesis. Shulman handles your other question, about why these two ways of computing the homotopy colimit agree (e.g., Section 5, drawing on Dwyer, Hirschhorn, Kan, Smith).