I'm trying to understand how to compute pushout in Spectra.
The reason it should satisfy it is because a stable category admitting finite limits, filtered colimits and $\Sigma$ (i.e pushout of $X \to *,*$) admit pushouts, does anyone have a reference or proof? I can't seem to prove it myself :(
As a sanity check we know that infinite suspension commutes with pushouts so we should be compatible with that.
I am working in a stable $\infty$ category (so that the solution would be I guess a series of applications of $\Sigma$ on various elements and filtered colimits and then saying that by the universal property it's the pushout of the initial diagram)
Given an additive $\infty$-category $C$ with homotopy pullbacks. Suppose that we do have suspensions, ie. homotopy pushouts of spans of the form $0 \leftarrow X \rightarrow 0$. Then they give rise to a loop-suspension adjunction $\Omega \dashv \Sigma$ of endofunctors of the $\infty$-category $C$.
If this is in fact an adjoint equivalence, ie. $\Omega = \Sigma^{-1}$, then $C$ is a stable $\infty$-category in the sense that it has all finite colimits and homotopy cartesian squares coincide with homotopy cocartesian squares.
The proof is a bit lengthy, so let me sketch the argument. In the end we want that homotopy cartesian squares and homotopy cocartesian squares coincide. So if we want to compute a h-pushout of a span $X \leftarrow Y \rightarrow Z$, we will have a square $$\begin{array}{ccc} Y &\overset{g}\rightarrow & Z\\ \downarrow_f&&\downarrow_z\\ X&\overset{x}\rightarrow&P \end{array}$$ which is both a h-pushout and a h-pullback. Since $\Omega$ is invertible, the homotopy category $\operatorname{ho} C$ consists of abelian group objects and is thus enriched in abelian groups. This means that we can find for a given morphism $g:Y\rightarrow Z$ a morphism $-g:Y\rightarrow Z$ such that the composite $Y \overset{(g,-g)}\rightarrow Z\oplus Z \overset{+}\rightarrow Z$ is nullhomotopic. One can then check that having a h-pullback / h-pushout as above is equivalent to having a h-pullback / h-pushout of the form $$\begin{array}{ccc} Y & \overset{(f,-g)}\longrightarrow & X\oplus Z\\ \downarrow&&\downarrow_{x+z}\\ 0&\longrightarrow&P \end{array}$$ so it suffices to show that we have all $h$-pushouts of the form $$\begin{array}{ccc} X&\overset{f}\rightarrow & Y\\ \downarrow &&\downarrow\\ 0&\rightarrow&C \end{array}$$ i.e. that we have $h$-cofibers.
Now given a morphism $f:X\rightarrow Y$ you can iterate taking the fiber to get a staircase of homotopy-pullbacks of the form $$\tag{$\star$}\begin{array}{ccccccc} \Omega F & \rightarrow &\Omega X & \rightarrow & 0\\ \downarrow&&\downarrow && \downarrow\\ 0&\rightarrow&\Omega Y & \rightarrow &F&\rightarrow &0\\ &&\downarrow &&\downarrow&&\downarrow\\ &&0&\rightarrow&X&\rightarrow & Y\\ \end{array}$$ and by applying the equivalence $\Sigma$ to it and flipping the resulting diagram we can this staircase of h-pullbacks to the right. In particular we obtain a homotopy cartesian square $$\tag{$\star\star$}\begin{array}{ccc} X &\rightarrow& Y\\ \downarrow&&\downarrow\\ 0&\rightarrow &\Sigma F \end{array}$$ and we are done, if we can show that this is a homotopy cocartesian square. We can do this by hand, using the fact that in the combined staircase of ($\star$) and ($\star\star$) any two consecutive squares combine to a homotopy cocartesian square (they are loop-suspension squares).
All of this shows that computing h-cofibers amounts to computing the suspension of h-fibers. Since h-pushouts are h-cofibers by the argument above, this gives you a recipe to compute them.