Suppose $\mathfrak{M}$ is the category of $S^1$-spectra of simplicial sheaves. I know its sequential homotopy colimits (colimit in $\mathbb{N}$ as usual) coincide with categorial colimits since stable equivalences are preserved under filtered colimits.


But I don't see why they coincide with the homotopy colimits in $\mathrm{Ho}(\mathfrak{M})$, as a triangulated category. That is,
$$\mathrm{Hocolim}(E\_i):=\mathrm{Cone}(1\text{-}\mathrm{shift}:\oplus E\_i\longrightarrow\oplus E\_i)$$.


Does anybody know the reason?



Definitions of sequential homotopy colimits
I've worked out.


For any two morphisms $f,g:A\to B$ in $\mathfrak{M}$, we have
$$Cone(f-g)=Cone(((id\_A,f),(id\_A,g)):A\oplus A\to A\oplus B)$$
by elementary transformations of matrices.


We prove that latter is the homotopy pushout of the diagram
$$A\xleftarrow{id+id}A\vee A\xrightarrow{f+g}B$$
, which is just $Hocoeq(f,g)$. So just assume we have a general pushout diagram
$\require{AMScd}$
\begin{CD}
W @>t>> V\\
@V s V V @VV a V\\
U @>>b> D
\end{CD}
in $\mathfrak{M}$, such that $W$ is cofibrant and $t$ is a cofibration. So $Cone(t)=Cone(b)=D/U=V/W$ by definition and left properness of $\mathfrak{M}$. By Lemma 1.4.3 in 'Neeman Triangulated Category', we have
$$Cone(W\xrightarrow{(s,t)}U\oplus V)=D$$
by possibly modifying $a$. Thus we have proved the claim. Hence we have proved
$$Cone(f-g)=Hocoeq(f,g)$$.


