In the accepted answer to this question, user Cary states "What made this spectral sequence tick is that homology/cohomology takes a cofiber sequence to a long exact sequence.".
However this doesn't really show in the construction of the spectral sequence c it is constructed "by hand" using the skeletal filtration of a specific model of the homotopy colimit.
My question is whether there is an abstract argument that uses something like the quoted bit (morally, "cofiber sequences are sent to exact triangles by the singular chains functor") to get to the spectral sequence, ideally without using specific models for the homotopy colimit, and without constructing a spectral sequence until the last possible moment.
If there is an answer which is clearer but using spectra instead of spaces (it's not unreasonable to expect that : the condition "cofiber sequences are sent to exact triangles" makes more sense in the context of triangulated categories, so it's reasonable to see it as a statement about the stable homotopy category and the derived category – although I can't clearly see how to adapt the singular chains functor to spectra), I'm interested in that as well (I might actually be more interested by thay, but I feel like hearing about spaces might be enlightening)
I'm also interested in references if there is no short answer (or if there is a short answer that requires some amount of machinery)
Best Answer
You might want to have a look here:
https://ncatlab.org/nlab/show/spectral+sequence+of+a+filtered+stable+homotopy+type
If you are not familiar with $(\infty,1)$-categories pretend you are in your preferred model structure of spectra and that colimits and cofibers should be thought of as homotopy colimits and homotopy cofibers etc...
It's basically from Lurie's Higher algebra but with more details.