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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)

Maxime Ramzi
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1 Answers1

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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.

Nikitas Nikandros
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  • Thank you I was looking for something along those lines ! (I'm not familiar with $\infty$-categories but I'm trying to learn). Is there anything relevant for other types of homotopy (co)limits ? It seems like the nLab only deals with ones indexed by $\mathbb N$ – Maxime Ramzi Feb 10 '20 at 11:42
  • One way to calculate a homotopy colimit of a diagram $\mathbf{X}:J\longrightarrow \mathrm{Spectra}$ is to first constuct the simplicial replacement $\mathrm{srep}(\mathbf{X}):\Delta^{\mathrm{op}}\longrightarrow \mathrm{Spectra}$ and then take geometric realization. The filtration in this case, is the skeletal filtration of the simplicial replacement. – Nikitas Nikandros Feb 10 '20 at 11:49
  • So you end up having to use that model for homotopy colimits and the skeletal filtration in the end anyway ? – Maxime Ramzi Feb 10 '20 at 12:12
  • Yeap. Unless you know another way to make a spectral sequence for a homotopy colimit... – Nikitas Nikandros Feb 10 '20 at 12:18
  • For that last comment, do you mean homotopy colimit in chain complexes ? Because if you meant in spaces/spectra well that's exactly what I was trying to ask about – Maxime Ramzi Feb 10 '20 at 12:50
  • Yeah I meant in spaces/spectra. But the link on nlab talks about infinity categories in general. (That could include stable model categories of chain complexes) . Hope i dind't confuse you ! – Nikitas Nikandros Feb 10 '20 at 15:53