I'm thinking about the Grothendieck spectral sequence. I'm trying to understand how it relates to exact couples. Can anyone help me prove in general that the Grothendieck Spectral sequence converges as it should using exact couples?

I'm also trying to relate the definition of an exact couple to the definition of an exact triangle in a triangulated category. It seems like an exact triangle $$N_* \rightarrow M_* \rightarrow L_* \rightarrow N_*[1]$$ Would give an example of an exact couple. Is this true? If so, what is a conceptual understanding of the couple that an exact couple converges to?

I'm also interested in general in a conceptual interpretation of the derived exact couple, especially from the point of view of spectra.

I'm thinking about the Grothendieck spectral sequence. I'm trying to understand how it relates to exact couples. Can anyone help me prove in general that the Grothendieck Spectral sequence converges as it should using exact couples?

I'm also trying to relate the definition of an exact couple to the definition of an exact triangle in a triangulated category. It seems like an exact triangle $$N_* \rightarrow M_* \rightarrow L_* \rightarrow N_*[1]$$ Would give an example of an exact couple. Is this true? If so, what is a conceptual understanding of the couple that an exact couple converges to?

I'm also interested in general in a conceptual interpretation of the derived exact couple, especially from the point of view of spectra.

It would be good to build up some intuition as to why exact couples would converge to something of interest, and as to the conceptual significance of the derived exact couple.