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In this article from 1990, Stover describes a specral sequence which converges to the higher homotopy groups of the homotopy colimit of a diagram $\underline{X}$ of topological spaces.

The second page terms $E^2_{p,*}$ are described as values of the $p^{th}$ derived functor of the colimit functor $\lim_p (\pi_*(\underline{X}))$ (where I use $\lim$ to mean the colimit functor) on the associated diagram of homotopy algebras $\pi_*(\underline{X})$. Here, for $p=0$ we take the normal colimit functor $\lim (\pi_*(\underline{X}))$.

This derived functor is defined as $\pi_p({\lim{F_*}})$ - that is, the $p^{th}$ homotopy group of the simplicial $\Pi$-algebra $\lim{F_*}$, where $F_*$ is a free simplicial resolution of the diagram $\pi_*(\underline{X})$.

In order to take the above $p^{th}$ homotopy group $\pi_p(\lim F_*)$, we need the face and degeneracy maps in the simplicial object $\lim F_*$. How are the face and degeneracy maps for $\lim F_*$ defined here?

More generally, are there examples in the literature of the usage of this spectral sequence in making concrete computations - or alternatively in aiding computations?

EDIT: Having offered a bounty on this question which has since expired, I would like to emphasise that a satisfactory answer would be one which answers my question about the face and degeneracy maps. An answer to the second, more general question, would be a bonus.

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  • $\begingroup$ A simplicial resolution of a diagram is a simplicial object in the category of diagrams. In particular, if we take colimits dimension-wise, we obtain a simplicial object. $\endgroup$ – Fernando Muro Jan 30 at 2:45
  • $\begingroup$ I am not aware of any computations using Stover's spectral sequence. $\endgroup$ – Ryan Budney Jan 30 at 4:29
  • $\begingroup$ Something related which has been developed in the meantime is the calculus of homotopy functors. This starts with the Blakers-Massey theorem, which can be stated as a theorem about homotopy groups of a colimit, in a range. My instinct is that Stover's spectral sequence on its own may be hard to wield, but enhancing it with the Goodwillie calculus may yield calculations. $\endgroup$ – Dev Sinha Feb 5 at 15:45

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