# Computations using “Stover's spectral sequence”

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.

• 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. – Fernando Muro Jan 30 at 2:45
• I am not aware of any computations using Stover's spectral sequence. – Ryan Budney Jan 30 at 4:29
• 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. – Dev Sinha Feb 5 at 15:45