  We study the mod-2 cohomology spectral sequence arising from delooping the
Bousfield-Kan cosimplicial space giving the 2-nilpotent completion of a
connective spectrum $X$. Under good conditions its $E_{2}$-term is computable
as certain non-abelian derived functors evaluated at $H^*(X)$ as a module over
the Steenrod algebra, and it converges to the cohomology of $\Omega^\infty X$.
We provide general methods for computing the $E_{2}$-term, including the
construction of a multiplicative spectral sequence of Serre type for
cofibration sequences of simplicial commutative algebras. Some simple examples
are also considered; in particular, we show that the spectral sequence
collapses at $E_{2}$ when $X$ is a suspension spectrum.
