Let
be a real linear
algebraic group and
a finitely generated cosimplicial group. We prove that the space of homomorphisms
has a homotopy stable
decomposition for each
.
When
is a compact Lie group, we show that the decomposition is
–equivariant
with respect to the induced action of conjugation by elements of
. In particular, under these
hypotheses on
, we obtain
stable decompositions for
and
, respectively,
where
are the finitely generated free nilpotent groups of nilpotency class
.
The spaces
assemble
into a simplicial space
.
When
we show that its
geometric realization
, has
a nonunital
–ring space
structure whenever
is
path connected for all
.
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