We proved in a previous article that the bar complex of an
–algebra inherits a
natural
–algebra
structure. As a consequence, a well-defined iterated bar construction
can be associated to any algebra
over an
–operad. In the case
of a commutative algebra
,
our iterated bar construction reduces to the standard iterated bar complex of
.
The first purpose of this paper is to give a direct effective definition of the iterated bar complexes
of
–algebras.
We use this effective definition to prove that the
–fold
bar construction admits an extension to categories of algebras over
–operads.
Then we prove that the
–fold
bar complex determines the homology theory associated to the category of algebras over an
–operad. In
the case
, we
obtain an isomorphism between the homology of an infinite bar construction and the usual
–homology
with trivial coefficients.
Keywords
iterated bar complex, $E_n$–operad, module over operad,
homology theory
