  We define a bar construction endofunctor on the category of commutative
augmented monoids $A$ of a symmetric monoidal category $\mathcal{V}$ endowed
with a left adjoint monoidal functor $F:s\mathbf{Set}\to \mathcal{V}$. To do
this, we need to carefully examine the monoidal properties of the well-known
(reduced) simplicial bar construction $B_\bullet(1,A,1)$. We define a geometric
realization $|-|$ with respect to the image under $F$ of the canonical
cosimplicial simplicial set. This guarantees good monoidal properties of $|-|$:
it is monoidal, and given a left adjoint monoidal functor $G:\mathcal{V}\to
\mathcal{W}$, there is a monoidal transformation $|G-|\Rightarrow G|-|$. We can
then consider $BA=|B_\bullet A|$ and the iterations $B^nA$. We establish the
existence of a graded multiplication on these objects, provided the category
$\mathcal{V}$ is cartesian and $A$ is a ring object. The examples studied
include simplicial sets and modules, topological spaces, chain complexes and
spectra.
