The cyclic bar construction on A_∞ H-spaces

We set up a general framework for enriching a category A over a symmetric monoidal category C using a non-Σ operad P in C. To do this we require A to come with a functor U : A → Δ Σ₊ to the category of noncommutative sets. By viewing the simplicial indexing category as a category over Δ Σ₊ in two different ways, we obtain two generalizations of simplicial objects. For the operad given by the Stasheff associahedra we obtain a model for the 2-sided bar construction in the first case and the cyclic bar and cobar construction in the second case. Using either the associahedra or the cyclohedra in place of the geometric simplices we can define the geometric realization of these objects.