Graded multiplications on iterated bar constructions

Abstract

We define a bar construction endofunctor on the category of commutative augmented monoids A of a symmetric monoidal category V endowed with a left adjoint monoidal functor F : sSet → V. To do this, we need to carefully examine the monoidal properties of the well-known (reduced) simplicial bar construction B • (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 : V → W, there is a monoidal transformation |G − | ⇒ G| − |. We can then consider BA = |B • A| and the iterations B n A. We establish the existence of a graded multiplication on these objects, provided the category V is cartesian and A is a ring object. The examples studied include simplicial sets and modules, topological spaces, chain complexes and spectra.