Lax Operad Actions and Coherence for Monoidal N -categories, A∞ Rings and Modules

We establish a general coherence theorem for lax operad actions on an n-category which implies that an n-category with such an action is lax equivalent to one with a strict action. This includes familiar coherence results (e.g. for symmetric monoidal categories) and many new ones. In particular, any braided monoidal n-category is lax equivalent to a strict braided monoidal n-category. We also obtain coherence theorems for A∞ and E∞ rings and for lax modules over such rings. Using these results we give an extension of Morita equivalence to A∞ rings and some applications to infinite loop spaces and algebraic K-theory.