Lax Operad Actions and Coherence for Monoidal n-Categories, A_{\infty} Rings and Modules

Gerald Dunn

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_{\infty} and E_{\infty} rings and for lax modules over such rings. Using these results we give an extension of Morita equivalence to A_{\infty} rings and some applications to infinite loop spaces and algebraic K-theory.

Keywords: braided monoidal n-category, operad, ring spectrum, A8 ring, Morita equivalence.

1991 MSC: 18C15, 18D05, 18D10, 19D23, 55P47, 55U40.

Theory and Applications of Categories, Vol. 3, 1997, No. 4, pp 50-84.

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