E n-monoidal categories and their group completions

Abstract

An E n -monoidal structure on a category A is a coherently associative and commutative multiplication on A with respect to which the classifying space BA has an n-fold delooping. When n = 1, 2 or ∞ an E n -monoidal structure is, upto equivalence, a strict monoidal, braided tensor or permutative strucuture respectively. Theconstruction of deloopings requires a careful analysis of higher homotopy commutativity for E n - monoidal categories A. There results a category W A such that BWA is an n-fold delooping of BA. We also construct an n-fold delooping of A as a sequence of 1-fold deloopings.