Tensor product of operads and iterated loop spaces

Abstract

A algebraic characterization of an n-fold loop space in terms of its n different 1-fold loop structures is established. This amounts to describing the higher homotopy commutativity for such a space as a strict partial commutativity of the 1-fold loop structures. The tensor product of operads (a special case of the construction for algebraic theories) is ideally suited for this. In particular we show that the operad of little n-cubes C n is homotopy equivalent to the n-fold tensor product C ⊗ n 1, i.e., ‘tensoring these A ∞-structures yields an iterated loop structure’. This is not true for arbitrary A ∞-operads.