Injective braids, braided operads and double loop spaces

Abstract

We construct the category of B-spaces, which is a braided monoidal diagram category. This category is Quillen equivalent to the category of simplicial sets. The induced equivalence of homotopy categories maps a commutative B-spaces monoid to a space that is weakly equivalent to a double loop space, if it is connected. If X is a connected space, we find a commutative B-space monoid, such that the homotopy colimit of it is weakly equivalent to doubleloops(doublesuspension(X)). Similarly we find a commutative B-space monoid that represents the nerve of a braided strict monoidal category.