Braided injections and double loop spaces

Abstract

We consider a framework for representing double loop spaces (and more generally E-2 spaces) as commutative monoids. There are analogous commutative rectifications of braided monoidal structures and we use this framework to define iterated double deloopings. We also consider commutative rectifications of E-infinity spaces and symmetric monoidal categories and we relate this to the category of symmetric spectra.