Braided injections and double loop spaces

TL;DR

This paper develops a framework for representing double loop spaces and related structures as commutative monoids, providing new methods for rectification and deloopings in algebraic topology.

Contribution

It introduces a novel framework for commutative rectifications of braided and E-infinity structures, enabling iterated double deloopings and connections to symmetric spectra.

Findings

Framework for representing double loop spaces as commutative monoids

Construction of iterated double deloopings using rectification

Relation between E-infinity spaces, symmetric monoidal categories, and symmetric spectra

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.