Braided injections and double loop spaces (Q2796093)

The authors prove a rigidification result for braided monoidal categories and for spaces equipped with an action of the little 2-disks operad. In more details, they introduce a category \(\mathfrak{B}\) of braided injections. This is equipped with the structure of a braided monoidal category. It follows that the category of \(\mathfrak{B}\)-spaces (i.e., functors from \(\mathfrak{B}\) to spaces) is equipped with a Day type braided monoidal tensor product structure and similarly for the category of \(\mathfrak{B}\)-categories. The rigidification result takes the form of an equivalence between the homotopy theory of braided monoidal categories and that of commutative monoids in \(\mathfrak{B}\)-categories and similarly an equivalence between the homotopy theory of a certain category of \(E_2\)-spaces (for a specific choice of \(E_2\)-operad) and that of commutative monoids in \(\mathfrak{B}\)-spaces. An application of this rigidification result is to produce an explicit braided monoidal category that models the \(E_2\)-monoidal category of spaces over a fixed \(E_2\)-space.