On the classifying space for braided monoidal categories (Q2912634)

The concept of braiding as a refinement of symmetry is the starting point of a rich interplay between geometry (knot theory) and algebra (representation theory). It was noticed by \textit{J. D. Stasheff} [Trans. Am. Math. Soc. 108, 275--292, 293--312 (1963; Zbl 0114.39402)] that the group completion of the classifying space \(B\mathcal{M}\) for a braided monoidal category \((\mathcal{M},\otimes,\mathbf{c})\) is a double loop space \(\Omega^2B(\mathcal{M},\otimes,\mathbf{c})\).NEWLINENEWLINEThe authors make use of some their previous results to state an existence of homotopy equivalences: NEWLINE\[NEWLINEB(\mathcal{M},\otimes)\simeq \Omega B(\mathcal{M},\otimes,\mathbf{c})NEWLINE\]NEWLINE and NEWLINE\[NEWLINEB(\mathcal{M},\otimes,\mathbf{c})\simeq |\text{Ner}(\mathcal{M},\otimes,\mathbf{c})|,NEWLINE\]NEWLINE where \(\text{Ner}(\mathcal{M},\otimes,\mathbf{c})\) is the geometric nerve of \((\mathcal{M},\otimes,\mathbf{c})\).NEWLINENEWLINEFor the entire collection see [Zbl 1243.00023].