In the situation of Corollary 2.4.2, the braid relation ([001P]) holds, because it can be proven using the naturality of \(\beta\). There are in fact two distinct proofs, namely by sliding either of the two highlighted crossings under the remaining strand: 
In a higher-categorical version of a prebraiding, these two witnesses for the braid relation need not be realized by the same 2-morphism. However, in the axiomatics of braided monoidal 2-categories, the cells witnessing these two proofs are equated by the so-called \(S_+=S_-\) relation of [BN96] (which was omitted in [KV94]). For a monoidal higher category, a prebraiding on the identity functor therefore does not imply the braid relations ([001P]), see also Remark 2.5.7, and in particular does not encode a braided monoidal (i.e. \(\mathbb E_2\)-)structure. In section 7, we will revisit this point and show that a prebraiding on the identity functor always encodes an \(\mathbb A_2 \otimes \mathbb E_1\)-structure, which differ in general from \(\mathbb E_2\)-structures.
Original source: arXiv:2401.02956v2
Original source · 2401.02956v2