Let \(\mathcal O\) be an \(\infty\)-operad, \(A\) an \(\mathbb E_1\)-algebra in \(\mathcal O\), and assume that the spaces \[\mathrm{Im}\left( S_n \rightarrow\mathrm{Mul}_{\mathcal O}(\underbrace{A, \ldots, A}_{n}; A) \right)\] are \(1\)-truncated (i.e. \(1\)-groupoids) for all \(n\geq 0\), where the map from \(S_n = \mathbb E_1(n)\) is induced by the \(\mathbb E_1\)-structure on \(A\). Then, the map of spaces \[\mathrm{Braid}_{\mathcal O}(A) \rightarrow\mathrm{PreBraid}_{\mathcal O}(A)\] is an equivalence.
Proof.
Fix a prebraiding on \(A\), represented by a lift of the map of \(\infty\)-operads \(\mathbb E_1 \rightarrow\mathcal O\) representing \(A\), to a map of \(\infty\)-operads \(\mathbb A_2 \otimes \mathbb E_1 \rightarrow\mathcal O\). The fiber of \(\mathrm{Braid}_{\mathcal O}(A) \rightarrow\mathrm{PreBraid}_{\mathcal O}(A)\) at this prebraiding is precisely the space of further lifts 
The operad map \(\mathbb E_1 \rightarrow\mathbb A_2 \otimes \mathbb E_1\) is \(0\)-surjective by proposition 7.6.1, and the composite \(\mathbb E_1 \rightarrow\mathbb A_2 \otimes \mathbb E_1 \rightarrow\mathbb E_2\) is \(0\)-surjective since all mapping spaces of \(\mathbb E_2\) are connected and all mapping spaces of \(\mathbb E_1\) are non-empty. Hence, it follows from lemma 8.3.3 applied to the \(\infty\)-category \(\mathrm{Op}\) (with its (\(0\)-surjective, \(0\)-faithful) factorization system) that this space of lifts is equivalent to the space of lifts 
Since all multi-hom spaces of \(\mathcal O|_{\mathbb E_1}\) are by assumption \(1\)-truncated, and hence \(\mathcal O|_{\mathbb E_1}\) is a \(2\)-operad (see definition 7.7.1), it follows from corollary 7.7.8 that this space of lifts is contractible. ◻
Original source: arXiv:2401.02956v2
Original source · 2401.02956v2