Let \(C\) be an \(\mathbb E_{\infty}\)-algebra in a symmetric monoidal \(\infty\)-category \(\mathcal V\), and let \(F\) be a morphism of \(\mathbb E_1\)-algebras in \(\mathcal V_{/C}\), i.e. equivalently a commuting diagram 
of \(\mathbb E_1\)-algebras in \(\mathcal V\). Then, the spaces \[\mathrm{PreBraid}_{\mathcal V_{/C}}(F) \simeq \mathrm{PreBraid}_{\mathcal V}(F)_{/C}\] are equivalent.
Proof.
Apply proposition 7.3.2 to the symmetric monoidal \(\infty\)-category \(\mathrm{Alg}_{\mathbb E_1}(\mathcal V_{/C}) \simeq \mathrm{Alg}_{\mathbb E_1}(\mathcal V)_{/C}\). ◻
Original source: arXiv:2401.02956v2
Original source · 2401.02956v2