Let \(\mathcal V\) be a symmetric monoidal \(\infty\)-category, or more generally an \(\infty\)-operad.
For an \(\mathbb E_1\)-algebra \(A\) in \(\mathcal V\), we write \[\mathrm{Braid}_{\mathcal V}(A):= \mathrm{Hom}_{\mathrm{Op}}(\mathbb E_2, \mathcal V) \times_{\mathrm{Hom}_{\mathrm{Op}}(\mathbb E_1, \mathcal V)} \{A\}\] for the space of \(\mathbb E_2\)-algebra structures on \(A\) compatible with the given \(\mathbb E_1\)-structure and refer to this space as the space of braidings on \(A\).
For an \(\mathbb E_1\)-algebra \(A\) in \(\mathcal V\), we write \[\mathrm{PreBraid}_{\mathcal V}(A):= \mathrm{Hom}_{\mathrm{Alg}_{\mathbb E_1}(\mathcal V)}(A\otimes A, A) \times_{\mathrm{Hom}_{\mathrm{Alg}_{\mathbb E_1}(\mathcal V)}(A, A)^{\times 2}}\{\mathrm{id}_A,\mathrm{id}_A\}\] and refer to this space as the space of prebraidings on \(A\).
For \(f \colon A \rightarrow B\) a morphism of \(\mathbb E_1\)-algebras in \(\mathcal V\), we write \[\mathrm{PreBraid}_{\mathcal V}(f) := \mathrm{Hom}_{\mathrm{Alg}_{\mathbb E_1}(\mathcal V)}(A \otimes A, B) \times_{\mathrm{Hom}_{\mathrm{Alg}_{\mathbb E_1}(\mathcal V)}(A, B)^{\times 2}} \{ f, f\}\] and refer to this space as the space of prebraidings on \(f\).