ScalingStacks

For an \(\mathbb E_1\)-algebra \(A\), it follows by definition that \(\mathrm{PreBraid}_{\mathcal V}(A) = \mathrm{PreBraid}_{\mathcal V}(\mathrm{id}_A)\). Recall from corollary 7.2.6 that analogous to \(\mathrm{Braid}_{\mathcal V}\), the spaces of prebraidings are also corepresented by certain \(\infty\)-operads: \[\begin{aligned} \mathrm{PreBraid}_{\mathcal V}(A) & = \mathrm{Hom}_{\mathrm{Op}}( \mathbb A_2 \otimes \mathbb E_1, \mathcal V) \times_{\mathrm{Hom}_{\mathrm{Op}}(\mathbb E_1, \mathcal V)} \{A\}\\ \mathrm{PreBraid}_{\mathcal V}(f:A \rightarrow B) &=\mathrm{Hom}_{\mathrm{Op}}(\mathbb T_2 \otimes \mathbb E_1, \mathcal V) \times_{\mathrm{Hom}_{\mathrm{Op}}([1] \otimes \mathbb E_1, \mathcal V)} \{f\} \end{aligned}\] Moreover, for any \(\mathbb E_1\)-algebra \(A\), composing with the operad map \(\mathbb A_2 \otimes \mathbb E_1 \rightarrow\mathbb E_1 \otimes \mathbb E_1 \simeq \mathbb E_2\) from example 7.2.7 defines a ‘forgetful’ map of spaces \[ \mathrm{Braid}_{\mathcal V}(A) \rightarrow\mathrm{PreBraid}_{\mathcal V}(A).\]