ScalingStacks

Fix a prebraiding on \(g\circ f\), corepresented by a lift of the operad map \([2] \otimes \mathbb E_1 \rightarrow\mathcal O\) to an operad map \(\mathbb T_2 \sqcup_{\{0<2\}}[2] \otimes \mathbb E_1 \rightarrow\mathcal O\). The fiber of \(\mathrm{PreBraid}_{\mathcal O}(g) \rightarrow\mathrm{PreBraid}_{\mathcal O}(g\circ f)\) is given by the space of lifts We now claim that both operad maps \[ [2] \otimes \mathbb E_1 \rightarrow\left(\mathbb T_2 \sqcup_{\{0<2\}}{[2]} \right) \otimes \mathbb E_1 \qquad [2] \otimes \mathbb E_1 \rightarrow\left( \mathbb T_2 \sqcup_{\{1<2\}}{[2]} \right) \otimes \mathbb E_1\] are \(0\)-surjective. Indeed, this follows since \([1] \otimes \mathbb E_1 \rightarrow\mathbb T_2 \otimes \mathbb E_1\) is \(0\)-surjective by proposition 7.6.1 and since both operad maps ([00HN]) are by definition given by a pushout of this \(0\)-surjective operad map against the operad maps \([1] \otimes \mathbb E_1 \rightarrow[2] \otimes \mathbb E_1\) induced by the inclusions \(\{0<2\} \rightarrow[2]\) and \(\{1<2\} \rightarrow[2]\), respectively (see observation 7.2.10).