ScalingStacks

Proof.

Let \(\mathcal V_{Q/^{\mathcal L}}\) denote the full subcategory of \(\mathcal V_{Q/}\) on the morphisms \(Q\rightarrow X\) which are in \(\mathcal L\). The factorization system induces a right adjoint of the inclusion \(\mathcal V_{Q/^{\mathcal L}} \hookrightarrow \mathcal V_{Q/}\) which sends \(Q\rightarrow X\) to its factorization \(Q\rightarrow X|_Q\). The statement then follows immediately from adjunction. ◻

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. ◻

Proof.

Consider the map of operads \([1] \otimes \mathbb E_1 \rightarrow\mathcal O\) representing the \(\mathbb E_1\)-algebra map \(f \colon A \rightarrow B\). The fiber of \(\mathrm{PreBraid}_{\mathcal O}(f) \rightarrow\mathrm{PreBraid}_{\mathcal P}(F(f))\) at a prebraiding represented by an operad map \(\mathbb T_2 \otimes \mathbb E_1 \rightarrow\mathcal P\) is precisely the space of (dashed) lifts of the following commuting square of operads: Since \([1] \otimes \mathbb E_1 \rightarrow\mathbb T_2 \otimes \mathbb E_1\) is \(0\)-surjective, it follows from lemma 8.3.3 that this space of lifts is equivalent to the space of lifts Hence, replacing \(\mathcal O\) by \(\mathcal O|_{[1] \otimes \mathbb E_1}\) and \(\mathcal P\) by \(\mathcal P|_{[1] \otimes \mathbb E_1}\) with multimapping spaces as in observation 8.3.2, we may without loss of generality assume that the maps \[\mathrm{Mul}_{\mathcal O}(a, \ldots, a; b) \rightarrow\mathrm{Mul}_{\mathcal P}(Fa, \ldots, Fa; Fb)\] are equivalences. It then follows from proposition 7.8.2.([00GH]) that the maps \[\mathrm{Mul}_{\mathrm{Alg}_{\mathbb E_1}(\mathcal O)}(a, \ldots, a; b) \rightarrow\mathrm{Mul}_{\mathrm{Alg}_{\mathbb E_1}(\mathcal P)}(Fa, \ldots, Fa; Fb)\] are equivalences. Hence, \[\begin{aligned} \mathrm{PreBraid}_{\mathcal O}(g) = &\mathrm{Mul}_{\mathrm{Alg}_{\mathbb E_1}(\mathcal O)}(a,a; b) \times_{\mathrm{Mul}_{\mathrm{Alg}_{\mathbb E_1}(\mathcal O)}(a, b)^{2}} \{g\} \\ &\longrightarrow \mathrm{Mul}_{\mathrm{Alg}_{\mathbb E_1}(\mathcal P)}(Fa,Fa; Fb) \times_{\mathrm{Mul}_{\mathrm{Alg}_{\mathbb E_1}(\mathcal P)}(Fa, Fb)^{2}} \{Fg\} = \mathrm{PreBraid}_{\mathcal O}(Fg) \end{aligned}\] is an equivalence. ◻

Proof.

The pair of composable morphisms of \(\mathbb E_1\)-algebras \(\{f,g\}\) may be corepresented by an operad map \([2] \otimes \mathbb E_1 \rightarrow\mathcal O\). Recall the operad maps \([2] \otimes \mathbb E_0 \rightarrow\mathbb T_2 \sqcup_{\{0<2\}} [2] \rightarrow\mathbb T_2 \sqcup_{\{1<2\}}\) from observation 7.2.10 corepresenting a pair of composable \(\mathbb E_0\)-morphisms with a \(\mathbb T_2\)-structure on \(g\) and on \(g\circ f\), respectively, and the construction of a \(\mathbb T_2\)-structure on \(g\) from a \(\mathbb T_2\)-structure on \(g\circ f\).

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).

Hence, it follows from lemma 8.3.3 applied to the \(\infty\)-category \(\mathrm{Op}\) that our space of lifts is equivalent to the space of lifts Therefore, without loss of generality, we may replace \(\mathcal O\) by \(\mathcal O|_{[2] \otimes \mathbb E_1}\) and hence with observation 8.3.2, we may assume that \[\mathrm{Mul}_{\mathcal O}(b, \ldots, b; c) \rightarrow\mathrm{Mul}_{\mathcal O}(a, \ldots, a; c)\] are equivalences. It therefore follows from proposition 7.8.2.([00GI]) that \[\mathrm{Mul}_{\mathrm{Alg}_{\mathbb E_1}(\mathcal O)}(b, \ldots, b; c) \rightarrow\mathrm{Mul}_{\mathrm{Alg}_{\mathbb E_1}{\mathcal O}}(a, \ldots, a; c)\] are equivalences, and hence as in the proof of corollary 8.3.5 that \[\mathrm{PreBraid}_{\mathcal O}(g) \rightarrow\mathrm{PreBraid}_{\mathcal O}(g\circ f)\] are equivalences. ◻

Proof.

This is an immediate corollary of lemma 7.2.9 applied to the adjunction ([00HQ]). ◻