ScalingStacks

Proof.

Apply proposition 7.3.2 to the symmetric monoidal \(\infty\)-category \(\mathrm{Alg}_{\mathbb E_1}(\mathcal V_{/C}) \simeq \mathrm{Alg}_{\mathbb E_1}(\mathcal V)_{/C}\). ◻

Proof.

Recall that the functor \({\mathbf K}^b_{\mathrm{loc}}(\mathrm{Sbim}) \rightarrow\mathrm{st}^{B\mathbb{Z}}_{k}\) factors by definition through the small full subcategory \(\mathrm{DMor}^{\mathrm{poly}, \mathrm{gr-perf}}(\mathrm{mod}_k^{\mathbb{Z}})\hookrightarrow \mathrm{DMor}^{\mathrm{flat}, \mathrm{gr-perf}}(\mathrm{mod}_{k}^{\mathbb{Z}})\hookrightarrow \mathrm{st}^{B\mathbb{Z}}_{k}\). Hence, the space ([00H6]) is equivalent to the space \[ \mathrm{Braid}_{\mathrm{Cat}[\mathrm{st}^{B\mathbb{Z}}_{k}]_{/\mathrm{DMor}^{\mathrm{poly}, \mathrm{gr-perf}}(\mathrm{mod}_k^{\mathbb{Z}})}}\left({\mathbf K}^b_{\mathrm{loc}}(\mathrm{Sbim})\right).\] We now invoke theorem 8.2.1 for \(\mathcal C= \mathrm{Sbim}\), \(\mathcal D= \mathrm{DMor}^{\mathrm{poly}, \mathrm{gr-perf}}(\mathrm{mod}_k^{\mathbb{Z}})\), and \(\mathcal B=\mathrm{BSbim}\), the functor \(H_{\mathrm{loc}}\colon \mathrm{Sbim}\rightarrow\mathrm{DMor}^{\mathrm{poly}, \mathrm{gr-perf}}(\mathrm{mod}_k^{\mathbb{Z}})\) from proposition 6.5.2 which is faitfhul by lemma 6.5.6, and the functor \(\iota\colon \mathrm{BSbim}\rightarrow\mathrm{Sbim}\) from ([00CU]) which satisfies the relevant conditions of theorem 8.2.1 by proposition 6.3.2. Thus, the space of braidings ([00H9]) is equivalent to the space of prebraidings \[\mathrm{PreBraid}_{(\mathrm{Cat}_{(1,1)})_{/h_1\mathrm{DMor}^{\mathrm{poly}, \mathrm{gr-perf}}(\mathrm{mod}_k^{\mathbb{Z}})}}(h_1 \mathrm{BSbim}\rightarrow h_1 {\mathbf K}^b_{\mathrm{loc}}(\mathrm{Sbim})).\] Both, \(h_1 \mathrm{BSbim}\rightarrow h_1 {\mathbf K}^b_{\mathrm{loc}}(\mathrm{Sbim})\) and \(h_1{\mathbf K}^b_{\mathrm{loc}}(\mathrm{Sbim}) \rightarrow h_1\mathrm{DMor}^{\mathrm{poly}, \mathrm{gr-perf}}(\mathrm{mod}_k^{\mathbb{Z}})\), agree with the respective functors from section 2, namely with ([001D]) by corollary 6.4.3 and with  ([001H]) by  corollary 6.5.4 respectively. Hence, it follows from example 8.1.8 that this space ([00HA]) is equivalent to the set \(\mathrm{PreBraid}_{/h_1\mathrm{DMor}^{\mathrm{poly}, \mathrm{gr-perf}}(\mathrm{mod}_k^{\mathbb{Z}})}\left(h_1 \mathrm{BSbim}\rightarrow h_1 \mathrm{K}^b_{\mathrm{loc}}(\mathrm{Sbim})\right)\) from theorem 2.6.4.

The second half of corollary 8.2.2 follows directly from the first: By corollary 2.5.3, the condition on the positive braiding \(\sigma_{1,1}\) fixes an element of the set ([00H7]) and hence a point in the space ([00H6]). Thus, there is a contractible space of braidings on \({\mathbf K}^b_{\mathrm{loc}}(\mathrm{Sbim})\) over \(\mathrm{st}^{B\mathbb{Z}}_{k}\) compatible with the given prebraiding on \(h_1\mathrm{BSbim}\rightarrow h_1\mathrm{DMor}^{\mathrm{poly}, \mathrm{gr-perf}}(\mathrm{mod}_k^{\mathbb{Z}})\). ◻

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]). ◻

Proof.

Since \(h_1 \colon \mathrm{Cat}_{(\infty, {2})}\rightarrow\mathrm{Cat}_{({1}, {1})}\) is (strongly) symmetric monoidal, this follows directly from corollary 5.5.7. ◻

Proof.

Since the (surjective on objects and dominant on 1-morphisms, faithful) factorization system is compatible with the monoidal structure on \(\mathrm{Cat}[\mathrm{add}_{k}^{B\mathbb{Z}}]\), as an \(n\)-fold tensor power of a functor in the left class, the functor \(\mathcal X^{\otimes n} \rightarrow\mathcal Y^{\otimes n}\) remains surjective on objects and dominant on \(1\)-morphisms, and hence the first map is an equivalence as a direct consequence of the orthogonality of surjective-on-objects-and-dominant-on-1-morphisms and faithful functors. ◻

Proof.

The square ([00HY]) factors as Hence, we may without loss of generality assume that \(\mathrm{Im}(f) = C\) and \(\mathrm{Im}(g) = D\), i.e. that \(f\) and \(g\) are surjective on \(\pi_0\) and prove that in this case \(h\) is an isomorphism. Working fiberwise (and identifying \(A\) with \(B\) via the given equivalence), it suffices to consider the case \(D={\sf pt}\); in other words, given a \((-1)\)-connected map \(f\colon A\twoheadrightarrow C\) so that for all \(c\in C\) the map \(\mathrm{fib}_c(f) \rightarrow A (=\mathrm{fib}_{\sf pt}(A\rightarrow{\sf pt}))\) is an isomorphism, we need to show that \(C\) is contractible. This follows directly from the long exact sequence of homotopy groups associated to the fiber sequence. ◻

Proof.

We first prove the part ([00I3]). Since the monoidal structure on \(\mathcal C\rightarrow{\mathbf K}^b_{\mathrm{loc}}(\mathcal C)\) arises from adjunction, the relevant maps from \(S_n\) all factor as: Since \(\mathcal C\rightarrow\mathcal D\) is faithful, it follows from proposition 8.4.2 that the top horizontal map is an equivalence. Moreover, for every \(f\in \mathrm{Hom}_{\mathrm{Cat}[\mathrm{add}^{B \mathbb{Z}}_k]_{/\mathcal D}}(\mathcal C^{\otimes n}, {\mathbf K}^b_{\mathrm{loc}}(\mathcal C))\), the induced map between the fibers of the vertical maps is \[\mathrm{Hom}_{\mathrm{Cat}[\mathrm{add}^{B \mathbb{Z}}_k]_{/{\mathbf K}^b_{\mathrm{loc}}(\mathcal C)}}(\mathcal C^{\otimes n}, \mathcal C) \rightarrow\mathrm{Hom}_{\mathrm{Cat}[\mathrm{add}_{k}^{B\mathbb{Z}}]_{/{\mathbf K}^b_{\mathrm{loc}}(\mathcal C)}}\left(\mathrm{Lin}_{k, \mathrm{loc}}^{\mathbb{Z}} (\mathcal B)^{\otimes n}, \mathcal C\right).\] It follows from proposition 4.3.2 that \(\mathcal C\rightarrow{\mathbf K}^b_{\mathrm{loc}}(\mathcal C)\) is faithful. Hence, this map between fibers is also an equivalence by proposition 8.4.2. Therefore, it follows from lemma 8.4.3 that the induced map between the full images of the vertical maps is an equivalence, and hence so is also the induced map between the full images of \(S_n\).

The proof of part ([00I3]) is entirely analogous: The relevant maps from \(S_n\) all factor as Since \(\mathcal C\rightarrow\mathcal D\) is faithful by assumption, it follows from corollary 8.4.1 that the top horizontal map is an equivalence. The induced map between the fibers of the vertical maps at an \(f\in \mathrm{Hom}_{{\mathrm{Cat}_{(\infty, {2})}}_{/\mathcal D}}(\mathcal B^{\otimes n}, {\mathbf K}^b_{\mathrm{loc}}(\mathcal C))\) is given by \[\mathrm{Hom}_{{\mathrm{Cat}_{(\infty, {2})}}_{/{\mathbf K}^b_{\mathrm{loc}}(\mathcal C)}}(\mathcal B^{\otimes n}, \mathcal C) \rightarrow\mathrm{Hom}_{{\mathrm{Cat}_{({1}, {1})}}_{/h_1{\mathbf K}^b_{\mathrm{loc}}(\mathcal C)}}\left(h_1 \mathcal B^{\times n}, h_1\mathcal C\right)\] and hence is also an equivalence by corollary 8.4.1 since also \(\mathcal C\rightarrow{\mathbf K}^b_{\mathrm{loc}}(\mathcal C)\) is faithful. Therefore, it follows from lemma 8.4.3 that the induced map between the full images of the vertical maps is an equivalence, and hence so is also the induced map between the full images of \(S_n\).

Part ([00I2]) follows from combining parts ([00I3]) and ([00I4]): It follows from adjunction and monoidality of \({\mathbf K}^b_{\mathrm{loc}}(-)\) that \[\mathrm{Hom}_{\mathrm{Cat}[\mathrm{st}^{B\mathbb{Z}}_{k}]_{/\mathcal D}}\left( {\mathbf K}^b_{\mathrm{loc}}(\mathcal C)^{\otimes n}, {\mathbf K}^b_{\mathrm{loc}}(\mathcal C) \right) \simeq \mathrm{Hom}_{\mathrm{Cat}[\mathrm{add}_{k}^{B\mathbb{Z}}]_{/\mathcal D}}\left( \mathcal C^{\otimes n}, {\mathbf K}^b_{\mathrm{loc}}(\mathcal C)\right).\] Similarly, it follows from adjunction and monoidality of \(\mathrm{Lin}_{k, \mathrm{loc}}^{\mathbb{Z}} (-)\) that \[\mathrm{Hom}_{\mathrm{Cat}[\mathrm{add}_{k}^{B\mathbb{Z}}]_{/\mathcal D}}\left( \mathrm{Lin}_{k, \mathrm{loc}}^{\mathbb{Z}} (\mathcal B)^{\otimes n}, {\mathbf K}^b_{\mathrm{loc}}(\mathcal C) \right) \simeq \mathrm{Hom}_{{\mathrm{Cat}_{(\infty, {2})}}_{/\mathcal D}} \left( \mathcal B^{\times n}, {\mathbf K}^b_{\mathrm{loc}}(\mathcal C)\right).\] Hence, combining the second and third statement, we find that \[\mathrm{Im}\left( S_n \rightarrow\mathrm{Hom}_{\mathrm{Cat}[\mathrm{st}^{B\mathbb{Z}}_{k}]_{/\mathcal D}}\left( {\mathbf K}^b_{\mathrm{loc}}(\mathcal C)^{\otimes n}, {\mathbf K}^b_{\mathrm{loc}}(\mathcal C) \right) \right) \simeq \mathrm{Im}\left( S_n \rightarrow\mathrm{Hom}_{{\mathrm{Cat}_{({1}, {1})}}_{/h_1\mathcal D}}\left(h_1\mathcal B^{\times n}, h_1 {\mathbf K}^b_{\mathrm{loc}}(\mathcal C) \right) \right)\] which is — as a mapping space of \(\mathrm{Cat}_{({1}, {1})}\) — a \(1\)-groupoid. ◻

Proof of theorem 8.2.1.

Given categories and functors as in part ([00H4]) of theorem 8.2.1, we will prove that the composite \[\begin{aligned} \mathrm{Braid}_{\mathrm{Cat}[\mathrm{st}^{B\mathbb{Z}}_{k}]_{/\mathcal D}}({\mathbf K}^b_{\mathrm{loc}}(\mathcal C)) &\rightarrow\mathrm{PreBraid}_{{\mathrm{Cat}_{({1}, {1})}}_{/h_1\mathcal D}}(h_1 \mathcal C\rightarrow h_1{\mathbf K}^b_{\mathrm{loc}}(\mathcal C)) \\ \nonumber &\rightarrow\mathrm{PreBraid}_{{\mathrm{Cat}_{({1}, {1})}}_{/h_1\mathcal D}}(h_1 \mathcal B\rightarrow h_1{\mathbf K}^b_{\mathrm{loc}}(\mathcal C)) \end{aligned}\] is an equivalence. Note that part ([00H2]) of theorem 8.2.1 then follows by taking \(\mathcal B\rightarrow\mathcal C\) to be the identity \(\mathcal C\rightarrow\mathcal C\) (which clearly satisfies the required conditions). Then, the second statement follows since the first map and the composite in ([00I7]) are equivalences, and hence so is the second map.

To prove that ([00I7]) is an equivalence, note that it follows from lemma 6.3.1 that the condition on \(\mathcal B\rightarrow\mathcal C\) in the statement of theorem 8.2.1.([00H4]) equivalently asserts that \(\mathrm{Lin}_{k, \mathrm{loc}}^{\mathbb{Z}} (\mathcal B) \rightarrow\mathcal C\) is surjective on objects and dominant on \(1\)-morphisms.

We now unpack ([00I7]) as a sequence of equivalences of spaces of (pre-)braidings:

Diagram references: 8.3.4 8.4.4 ([00I2]) 8.3.7 8.3.6 8.4.4 ([00I3]) 8.3.7 8.3.5 8.4.4 ([00I4]) This completes the proof of theorem 8.2.1. ◻