Proof. The only non-trivial parts are: the construction of \((-)^\mathrm{an}_{\mathrm{Str},\mathrm{lft}}\) and checking that the square involving \(\lambda_\mathrm{Str}, (-)^\mathrm{an}_{\mathrm{PSh}\mathrm{Str},\mathrm{lft}}\) and \((-)^\mathrm{an}_{\mathrm{PSh}\mathrm{Str}}\) commutes. The rest of the statement follows by properties of left and right Kan extensions along fully faithful functors.

So, let \((Y,P,s:Y^{{\mathrm{Zar}}}\to \mathrm{Alex}(P))\) be an element of \(\mathrm{Str}\mathrm{Sch}^{\mathrm{lft}}_\mathbb{C}\). The morphism \(\varphi_Y: \mathrm{an}(Y)\to Y\) induces a map of topological spaces \(\varphi_Y^{\mathrm{top}} : Y^\mathrm{an}\to Y^{\mathrm{Zar}}\). Define \(s^{\mathrm{an}}\) to be the composite \[s^\mathrm{an}= s\circ \varphi_Y^{\mathrm{top}}: Y^{\mathrm{an}}\to Y^{{\mathrm{Zar}}} \to \mathrm{Alex}(P).\] Let \((f,r):(Y,s)\to (W,s')\) be a stratified map. Consider the map \(\mathrm{an}(f): \mathrm{an}(Y)\to \mathrm{an}(W)\): by definition the map \(\mathrm{an}(f)\) fits in the commutative diagram of ringed spaces \[\begin{tikzcd} \mathrm{an}(Y) \arrow[r, "{\mathrm{an}(f)}"] \arrow[d, "{\varphi_Y}"'] & \mathrm{an}(W) \arrow[d, "{\varphi_W}"] \\ Y \arrow[r, "f"] & W. \end{tikzcd}\] By forgetting the sheaves, we have the commutative diagram \[\begin{tikzcd} Y^{\mathrm{an}} \arrow[r, "{f^{\mathrm{an}}}"] \arrow[d, "{\varphi_Y^{\mathrm{top}}}"'] & \mathrm{an}(W) \arrow[d, "{\varphi_W^{\mathrm{top}}}"] \\ Y^{{\mathrm{Zar}}} \arrow[r, "f^{{\mathrm{Zar}}}"] \arrow[d, "{s}"'] & W^{{\mathrm{Zar}}} \arrow[d, "{s'}"] \\ \mathrm{Alex}(P) \arrow[r, "r"] & \mathrm{Alex}(Q). \end{tikzcd}\] Therefore \((f^{\mathrm{an}},r)\) is a map of stratified spaces \((Y^{\mathrm{an}},s^\mathrm{an})\to (W^{\mathrm{an}}, {s'}^\mathrm{an})\). This defines a functor \[(-)^{\mathrm{an}}_{\mathrm{Str},\mathrm{lft}}: \mathrm{Str}\mathrm{Sch}^{\mathrm{lft}}_{\mathbb{C}}\to \mathrm{Str}{\mathrm{Top}}, \quad \quad (Y,s)\mapsto (Y^\mathrm{an}, s^\mathrm{an}), \;\mathrm{ and }\;(f,r) \mapsto (f^{\mathrm{an}},r),\] which enhances \((-)^\mathrm{an}: \mathrm{Sch}^{\mathrm{lft}}_\mathbb{C}\to {\mathrm{Top}}\), in the sense that the top square in ?? commutes. This functor still preserves finite limits: indeed, given a finite diagram \(F\to \mathrm{Str}\mathrm{Sch}^{\mathrm{lft}}_\mathbb{C}\), by 7, the limit \(\lim_{\alpha \in F} (Y_\alpha, P_\alpha, s_\alpha: Y_{\alpha}^{\mathrm{Zar}}\to \mathrm{Alex}(P_\alpha) )\) is \[\left(\lim_{\alpha \in F} Y_\alpha , \lim_{\alpha \in F} P_\alpha, s : \left( \lim_{\alpha \in F} Y_{\alpha} \right)^{\mathrm{Zar}}\to \lim_{\alpha \in F} Y_{\alpha}^{\mathrm{Zar}}\to \mathrm{Alex}( \lim_{\alpha\in F} P_\alpha) \right).\] By the definition of \((-)^\mathrm{an}_{\mathrm{Str},\mathrm{lft}}\) and by the fact that the original \((-)^\mathrm{an}\) preserves finite limits, this in turns is equal to \[\left( \lim_{\alpha \in F} Y_\alpha^\mathrm{an}, \lim_{\alpha \in F} P_\alpha, s^\mathrm{an}: \lim_{\alpha \in F} Y_{\alpha}^\mathrm{an} \to \left( \lim_{\alpha \in F} Y_{\alpha} \right)^{\mathrm{Zar}} \xrightarrow{} \lim_{\alpha \in F} Y_{\alpha}^{\mathrm{Zar}} \to \mathrm{Alex}( \lim_{\alpha\in F} P_\alpha) \right).\] By the universal property of limits, the map \(\lim_{\alpha \in F} Y_{\alpha}^\mathrm{an} \to \left( \lim_{\alpha \in F} Y_{\alpha} \right)^{\mathrm{Zar}} \xrightarrow{} \lim_{\alpha \in F} Y_{\alpha}^{\mathrm{Zar}}\) coincides with the limit map \(\lim_{\alpha \in F} Y_{\alpha}^\mathrm{an} \to \lim_{\alpha \in F} Y_{\alpha}^{\mathrm{Zar}}\), and we conclude.

For what concerns the commutativity of the square with the diagonal dashed arrows in ?? , note that any element of \(\mathrm{PSh}^\mathrm{small}(\mathrm{Str}\mathrm{Sch}_\mathbb{C}^\mathrm{lft})\) is a colimit \(\mathop{\mathrm{colim}}_i (Y_i,s_i)\) of objects in \(\mathrm{Str}\mathrm{Sch}_\mathbb{C}^\mathrm{lft}\). We thus have the assignments \[\begin{tikzcd} \mathop{\mathrm{colim}}_i (Y_i,s_i) \arrow[d, mapsto, "\lambda_\mathrm{Str}"] \arrow[rr, mapsto, "(-)^\mathrm{an}_{\mathrm{PSh}\mathrm{Str},\mathrm{lft}}"] & & \mathop{\mathrm{colim}}_i (Y_i,s_i)^\mathrm{an}_{\mathrm{Str},\mathrm{lft}} \arrow[d, equal] \\ \mathop{\mathrm{colim}}_i \!\text{\usefont{U}{min}{m}{n}\symbol{'207}}\!\circ i_{\mathrm{Str},\mathrm{lft}} (Y_i,s_i) \arrow[rr, mapsto, "(-)^\mathrm{an}_{\mathrm{PSh}\mathrm{Str}}"] & & \mathop{\mathrm{colim}}_i \left(i_{\mathrm{Str},\mathrm{lft}} (Y_i,s_i) \right)^\mathrm{an}_\mathrm{Str}, \end{tikzcd}\] hence the claim. ◻

Proof. We want to apply [14]. In order to do this we note that

1. since \(\mathrm{Fgt}_{\mathrm{str}, \mathrm{top}}\) preserves colimits, the composite \(\mathrm{Fgt}_{\mathrm{str}, \mathrm{top}} \circ (-)^\mathrm{an}_{\mathrm{PSh}\mathrm{Str}, \mathrm{lft}}\) coincides with the left Kan extension \[\begin{tikzcd}[column sep=2cm] \mathrm{Str}\mathrm{Sch}_\mathbb{C}^\mathrm{lft} \arrow[r, "(-)^\mathrm{an}_{\mathrm{Str}, \mathrm{lft}}"] \arrow[d, "{\!\text{\usefont{U}{min}{m}{n}\symbol{'207}}\!}"] & \mathrm{Str}{\mathrm{Top}}\arrow[r, "{\mathrm{Fgt}_{\mathrm{str}, \mathrm{top}}}"] & \mathrm{Set}, \\ \mathrm{PSh}^\mathrm{small}(\mathrm{Str}\mathrm{Sch}_\mathbb{C}^\mathrm{lft}) \arrow[rru, "{LKE}"', dashed] & & \end{tikzcd}\]

2. the categories \(\mathrm{Str}\mathrm{Sch}^\mathrm{lft}_\mathbb{C}\) and \(\mathrm{PSh}^\mathrm{small}(\mathrm{Str}\mathrm{Sch}^\mathrm{lft}_\mathbb{C})\) have finite limits (respectively by 7, and because the presheaf category \(\mathrm{PSh}(\mathrm{Str}\mathrm{Sch}^\mathrm{lft}_\mathbb{C})\) has all limits),

3. \(\mathrm{Str}\mathrm{Sch}^\mathrm{lft}_\mathbb{C}\) is small,

4. both \((-)^\mathrm{an}_{\mathrm{Str}, \mathrm{lft}}\) and \(\mathrm{Fgt}_{\mathrm{str}, \mathrm{top}}\) preserve finite limits (respectively by 13 and 5).

Hence the statement. ◻

Proof. Each morphism \(\pi_{X,i}\) is of the form \((\pi_{X,i}^\mathrm{top},\pi_{X,i}^\mathrm{pset})\). Same for \(\pi_{Y,i}\). 18 tells us that \(\underset{i\in \mathbb{N}}{\mathop{\mathrm{colim}}}(\pi_{X,i}^\mathrm{top}) \times \underset{i\in \mathbb{N}}{\mathop{\mathrm{colim}}}(\pi_{Y,i}^\mathrm{top})\) is an isomorphism. Since \(R_i \xrightarrow{\sim} Q_i\), the map \[\underset{i\in \mathbb{N}}{\mathop{\mathrm{colim}}}(\pi_{X,i}^\mathrm{pset}) \times \underset{i\in \mathbb{N}}{\mathop{\mathrm{colim}}}(\pi_{Y,i}^\mathrm{pset}): \underset{i\in \mathbb{N}}{\mathop{\mathrm{colim}}} (R_i\times_{Q_i} P_i) \rightarrow \underset{i\in \mathbb{N}}{\mathop{\mathrm{colim}}}(R_i) \underset{\underset{i\in \mathbb{N}}{\mathop{\mathrm{colim}}} Q_i}{\times} \underset{i\in \mathbb{N}}{\mathop{\mathrm{colim}}} (P_i)\] is an isomorphism. ◻

Proof. Consider the stratified actions \[\label{40N41-actions} \begin{gather} (\mathrm{L}^+G,\mathrm{triv})^\mathrm{an}_\mathrm{Str}\times (\mathrm{Gr}_G^{(N)}, \mathfrak{s}^{(N)})^\mathrm{an}_{\mathrm{Str},\mathrm{lft}}\to (\mathrm{Gr}_G^{(N)}, \mathfrak{s}^{(N)})^\mathrm{an}_{\mathrm{Str},\mathrm{lft}}, \\ (\mathrm{L}^+G_{X^I},\mathrm{Inc}_I)^\mathrm{an}_\mathrm{Str}\times (\mathrm{Gr}_{G,X^I}^{(N)}, \mathfrak{s}_I^{(N)})^\mathrm{an}_{\mathrm{Str},\mathrm{lft}}\to (\mathrm{Gr}_{G,X^I}^{(N)}, \mathfrak{s}_I^{(N)})^\mathrm{an}_{\mathrm{Str},\mathrm{lft}}. \end{gather}\tag{3}\] Notice that each \(\mathrm{Gr}_{G}^{(N), \mathrm{an}}\), resp. \(\mathrm{Gr}_{G,X^I}^{(N), \mathrm{an}}\), is locally compact Hausdorff, being the analytification of a projective variety, resp. a projective variety over \(X^I\). The same holds for \(\mathrm{L}^{m}G^\mathrm{an}, \mathrm{L}^{m}G_{X^I}^\mathrm{an}\), and hence for \(\mathrm{L}^+G^\mathrm{an}, \mathrm{L}^+G_{X^I}^\mathrm{an}\), since limits of locally compact Hausdorff spaces are locally compact Hausdorff.

To get the wanted actions on \((\mathrm{Gr}_G, \mathfrak s)_{\mathrm{PSh}\mathrm{Str}, \mathrm{lft}}^\mathrm{an}, (\mathrm{Gr}_{G,X^I},\mathfrak s_I)^\mathrm{an}_{\mathrm{PSh}\mathrm{Str},\mathrm{lft}}\), it suffices to apply 19 and pass to the colimit in \(N\) in the expressions 3 . ◻

Proof. Fix \(I\in \mathrm{Fin}_{\geq 1, \mathrm{surj}}\). Consider the unordering functor \[\mathcal{U}_I: X^I \to \mathrm{Ran}(X),\quad \quad x_I=(x_1,\dots, x_{|I|})\mapsto \{ x_1',\dots, x_k' \}\] where we forget the order of the \(x_i\)’s and we do not repeat maps that are equal (so \(k\) is the number of different maps in \(x_I\)). Notice that for any \(J,\phi:I\twoheadrightarrow J\), we have \(\mathcal{U}_J=\mathcal{U}_I\circ \Delta_\phi\). Hence we get a well-defined surjective map \[\mathcal{U}: \underset{I \in \mathrm{Fin}_{\geq 1, \mathrm{surj}}^{\mathrm{op}} }{\mathop{\mathrm{colim}}} X^I \to \mathrm{Ran}(X).\] Let us check that it is injective as well. Suppose that \(x_I\in X^I\) and \(y_{I'}\in X^{I'}\) are sent to the same \(\{x_1',\dots,x_k'\}\). Fix an order on \(\{x_1',\dots,x_k'\}\): \((x_{1}',\dots,x_{|J|}')\) where \(J\) has cardinality \(k\). Define \[\begin{align} \psi: I\twoheadrightarrow J,\quad \quad & \psi(i)=j \iff x_i= x_{j}' \\ \psi':I'\twoheadrightarrow J,\quad \quad & \psi'(i')=j \iff y_{i'}= x_{j}'. \end{align}\] Consider now the fiber product \(I\times_J I'\) \[\begin{tikzcd} I\times_J I' \arrow[r, "{p_2}"] \arrow[d, "{p_1}"'] & I' \arrow[d, "{\psi'}"] \\ I \arrow[r, "{\psi}"'] & J \end{tikzcd}\] and the element \(z_{I\times_J I'}\) in \(X^{I\times_J I'}\) defined as \(z_{(i,i')}=x_i=y_{i'}\): then \(\Delta_{p_1}(x_I)=\Delta_{p_2}(y_{I'})=z_{I\times_J I'}\), making \(x_I\) and \(y_{I'}\) the same element in the colimit. This proves that \(\mathcal{U}\) is an isomorphism in \(\mathrm{PSh}(\mathrm{Aff}_\mathbb{C})\). Finally, \(X^I\) coincides with the small colimit of all its affine open subschemes. Since composition of small colimits is small, we have that \(\mathrm{Ran}(X)\) is actually an element of \(\mathrm{PSh}^\mathrm{small}(\mathrm{Aff}_\mathbb{C})\). ◻

Proof. The first part follows directly from the definition, since for any \(x_I\in X^I(R)\), \(\Gamma_{x_I}\) only depends on \(\mathcal{U}_I(x_I)\). By universality of small colimits, we get \[\underset{I\in \mathrm{Fin}_{\geq 1, \mathrm{surj}}^\mathrm{op}}{\mathop{\mathrm{colim}}} \mathrm{Gr}_{G,X^I}\simeq \underset{I\in \mathrm{Fin}_{\geq 1, \mathrm{surj}}^\mathrm{op}}{\mathop{\mathrm{colim}}} \left( {X^I}\times_{\mathrm{Ran}(X)}\mathrm{Gr}_{G,\mathrm{Ran}(X)} \right) \simeq \left(\underset{I\in \mathrm{Fin}_{\geq 1, \mathrm{surj}}^\mathrm{op}}{\mathop{\mathrm{colim}}} X^I \right)\times_{\mathrm{Ran}(X)}\mathrm{Gr}_{G,\mathrm{Ran}(X)}\] which is isomorphic to \(\mathrm{Gr}_{G,\mathrm{Ran}(X)}\) by 23. As observed at the end of the proof of 23, both \(\mathrm{Gr}_{G, X^I}^{(N)}\)’s and \(\mathrm{Gr}_{G, X^I}\) can be viewed as small presheaves because they are ind-schemes. Thus so is \(\mathrm{Gr}_{G,\mathrm{Ran}(X)}\). ◻

Proof. Recall that the \(X^I\)’s are endowed with the incidence stratification, with respect to which the maps \(\Delta_\phi\)’s are indeed stratified. Consider the stratum \(\mathrm{Gr}_{G, X^{\psi}, \underline{\nu}}\) inside \(\mathrm{Gr}_{G,X^J}\) indexed by \(([J\overset{\psi}{\twoheadrightarrow} J'],\underline{\nu} \in (\mathbb{X}_\bullet(T)^+)^{|J'|})\). The map \(\delta_\phi\) sends \(\mathrm{Gr}_{G, X^{\psi}, \underline{\nu}}\) into the stratum \(\mathrm{Gr}_{G, X^{\psi\circ \phi},\underline{\nu}}\) of \(\mathrm{Gr}_{G,X^I}\) indexed by \(([I\overset{\psi\circ \phi}{\twoheadrightarrow} J'], \underline{\nu}\in ( \mathbb{X}_\bullet(T)^+)^{|J'|})\). Thus, the \(\delta_\phi\)’s are stratified. ◻

Proof. Both \((X^I,\mathrm{Inc}_I)\)’s and \((\mathrm{Gr}_{G,X^I}^{(N)}, \mathfrak{s}_{I})\)’s are objects of \(\mathrm{Str}\mathrm{Sch}^\mathrm{lft}_\mathbb{C}\) and \(\Delta_\phi, \delta_\phi\)’s are stratified maps (29). Consider then the following colimits in \(\mathrm{PSh}^\mathrm{small}(\mathrm{Str}\mathrm{Sch}^\mathrm{lft}_\mathbb{C})\) \[\label{stratified-Ran-forgets} \underset{I\in \mathrm{Fin}_{\geq 1, \mathrm{surj}}^\mathrm{op}}{\mathop{\mathrm{colim}}} (X^I, \mathrm{Inc}_I), \quad \underset{ I\in \mathrm{Fin}_{\geq 1, \mathrm{surj}}^\mathrm{op}}{\mathop{\mathrm{colim}}} (\mathrm{Gr}_{G, X^I}, \mathfrak{s}_I) \simeq \underset{I\in \mathrm{Fin}_{\geq 1, \mathrm{surj}}^\mathrm{op}, N\in\mathbb{N}}{\mathop{\mathrm{colim}}} (\mathrm{Gr}^{(N)}_{G, X^I}, \mathfrak{s}_{I}).\tag{4}\] Since the forgetful functor preserves colimits, by 23 and 26 we have \[\mathrm{Fgt}_\mathrm{str} \left( \underset{I\in \mathrm{Fin}_{\geq 1, \mathrm{surj}}^\mathrm{op}}{\mathop{\mathrm{colim}}} (X^I,\mathrm{Inc}_I) \right) \simeq \mathrm{Ran}(X),\quad \mathrm{Fgt}_\mathrm{str} \left( \underset{I\in \mathrm{Fin}_{\geq 1, \mathrm{surj}}^\mathrm{op}}{\mathop{\mathrm{colim}}} (\mathrm{Gr}_{G,X^I},\mathfrak{s}_I) \right) \simeq \mathrm{Gr}_{G,\mathrm{Ran}(X)}.\] Finally, since both stratified presheaves are defined as colimits, pulling back along the colimit map \(\mathcal{U}_I\) recovers the \(I\)-th level by universality of colimits in the category of stratified presheaves. ◻

Proof. Analogous to the proof of 26. ◻

Proof. Forgetting the group structure, \((\mathrm{L}^+G_{X^I}, \mathrm{Inc}_I)\) is an element of \({\mathrm{PSh}^\mathrm{small}(\mathrm{Str}\mathrm{Sch}_\mathbb{C})/}_{(X^I,\mathrm{Inc}_I)}\). Via the composite \[(\mathrm{L}^+G_{X^I},\mathrm{Inc}_I) \to (X^I,\mathrm{Inc}_I) \to (\mathrm{Ran}(X), \mathrm{Inc}_\mathrm{Ran})\] we actually have that \((\mathrm{L}^+G_{X^I},\mathrm{Inc}_I) \in {\mathrm{PSh}^\mathrm{small}(\mathrm{Str}\mathrm{Sch}_\mathbb{C})/}_{(\mathrm{Ran}(X),\mathrm{Inc}_{\mathrm{Ran}})}\). Therefore, with the same argument done in proof of 30, by 32 there exists a stratified small presheaf \[(\mathrm{L}^+G_\mathrm{Ran}(X), \mathrm{Inc}_\mathrm{Ran}) \simeq\underset{I\in \mathrm{Fin}_{\geq 1, \mathrm{surj}}}{\mathop{\mathrm{colim}}} (\mathrm{L}^+G_{X^I}, \mathrm{Inc}_I)\] which recovers \((\mathrm{L}^+G_{X^I}, \mathrm{Inc}_I)\)’s by pull-back.

We now want to see that there is a multiplication law on \((\mathrm{L}^+G_{\mathrm{Ran}(X)}, \mathrm{Inc}_\mathrm{Ran})\) that respects this pullback. At the level of the underlying presheaf \(\mathrm{L}^+G_{\mathrm{Ran}(X)}\), it is defined as \[\label{eq-multiplication-Ran} \mathrm{L}^+G_{\mathrm{Ran}(X)}\times_{\mathrm{Ran}(X)}\mathrm{L}^+G_{\mathrm{Ran}(X)} \to \mathrm{L}^+G_{\mathrm{Ran}(X)}, \quad (\underline{x},g).(\underline{x},h)\mapsto (\underline{x}, gh).\tag{5}\] To check that it is stratified, we describe it in a different way. Consider the colimit \[\mathop{\mathrm{colim}}_{I\in \mathrm{Fin}_{\geq 1, \mathrm{surj}}^\mathrm{op}} \left( (\mathrm{L}^+G_{X^I},\mathrm{Inc}_I)\times_{(X^I,\mathrm{Inc}_I)} (\mathrm{L}^+G_{X^I}, \mathrm{Inc}_I), \right)\quad \mathrm{taken in } {\mathrm{PSh}^\mathrm{small}(\mathrm{Str}\mathrm{Sch}_\mathbb{C})}/_{(\mathrm{Ran}(X),\mathrm{Inc}_\mathrm{Ran})}.\] By replacing \((\mathrm{L}^+G_{X^I}, \mathrm{Inc}_I)\) with \((\mathrm{L}^+G_{\mathrm{Ran}(X)}, \mathrm{Inc}_\mathrm{Ran})\times_{(\mathrm{Ran}(X),\mathrm{Inc}_\mathrm{Ran})} (X^I,\mathrm{Inc}_I)\), the previous colimit can be written as \[\mathop{\mathrm{colim}}_{I\in \mathrm{Fin}_{\geq 1, \mathrm{surj}}^\mathrm{op}}\left( ( \mathrm{L}^+G_{\mathrm{Ran}(X)}, \mathrm{Inc}_\mathrm{Ran}) \underset{(\mathrm{Ran}(X), \mathrm{Inc}_\mathrm{Ran})}{\times} ( \mathrm{L}^+G_{\mathrm{Ran}(X)}, \mathrm{Inc}_\mathrm{Ran}) \underset{(\mathrm{Ran}(X), \mathrm{Inc}_\mathrm{Ran})}{\times} (X^I,\mathrm{Inc}_I) \right).\] By universality of colimits in \({\mathrm{PSh}^\mathrm{small}({\mathrm{Str}\mathrm{Sch}_\mathbb{C}})}/_{(\mathrm{Ran}(X),\mathrm{Inc}_{\mathrm{Ran}})}\), this is exactly \[( \mathrm{L}^+G_{\mathrm{Ran}(X)}, \mathrm{Inc}_\mathrm{Ran}) \underset{(\mathrm{Ran}(X), \mathrm{Inc}_\mathrm{Ran})}{\times} ( \mathrm{L}^+G_{\mathrm{Ran}(X)}, \mathrm{Inc}_\mathrm{Ran}).\] In this way, the multiplication law 5 can be presented as a colimit of the multiplication laws of \((\mathrm{L}^+G_{X^I},\mathrm{Inc}_I)\)’s, and hence it is stratified. Applying universality of colimits in the other direction we see that it recovers the multiplication on \(\mathrm{L}^+G_{X^I}\) when pull-backed. ◻

Proof. By universality of small colimits, it is enough to give \(\delta_\phi\)-compatible actions \[\mathrm{act}_{\mathrm{Ran}, I} : (\mathrm{L}^+G_{\mathrm{Ran}(X)}, \mathrm{Inc}_\mathrm{Ran}) \underset{(\mathrm{Ran}(X), \mathrm{Inc}_\mathrm{Ran})}{\times} (\mathrm{Gr}_{G,X^I}, \mathfrak{s}_I) \to (\mathrm{Gr}_{G,X^I}, \mathfrak{s}_I),\] and then pass to the colimit on both sides. The LHS is the same as \[(\mathrm{L}^+G_{\mathrm{Ran}(X)}, \mathrm{Inc}_\mathrm{Ran}) \underset{(\mathrm{Ran}(X), \mathrm{Inc}_\mathrm{Ran})}{\times} (X^I, \mathrm{Inc}_I) \underset{(X^I, \mathrm{Inc}_I)}{\times}(\mathrm{Gr}_{G,X^I}, \mathfrak{s}_I),\] which is isomorphic to \((\mathrm{L}^+G_{X^I}, \mathrm{Inc}_I)\times_{(X^I, \mathrm{Inc}_I)}(\mathrm{Gr}_{G,X^I}, \mathfrak{s}_I)\) by 34. Hence we can define \(\mathrm{act}_{\mathrm{Ran}, I}\) as \(\mathrm{act}_I\) (see 104 and 105). We now only need to show that the \(\mathrm{act}_I\)’s are \(\delta_\phi\)-compatible.

This follows from noticing that, for any locally closed subscheme \(\mathrm{Gr}_{G, X^{\psi}}\), \(\psi: J \twoheadrightarrow L\), the map \(\delta_\phi\) becomes the identity via the isomorphism 25 of the factorization property \[\begin{tikzcd} \left ( \prod_{l=1}^{|L|}\mathrm{Gr}_{G,X} \right )_{\mathrm{disj}} \arrow[r, "\mathrm{id}"] \arrow[d, "\mathfrak{f}_{\psi}"] & \left ( \prod_{l=1}^{|L|}\mathrm{Gr}_{G,X} \right )_{\mathrm{disj}} \arrow[d, "\mathfrak{f}_{\psi\circ\phi}"] \\ \mathrm{Gr}_{G, X^\psi} \arrow[r, "\delta_\phi"] & \mathrm{Gr}_{G, X^{\psi\circ \phi}}. \end{tikzcd}\] ◻

Proof. Statements \((1)\) and \((3)\) follow from the fact that \((-)^\mathrm{an}_{\mathrm{PSh}\mathrm{Str}, \mathrm{lft}}\) preserves small colimits and Statement \((2)\) from the fact that \((-)^\mathrm{an}_{\mathrm{PSh}\mathrm{Str}}\) preserves small colimits of stratified schemes. ◻

Proof. Perfect maps are stable under products by [18], and so are surjections. Since a finite limit of topological spaces is always a closed subspace of a product when the spaces involved are Hausdorff, one can deduce the statement from [18] and from the fact that surjections are stable under pullback. ◻

Proof. Let us first show that \(\widetilde{\mathcal{U}}_n^{(N), \mathrm{an}}\) is closed. Let \(A\subseteq \mathrm{Gr}_{G, X^n}^{(N), \mathrm{an}}\) be a closed subset. By definition of the colimit topology, \(\widetilde{\mathcal{U}}^{(N), \mathrm{an}}_n(A)\) is closed in \(\underset{I\in \mathrm{Fin}_{\geq 1, \mathrm{surj}}, |I|\leq n}{\mathop{\mathrm{colim}}} \mathrm{Gr}^{(N),\mathrm{an}}_{G, X^I}\) if and only if \((\widetilde{\mathcal{U}}^{(N), \mathrm{an}}_I)^{-1}( \widetilde{\mathcal{U}}_n^{(N), \mathrm{an}}(A) )\) is closed in \(\mathrm{Gr}_{G, X^I}^{(N), \mathrm{an}}\) for any \(I\in \mathrm{Fin}_{\geq 1, \mathrm{surj}}, |I|\leq n\). For any \(\{1,\dots,n\}\overset{\phi}{\twoheadrightarrow} I\) we have \(\widetilde{\mathcal{U}}^{(N), \mathrm{an}}_I = \widetilde{\mathcal{U}}^{(N), \mathrm{an}}_n\circ \delta_\phi\), and hence it is enough to check that \((\widetilde{\mathcal{U}}^{(N), \mathrm{an}}_I)^{-1}\circ \widetilde{\mathcal{U}}^{(N), \mathrm{an}}_n(A)\) is closed for \(I=\{1,\dots,n\}\). This is done by induction on \(n\) as in the proof of [19]. Note that \(\widetilde{\mathcal{U}}^{(N)}_n(\mathbb{C})\) has finite nonempty fibers (for instance, this follows easily by taking complex points in 9 and using 14 to argue that \(\widetilde{\mathcal{U}}^{(N)}_n(\mathbb{C})\) is a pullback of \(\mathcal{U}_n(\mathbb{C})\)). Therefore \(\widetilde{\mathcal{U}}^{(N), \mathrm{an}}_n\) has finite nonempty fibers as well. Because \(\mathrm{Gr}^{(N), \mathrm{an}}_{G, X^n}\) is Hausdorff, \(\widetilde{\mathcal{U}}^{(N), \mathrm{an}}_n\) is a perfect quotient. An analogous proof shows the statement for \(\widetilde{\mathcal{U}}^+_{n}\) (recall that \(\mathrm{L}^+G_{X^n}^\mathrm{an}\) is Hausdorff because limit of analytifications of quasi-projective complex schemes). ◻

Proof. By 14, it does after applying \(\mathrm{Fgt}_{\mathrm{str}, \mathrm{top}}\circ (-)^\mathrm{an}_{\mathrm{PSh}\mathrm{Str}, \mathrm{lft}}\): so \[\label{underlying-sets-square} \begin{tikzcd}[column sep=2cm] {\mathrm{Gr}^{(N)}_{G, X^n}(\mathbb{C})} \arrow[r, "{\tilde{\mathcal{U}}_n^{(N)}(\mathbb{C})}"] \arrow[d, "{p_{n}^{(N)}(\mathbb{C})}"] & {\mathrm{Gr}^{(N)}_{G, \mathrm{Ran}_{\leq n}(X)}(\mathbb{C})} \arrow[d, "{p_{\mathrm{Ran}_{\leq n}}^{(N)}(\mathbb{C})}"] \\ {X^n(\mathbb{C})} \arrow[r, "{\mathcal{U}_n(\mathbb{C})}"] & {\mathrm{Ran}_{\leq n}(X)(\mathbb{C})} \end{tikzcd}\tag{10}\] is cartesian in \(\mathrm{Set}\). To show that it was already cartesian in \({\mathrm{Top}}\) (so before forgetting the topology), it suffices to prove that \({\mathrm{Gr}^{(N)}_{G, X^n}(\mathbb{C})}\) was endowed with the fiber product topology. Namely that a subset \(A\) of \(\mathrm{Gr}^{(N)}_{G, X^n}(\mathbb{C})\) is closed if and only if \(p_n^{(N), \mathrm{an}}(A)\) and \(\widetilde{\mathcal{U}}_n^{(N),\mathrm{an}}(A)\) are both closed. This is true because:

• \(p_n^{(N),\mathrm{an}}\) is a proper map by [16], hence its analytification is a closed map;

• \(\widetilde{\mathcal{U}}_n^{(N),\mathrm{an}}(A)\) is closed by 46.

At the level of the stratifying posets, the diagram is \[\begin{tikzcd} \mathfrak{s}^{(N), \mathrm{an}}_n \arrow[r, hook] \arrow[d] & \mathop{\mathrm{colim}}_{|I|\leq n} \mathfrak{s}^{(N), \mathrm{an}}_I \arrow[d] \\ \mathrm{Inc}_n \arrow[r, hook, dashed] & \mathop{\mathrm{colim}}_{|I|\leq n} \mathrm{Inc}_I. \end{tikzcd}\] This is cartesian by 9 . ◻

Proof. We noticed and used already that \(X^{n, \mathrm{an}}, {\mathrm{Gr}^{(N),\mathrm{an}}_{G,X^n}}\) and \((\mathrm{L}^+G_{X^n}, \mathrm{Inc}_n)^\mathrm{an}_{\mathrm{Str}\mathrm{Sch}}\) are locally compact Hausdorff spaces, because they are (limits of) analytifications of quasi-projective complex schemes. Since \(\mathcal{U}^\mathrm{an}_n, \widetilde{\mathcal{U}}^{(N),\mathrm{an}}_n\) and \(\widetilde{\mathcal{U}}^{+,\mathrm{an}}_n\) are perfect quotients by 46, we can conclude applying 42. ◻

Proof. By 48 we can apply 19 to the fiber product \[(\mathrm{L}^+G_{\mathrm{Ran}(X)}, \mathrm{Inc}_\mathrm{Ran})^\mathrm{an}_{\mathrm{PSh}\mathrm{Str}} \underset{(\mathrm{Ran}(X), \mathrm{Inc}_\mathrm{Ran})^\mathrm{an}_{\mathrm{PSh}\mathrm{Str},\mathrm{lft}}}{\times} (\mathrm{L}^+G_{\mathrm{Ran}(X)}, \mathrm{Inc}_\mathrm{Ran})^\mathrm{an}_{\mathrm{PSh}\mathrm{Str}}\] and by 44 we can rewrite it as \[\label{application-of-Harpaz} \mathop{\mathrm{colim}}_{n\geq 1} \left( (\mathrm{L}^+G_{\mathrm{Ran}_{\leq n}(X)} , \mathrm{Inc}_{\leq n})^\mathrm{an}_{\mathrm{PSh}\mathrm{Str}} \underset{{(\mathrm{Ran}_{\leq n}(X), \mathrm{Inc}_{\leq n})^\mathrm{an}_{\mathrm{PSh}\mathrm{Str}, \mathrm{lft}}}}{\times} (\mathrm{L}^+G_{\mathrm{Ran}_{\leq n}(X)} , \mathrm{Inc}_{\leq n})^\mathrm{an}_{\mathrm{PSh}\mathrm{Str}} \right).\tag{11}\]

Let \(\mu^+_n\) be the multiplication on \((\mathrm{L}^+G_{X^n},\mathrm{Inc}_n)\). Consider the diagram \[\begin{tikzcd} (\mathrm{L}^+G_{X^n}, \mathrm{Inc}_n)^\mathrm{an}_{\mathrm{PSh}\mathrm{Str}} \underset{(X^{n, \mathrm{an}}, \mathrm{Inc}_n)}{\times} (\mathrm{L}^+G_{X^n}, \mathrm{Inc}_n)^\mathrm{an}_{\mathrm{PSh}\mathrm{Str}} \arrow[r, "{\mu^{+, \mathrm{an}}_n}"] \arrow[d, "\widetilde{\mathcal{U}}^{+,\mathrm{an}}_n \times_{\mathcal{U}_n^\mathrm{an}}\, \widetilde{\mathcal{U}}^{+,\mathrm{an}}_n"] & (\mathrm{L}^+G_{X^n}, \mathrm{Inc}_n)^\mathrm{an}_{\mathrm{PSh}\mathrm{Str}} \arrow[d, "\widetilde{\mathcal{U}}^{+,\mathrm{an}}_n"] \\(\mathrm{L}^+G_{\mathrm{Ran}_{\leq n}(X)}, \mathrm{Inc}_{\leq n})^\mathrm{an}_{\mathrm{PSh}\mathrm{Str}} \underset{\mathrm{Ran}^\mathrm{an}_{\leq n}}{\times} (\mathrm{L}^+G_{\mathrm{Ran}_{\leq n}(X)}, \mathrm{Inc}_{\leq n})^\mathrm{an}_{\mathrm{PSh}\mathrm{Str}} & (\mathrm{L}^+G_{\mathrm{Ran}_{\leq n}(X)}, \mathrm{Inc}_{\leq n})^\mathrm{an}_{\mathrm{PSh}\mathrm{Str}}, \end{tikzcd}\] where \(\mathrm{Ran}^\mathrm{an}_{\leq n}\) denotes \((\mathrm{Ran}_{\leq n}(X),\mathrm{Inc}_{\leq n})^\mathrm{an}_{\mathrm{PSh}\mathrm{Str}, \mathrm{lft}}\). By 43 the map \(\widetilde{\mathcal{U}}^{+, \mathrm{an}}_n \times_{\mathcal{U}_n^\mathrm{an}} \widetilde{\mathcal{U}}^{+, \mathrm{an}}_n\) is a perfect quotient. Therefore there is an arrow \(\mu^{+, \mathrm{an}}_{\leq n}\) completing the above diagram into a commutative square in \({\mathrm{Top}}\). Since at the level of the posets the diagram is \[\begin{tikzcd} \mathrm{Inc}_n \arrow[r, equal] \arrow[d] & \mathrm{Inc}_n \arrow[d] \\ \mathop{\mathrm{colim}}_{|I|\leq n} \mathrm{Inc}_I \arrow[r, equal, dashed] & \mathop{\mathrm{colim}}_{|I|\leq n} \mathrm{Inc}_I. \end{tikzcd}\] the map \(\mu^{+, \mathrm{an}}_{\leq n}\) is also stratified.

By 19, taking the colimit in \(n\), one recovers a well-defined continuous stratified group law on \((\mathrm{L}^+G_{\mathrm{Ran}(X)},\mathrm{Inc}_\mathrm{Ran})^\mathrm{an}_{\mathrm{PSh}\mathrm{Str}}\) over \((\mathrm{Ran}(X), \mathrm{Inc}_\mathrm{Ran})^\mathrm{an}_{\mathrm{PSh}\mathrm{Str}, \mathrm{lft}}\). Analogously, if we apply 19 twice, the fiber product \[(\mathrm{L}^+G_{\mathrm{Ran}(X)}, \mathrm{Inc}_\mathrm{Ran})^\mathrm{an}_{\mathrm{PSh}\mathrm{Str}} \underset{(\mathrm{Ran}(X), \mathrm{Inc}_\mathrm{Ran})^\mathrm{an}_{\mathrm{PSh}\mathrm{Str},\mathrm{lft}}}{\times} (\mathrm{Gr}_{G,\mathrm{Ran}(X)}, \mathfrak{s}_\mathrm{Ran})^\mathrm{an}_{\mathrm{PSh}\mathrm{Str},\mathrm{lft}}\] is isomorphic to \[\mathop{\mathrm{colim}}_{n\geq 1, N\geq 0} \left( (\mathrm{L}^+G_{\mathrm{Ran}_{\leq n}(X)}, \mathrm{Inc}_{\leq n})^\mathrm{an}_{\mathrm{PSh}\mathrm{Str}} \underset{(\mathrm{Ran}_{\leq n}(X), \mathrm{Inc}_{\leq n})^\mathrm{an}_{\mathrm{PSh}\mathrm{Str},\mathrm{lft}}}{\times} (\mathrm{Gr}_{G,\mathrm{Ran}_{\leq n}(X)}^{(N)}, \mathfrak{s}_{\leq n}^{(N)})^\mathrm{an}_{\mathrm{PSh}\mathrm{Str},\mathrm{lft}} \right).\] And in the same way, using that \(\widetilde{\mathcal{U}}^{+, \mathrm{an}}_n \times_{\mathcal{U}^\mathrm{an}_n}\widetilde{\mathcal{U}}^{(N), \mathrm{an}}_n\) is a perfect quotient, one recovers a continuous stratified group action of \((\mathrm{L}^+G_{\mathrm{Ran}(X)},\mathrm{Inc}_\mathrm{Ran})^\mathrm{an}_{\mathrm{PSh}\mathrm{Str}}\) on \((\mathrm{Gr}_{G,\mathrm{Ran}(X)}, \mathfrak{s}_\mathrm{Ran})^\mathrm{an}_{\mathrm{PSh}\mathrm{Str},\mathrm{lft}}\) over \(( \mathrm{Ran}(X), \mathrm{Inc}_{\mathrm{Ran}} )^\mathrm{an}_{\mathrm{PSh}\mathrm{Str},\mathrm{lft}}\). ◻

Proof. Let \(A\) be an \(R\)-algebra locally of finite type, \(\tau: \mathop{\mathrm{\mathrm{Spec}}}A \rightarrow \mathop{\mathrm{\mathrm{Spec}}}R\). Denote by \(f_A\) the base change of \(f\) to \(X_A\) by \(\tau\). If \(y_I\) is the composition \[\mathop{\mathrm{\mathrm{Spec}}}A \xhookrightarrow{x_I \times \mathrm{id}_A }X^I_A\xrightarrow{(f_A^{-1})^I} X_A^I \xrightarrow{\mathrm{pr}_{X^I}} X^I,\] define \[\Phi_{f,A}: (\mathrm{Gr}_{G,X^I})_R (A) \to (\mathrm{Gr}_{G,X^I})_R (A), \quad ( x_I, \mathcal{F},\alpha, \tau ) \mapsto (y_I,f^*_A\mathcal{F},f^*_A\alpha,\tau).\] This is well-defined because \(f_A^*(\mathcal{F}|_{X_A\setminus \Gamma_{x_I}})\simeq (f_A^*\mathcal{F})|_{X_A\setminus \Gamma_{y_I}}\) and \(f^*_A\mathcal{T}_{G,X_A}\simeq\mathcal{T}_{G,X_A}\). Since the formation of \(f_A\) is natural in \(A\), so is \(\Phi_{f, A}\). ◻

Proof. For the sake of notation, we write the proof for \(R=\mathbb{C}\). The general case is analogous. The map \(f^I:X^I\to X^I\) respects the incidence stratification on \(X^I\), so \(\Phi_f\) restricts to \(\left ( \prod_{j=1}^{|J|}\mathrm{Gr}_{G,X} \right )_\mathrm{disj}\). Moreover, since pulling back along an automorphism commutes with the operation of gluing torsors, for any \(X^\phi\subset X^I\), \(\Phi_f\) commutes with the factorization isomorphism 25 \[\mathfrak{f}_\phi : \left ( \prod_{j=1}^{|J|} \mathrm{Gr}_{G,X} \right )_{\mathrm{disj}} \xrightarrow{\sim} \mathrm{Gr}_{G, X^\phi}.\] Since this isomorphism restricts to each \(N\)-level, it is enough to check the statement for \(\mathrm{Gr}_{G, X}^{(N)}\).

Let us then consider the stratum \(\mathrm{Gr}_{G,X,\nu}\), \(\nu\leq N\), together with the isomorphism \[\mathfrak{bl} : \widehat{X}\times^{\underline{\mathrm{Aut}}_\mathbb{C}\mathbb{C}\llbracket t \rrbracket} \mathrm{Gr}_{G,\nu} \xrightarrow{\sim} \mathrm{Gr}_{G,X,\nu}\] defined in 93. Let \(A\) be a \(\mathbb{C}\)-algebra and pick an \(A\)-point \[[(x,\eta,\widetilde{\mathcal{F}},\widetilde{\alpha})] \in \left( (\widehat{X}\times \mathrm{Gr}_{G,\nu} )/ \underline{\mathrm{Aut}}_\mathbb{C}\mathbb{C}\llbracket t \rrbracket\right) (A).\] Let \((x,\mathcal{F},\alpha)\) be the image of \([(x,\eta,\widetilde{\mathcal{F}},\widetilde{\alpha})]\) in \(\mathrm{Gr}_{G,X,\nu}\), characterized by 23 as the pair \((\mathcal{F},\alpha)\) such that (see the notation in [formalcoordandtorsorrecall]) \[\eta^*i_{\widehat{x}}^*\mathcal{F} \simeq\widetilde{\mathcal{F}}, \quad (i_{\widehat{x}}\circ \eta)|_{\mathop{\mathrm{\mathrm{Spec}}}A(\!(\!t\!)\!)}^* \alpha \simeq\widetilde{\alpha}.\] Now \(\Phi_{f,A}(x,\mathcal{F},\alpha)=( f^{-1} x, f_A^*\mathcal{F},f_A^*\alpha )\). In particular \[\label{isomorphismonpairs} (f_A^{-1}\circ i_{\widehat{x}}\circ \eta)^* f_A^*\mathcal{F} \simeq\widetilde{\mathcal{F}}, \quad (f_A^{-1}\circ i_{\widehat{x}}\circ \eta)|_{\mathop{\mathrm{\mathrm{Spec}}}A(\!(\!t\!)\!)}^* f_A^*|_{\mathop{\mathrm{\mathrm{Spec}}}A(\!(\!t\!)\!)} \alpha \simeq\widetilde{\alpha}.\tag{12}\] Using the cartesian diagram \[\label{diagramforeta} \begin{tikzcd} \mathop{\mathrm{\mathrm{Spec}}}A\llbracket t \rrbracket \arrow[d, equal] \arrow[r, "{\widehat{f}_{A, x}^{-1}\circ \eta}"] & \underline{\mathop{\mathrm{\mathrm{Spec}}}}_{X_A}( \widehat{ \mathcal{O}}_{\Gamma_{f^{-1} x} }) \arrow[d, "{\widehat{f}_{A, x}}"] \arrow[r,"{i_{\widehat{f^{-1} x}}}"] & X_A \arrow[d,"{f_A}"] \\ \mathop{\mathrm{\mathrm{Spec}}}A\llbracket t \rrbracket \arrow[r,"\eta"] & \underline{\mathop{\mathrm{\mathrm{Spec}}}}_{X_A}( \widehat{ \mathcal{O}}_{\Gamma_x}) \arrow[r,"{i_{\widehat{x}}}"] & X_A, \end{tikzcd}\tag{13}\] equalities in 12 can be rewritten as \[(\widehat{f}_{A,x}^{-1}\circ\eta)^* i_{\widehat{f^{-1}x}}^* ( f_A^*\mathcal{F}) \simeq\widetilde{\mathcal{F}} , \quad (\widehat{f}_{A,x}^{-1}\circ\eta|_{\mathop{\mathrm{\mathrm{Spec}}}R(\!(\!t\!)\!)} )^* i_{\widehat{f^{-1}x}}^* (f_A^*\alpha ) \simeq\widetilde{\alpha},\] which means that \(\Phi_f\) acts on \(\widehat{X} \times \mathrm{Gr}_{G,\nu} / \underline{\mathrm{Aut}}_\mathbb{C}\mathbb{C}\llbracket t \rrbracket\) sending \[\label{mapPhifatloclevel} [(x,\eta, \widetilde{\mathcal{F}}, \widetilde{\alpha})] \mapsto [( \mathrm{pr}_{X}f_A^{-1}x, \widehat{f}_{A,x}^{-1}\circ\eta, \widetilde{\mathcal{F}}, \widetilde{\alpha} )]\tag{14}\] Passing to the sheafification, this implies that \(\Phi_f\) only modifies the first component of \(\widehat{X}\times^{\underline{\mathrm{Aut}}_\mathbb{C}\mathbb{C}\llbracket t \rrbracket} \mathrm{Gr}_{G,\nu}\) and therefore preserves the stratification. ◻

Proof. Consider the stratified map \(\Psi_{[0,1]}\) from 17 . By condition 1, for any \(t\in [0,1]\) the image of \(\Psi_{[0,1]}(t,-)|_{\mathrm{Gr}_{G, D^I}}\) lies all in \(\mathrm{Gr}_{G,D^I}\). Moreover, condition 2 reads as \[\Psi_{[0,1]}(0,-)|_{\mathrm{Gr}_{G, D^I}} = \mathrm{id}_{\mathrm{Gr}_{G,D^I}}, \quad \mathrm{and} \quad \mathrm{Im}\left(\Psi_{[0,1]}(1,-)|_{\mathrm{Gr}_{G,{D}^I}}\right) \subset \mathrm{Gr}_{G,{D'}^I} \xhookrightarrow{i_I} \mathrm{Gr}_{G,D^I}.\] Therefore, the map \(\Psi_{[0,1]}|_{\mathrm{Gr}_{G,D^I}}\) gives a stratified isotopy between \(\mathrm{id}_{\mathrm{Gr}_{G,D^I}}\) and \(i_I \circ \Psi_{[0,1]}(1,-)|_{\mathrm{Gr}_{G,D^I}}\).

Consider now \(\Psi_{[0,1]}(1,-)|_{\mathrm{Gr}_{G,D^I}} \circ i_I\) which is the same as \(\Psi_{[0,1]}(1,-)|_{\mathrm{Gr}_{G,{D'}^I}}\). Again by condition 1, for any \(z\) the image of \(\Psi_{[0,1]}(t,-)|_{\mathrm{Gr}_{G,{D'}^I}}\) is all contained in \(\mathrm{Gr}_{G,{D'}^I}\). Then \[\Psi_{[0,1]}|_{\mathrm{Gr}_{G,{D'}^I}}: \left( [0,1]\times \mathrm{Gr}_{G,{D'}^I}, \mathrm{triv}\times \mathfrak{s}_I^\mathrm{an}\right) \to \left(\mathrm{Gr}_{G,{D'}^I}, \mathfrak{s}_I^\mathrm{an}\right)\] gives a stratified isotopy between \(\mathrm{id}_{\mathrm{Gr}_{G,{D'}^I}}\) and \(\Psi_{[0,1]}(1,-)|_{\mathrm{Gr}_{G,D^I}} \circ i_I\).

Therefore \(\Psi_{[0,1]}(1,-)|_{\mathrm{Gr}_{G, D^I}} : \mathrm{Gr}_{G,D^I}\to \mathrm{Gr}_{G,{D'}^I}\) is a stratified homotopy inverse of the inclusion \(i_I:\mathrm{Gr}_{G,{D'}^I}\hookrightarrow \mathrm{Gr}_{G,D^I}\).

The proof for \(i_I^{(N)}\) is analogous (thanks to 51). ◻

Proof. Consider the map \[F: \mathbb{A}^1_\mathbb{C}\to \underline{\mathrm{End}}_\mathbb{C}(\mathbb{C}[z]) \simeq\underline{\mathrm{End}}_{\mathbb{C}}(\mathbb{A}^1_\mathbb{C})\] defined at the level of \(R\)-points as \[t\in R \mapsto F_t, \mathrm{ where } F_t(z) = z\Big(\frac{r'}{r}t + (1-t) \Big)+t\Big(z_0'-\frac{r'}{r}z_0\Big).\] Note that \(F_t^*\) is an automorphism of \(\mathbb{A}^1_R\) if and only if the scaling factor \(\lambda(t)= \frac{r'}{r}t + (1-t)\) is in \(R^{\times}=\mathbb{G}_{m,\mathbb{C}}(R)\). This happens if and only if \(\lambda(t)\) belongs to the open \(U\subseteq \mathbb{A}^1_\mathbb{C}\) obtained as the fiber product \[\begin{tikzcd} U \arrow[r] \arrow[d, hook] & \mathbb{G}_{m, \mathbb{C}} \arrow[d, hook] \\ \mathbb{A}^1_\mathbb{C} \arrow[r, "{\lambda}"] & \mathbb{A}^1_\mathbb{C}. \end{tikzcd}\] If \(t\in \mathbb{C}\), then \(\lambda(t)\notin \mathbb{C}^{\times}\) if and only if \(t=\frac{r}{r-r'}\): since \(r>r'\), then \([0,1]\subset U^{\mathrm{an}}\). Then \(F|_U\) is an algebraic isotopy in the sense of 52 and it satisfies the hypotheses of 56. ◻

Proof. With 58 at hand, one can argue as in 53. That is, given any algebraic isotopy \(U\to \underline{\mathrm{Aut}}_\mathbb{C}(X)\), we obtain a stratified map \[\left( U^\mathrm{an}\times\mathrm{Gr}_{G,\mathrm{Ran}(X)}^\mathrm{an}, \mathrm{triv}\times\mathfrak{s}_\mathrm{Ran}^\mathrm{an} \right) \to \left( \mathrm{Gr}_{G,\mathrm{Ran}(X)}^\mathrm{an}, \mathrm{triv}\times\mathfrak{s}_\mathrm{Ran}^\mathrm{an}\right)\] and hence a stratified isotopy \[\Psi^\mathrm{Ran}_{[0,1]}: \left( [0,1]\times\mathrm{Gr}_{G,\mathrm{Ran}(X)}^\mathrm{an}, \mathrm{triv}\times\mathfrak{s}_\mathrm{Ran}^\mathrm{an} \right) \to \left( \mathrm{Gr}_{G,\mathrm{Ran}(X)}^\mathrm{an}, \mathfrak{s}_\mathrm{Ran}^\mathrm{an}\right).\] Let \(X\) be \(\mathbb{A}^1_\mathbb{C}\): the analogues of 56 and 57 for \(\Psi^\mathrm{Ran}_{[0,1]}\) are deduced in the same way as above. ◻

Proof. By their definitions, the automorphism \(\Phi\) acts on the \(X^I\)-coordinate of \(\mathrm{Gr}_{G, X^I}\) in the same way as \(\Phi^{\mathrm{L}^+G}\) acts on the \(X^I\)-coordinate of \(\mathrm{L}^+G_{X^I}\). Therefore they can be combined together to obtain \[\begin{gather} \Phi^{\mathrm{L}^+G}\times_{X^I} \Phi: \underline{\mathrm{Aut}}_\mathbb{C}(X)\to \underline{\mathrm{Aut}}_\mathbb{C}(\mathrm{L}^+G_{X^I}\times_{X^I}\mathrm{Gr}_{G,X^I}). \end{gather}\] Similarly, for any \(N\in\mathbb{N}, m\geq m_{N,I}\), we have \[\begin{gather} \Phi^{\mathrm{L}^{m}G}\times_{X^I} \Phi: \underline{\mathrm{Aut}}_\mathbb{C}(X)\to \underline{\mathrm{Aut}}_\mathbb{C}(\mathrm{L}^{m}G_{X^I}\times_{X^I}\mathrm{Gr}_{G,X^I}^{(N)}). \end{gather}\] Let \(F\) and \(U\) be as in the proof of 57 and consider the evaluation morphism for \(\mathrm{L}^+G_{X^I}\times_{X^I}\mathrm{Gr}_{G,X^I}\). Then we get \[\label{mapforequivariantstatement} \mathrm{ev}\circ \left(\left(\Phi^{\mathrm{L}^+G}\times_{X^I} \Phi\right)\circ F, \mathrm{id}\right): U \times \mathrm{L}^+G_{X^I}\times_{X^I}\mathrm{Gr}_{G,X^I} \rightarrow \mathrm{L}^+G_{X^I}\times_{X^I}\mathrm{Gr}_{G,X^I}.\tag{18}\] In particular, since the stratification of \(\mathrm{L}^+G_{X^I}\) is controlled by \(\mathrm{Inc}_I\), we have that the map 18 respects the stratifications. Therefore when we pass to the analytifications, by applying 19 in the usual way we get \[\left( U^\mathrm{an}\times \mathrm{L}^+G_{X^I}^\mathrm{an}\times_{(X^\mathrm{an})^I}\mathrm{Gr}_{G,X^I}^\mathrm{an}, \mathrm{triv}\times \mathfrak{s}_I^\mathrm{an}\right) \rightarrow \left( \mathrm{L}^+G_{X^I}^\mathrm{an}\times_{(X^\mathrm{an})^I}\mathrm{Gr}_{G,X^I}^\mathrm{an}, \mathfrak{s}_I^\mathrm{an}\right).\]

Let \(X\) be the affine line. Restricting to \([0,1]\subset U^\mathrm{an}\), we finally get \[\Psi^{\mathrm{equiv}}_{[0,1]} \mathrel{\vcenter{:}}= \mathrm{ev} \circ \left( \Phi^{\mathrm{L}^+G, \mathrm{an}}_{[0,1]} \times_{\mathbb{C}^I} \Phi^\mathrm{an}_{[0,1]} , \mathrm{id}\right).\] By its definition, \(\Psi^{\mathrm{equiv}}_{[0,1]}(1,-)\) restricts to \[\left(\mathrm{L}^+G_{D^I} \times_{D^I} \mathrm{Gr}_{G,D^I},\mathfrak{s}_I^\mathrm{an}\right) \to \left( \mathrm{L}^+G_{{D'}^I} \times_{{D'}^I}\mathrm{Gr}_{G,{D'}^I},\mathfrak{s}_I^\mathrm{an}\right)\] and, by the same proof of 56 and 57, it gives a stratified homotopy inverse to \(i^+_I\times i_I\).

Therefore it remains to show that, for any \(t\in [0,1]\), \(\Psi^{\mathrm{equiv}}_{[0,1]}(t,-)|_{\mathrm{Gr}_{G, D^I}}\) and \(\Psi_{[0,1]}(t,-)|_{\mathrm{Gr}_{G, D^I}}\) fit in the commutative diagram \[\begin{tikzcd}[column sep=3cm] \mathrm{L}^+G_{D^I}\times_{D^I}\mathrm{Gr}_{G, D^I} \arrow[r, "{\Psi^{\mathrm{equiv}}_{[0,1]}(t,-)|_{\mathrm{Gr}_{G, D^I}}}"] \arrow[d, "{\mathrm{act}_I}"] & \mathrm{L}^+G_{D^I}\times_{D^I}\mathrm{Gr}_{G,D^I} \arrow[d, "{\mathrm{act}_I}"] \\ \mathrm{Gr}_{G,D^I} \arrow[r, "{\Psi_{[0,1]}(t,-)|_{\mathrm{Gr}_{G, D^I}}}"] & \mathrm{Gr}_{G, D^I}. \end{tikzcd}\] This, in turn, is implied by checking that for any \(f\in \mathrm{Aut}_\mathbb{C}(X)\) and each \(X^\phi\), the diagram \[\begin{tikzcd}[column sep=3cm] \mathrm{L}^+G_{X^I}|_{X^\phi}\times_{X^\phi}\mathrm{Gr}_{G, X^\phi} \arrow[r, "{(\Phi^{\mathrm{L}^+G}_f \times_{X^I} \Phi_f)|_{X^\phi}}"] \arrow[d] & \mathrm{L}^+G_{X^I}|_{X^\phi}\times_{X^\phi}\mathrm{Gr}_{G, X^\phi} \arrow[d] \\ \mathrm{Gr}_{G, X^\phi} \arrow[r, "{\Phi_f|_{X^\phi}}"] & \mathrm{Gr}_{G, X^\phi}, \end{tikzcd}\] is well-defined and commutes. As done in the proof of 51, by the factorization property 25 , it is enough to deal with the case \(I=\{*\}\) using the formal coordinates \[\mathfrak{bl} : \widehat{X}\times^{\underline{\mathrm{Aut}}_\mathbb{C}\mathbb{C}\llbracket t \rrbracket}\mathrm{Gr}_{G} \xrightarrow{\sim}\mathrm{Gr}_{G,X}.\] Recall that at the level of the presheaf quotient \(( \widehat{X}\times\mathrm{Gr}_{G} )/\underline{\mathrm{Aut}}_\mathbb{C}\mathbb{C}\llbracket t \rrbracket\), the map \(\Phi_f\) sends \[[(x,\eta,\widetilde{\mathcal{F}},\widetilde{\alpha})]\mapsto [(f^{-1}x, \widehat{f}^{-1}_x\circ \eta, \widetilde{\mathcal{F}},\widetilde{\alpha})]\] (see equation 14 ). Therefore given \((x,g)\in \mathrm{L}^+G_X\), on one side we have \[\begin{tikzcd}[column sep=3cm] (x,g), [(x,\eta,\widetilde{\mathcal{F}},\widetilde{\alpha})] \arrow[d, "{\mathrm{act}_{\{*\}}}"] & \\ {[(x,\eta,\widetilde{\mathcal{F}}, \eta^* g|_{\widetilde{\Gamma}_x\setminus \Gamma_x}\circ (\eta^{-1})^*\widetilde{\alpha})]} \arrow[r, "{\Phi_f}"] & (f^{-1}x, \widehat{f}^{-1}_x\circ \eta, \widetilde{\mathcal{F}}, \eta^* g|_{\widetilde{\Gamma}_x\setminus \Gamma_x}\circ (\eta^{-1})^*\widetilde{\alpha}). \end{tikzcd}\] On the other side, we have \[\begin{tikzcd}[column sep=3cm] (x,g),(x,\eta,\widetilde{\mathcal{F}},\widetilde{\alpha}) \arrow[r, "{\Phi_f^{\mathrm{L}^+G}\times_X \Phi_f}"] & (f^{-1}x, \widehat{f}_x^*g),(f^{-1}x, \widehat{f}^{-1}_x\circ \eta, \widetilde{\mathcal{F}},\widetilde{\alpha}) \arrow[d, "{\mathrm{act}_{\{*\}}}"] \\ & (f^{-1}x, \widehat{f}^{-1}_x\circ \eta, \widetilde{\mathcal{F}}, (\widehat{f}^{-1}_x\circ \eta)^* (\widehat{f}_x^*g)|_{\widetilde{\Gamma}_{f^{-1}x}\setminus \Gamma_{f^{-1}x}}\circ ((\widehat{f}^{-1}_x\circ \eta)^{-1})^*\widetilde{\alpha}). \end{tikzcd}\] One concludes computing explicitly the last term: \[\begin{align} (\widehat{f}^{-1}_x\circ \eta)^* (\widehat{f}_x^*g)|_{\widetilde{\Gamma}_{f^{-1}x}\setminus \Gamma_{f^{-1}x}}\circ ((\widehat{f}^{-1}_x\circ \eta)^{-1})^*\widetilde{\alpha} = & \; \eta^* (\widehat{f}^{-1}_x)^*(\widehat{f}_x^*g)|_{\widetilde{\Gamma}_{f^{-1}x}\setminus \Gamma_{f^{-1}x}}\circ \widehat{f}_x^* (\eta^{-1})^*\widetilde{\alpha} \\ =& \;\eta^* g|_{\widetilde{\Gamma}_{f^{-1}x}\setminus \Gamma_{f^{-1}x}}\circ (\eta^{-1})^*\widetilde{\alpha}. \end{align}\] The analogous statement holds for the \((N,m)\)-truncated objects by an identical argument. ◻

Proof. The only difference with respect to the previous proof is that one builds the map \(\Psi^{\mathrm{equiv},\mathrm{Ran}}_{[0,1]}\) in the same way as 49, by filtering \(\mathrm{Gr}_{G,\mathrm{Ran}(X)}^\mathrm{an}\) and then inducing maps on perfect quotients. Therefore, by construction, \(\Psi^{\mathrm{equiv}, \mathrm{Ran}}_{[0,1]}\) agrees with the action of \(\mathrm{L}^+G_{\mathrm{Ran}(X)}^\mathrm{an}\). ◻

Proof. Let \(x\) and \(y\) be two objects of \(\mathcal{C}\). Recall that the topological space \(\mathop{\mathrm{Hom}}^R_{\mathrm{N}(\mathcal{C})}(x,y)\) of the right homomorphisms (see its definition at [21]) describes the homotopy type \(\mathrm{Map}_{\mathrm{N}(\mathcal{C})}(x,y)\). Furthermore the topological space \(\mathop{\mathrm{Hom}}^R_{\mathrm{N}(\mathcal{C})}(x,y)\) is a discrete space in bijection with \(\mathop{\mathrm{Hom}}_{\mathcal{C}}(x,y)\). In particular, the conditions on (products of) mapping subspaces involved in the definition of \(\infty\)-operad for \(\mathrm{N}(\mathcal{C})\rightarrow \mathrm{N}(\mathrm{Fin}_*)\) all translate in conditions on (product of) subsets of morphisms in \(\mathcal{C}\). ◻

Proof. The aforementioned [3] rests upon [3]. Both lemmas hold if one replaces \(\mathrm{Disk}(\mathbb{R}^2)^\otimes\) with \(\mathrm{MDisk}(\mathbb{R}^2)^\otimes\): indeed, they rely on the categorical Seifert-Van Kampen Theorem [3], and therefore one can consider any subbase of the collection of all disks of \(\mathbb{R}^2\). This means that [3] holds with \(\mathrm{MDisk}(\mathbb{R}^2)^\otimes_\mathrm{nu}\) in place of \(\mathrm{Disk}(\mathbb{R}^2)^\otimes_\mathrm{nu}\). ◻

Proof. First of all, let us define a functor of 1-categories \(\mathfrak{G}: \mathrm{MDisk}(\mathbb{R}^2)^\otimes_\mathrm{nu}\to \mathrm{Str}{\mathrm{Top}}\), sending \[(D_1,\dots, D_n)\mapsto \prod_{i=1}^n\left( \mathrm{Gr}_{\mathrm{Ran}(D_i)}, \mathfrak s_{\mathrm{Ran}}^\mathrm{an}\right).\] On morphisms, we define it by steps. For maps \((D_1,\dots, D_n)\rightarrow D'\) over the inert morphism \(\rho_i\), it is defined as the projection on the \(i\)-th component followed by the inclusion \(i_{\mathrm{Ran}}\): \[\prod_{j=1}^n \left(\mathrm{Gr}_{G,\mathrm{Ran}(D_j)}, \mathfrak{s}_{\mathrm{Ran}}^\mathrm{an}\right) \xrightarrow{\pi_i} \left(\mathrm{Gr}_{G,\mathrm{Ran}(D_i)}, \mathfrak{s}_{\mathrm{Ran}}^\mathrm{an}\right) \xrightarrow{i_{\mathrm{Ran}}} \left( \mathrm{Gr}_{G,\mathrm{Ran}(D')}, \mathfrak{s}_{\mathrm{Ran}}^\mathrm{an}\right).\] Consider now maps \((D_1,\dots, D_n)\rightarrow D'\) over the active morphism \[a_n: \langle n\rangle \to \langle 1 \rangle, \quad *\mapsto *, \langle n\rangle\setminus \{*\} \mapsto 1,\] where \(D_i\)’s are then all disjoint and contained in \(D'\). Let \(I_1,\dots,I_n\in \mathrm{Fin}_{\geq 1, \mathrm{surj}}\), and consider \(({\mathbb{C}}^{I_1}\times \dots \times \mathbb{C}^{I_n})_\mathrm{disj}\) (see definition in 96). Fix \(N\geq 0\). By using the factorization property 27 and then analytifying (recall that \((-)^{\mathrm{an}}_{\mathrm{Str}, \mathrm{lft}}\) preserves finite limits), consider the isomorphism \[\mathfrak{f}_{(I_i)_{i=1}^n}^{(N), \mathrm{an}} : \prod_{i=1}^n \left( \mathrm{Gr}_{G,\mathbb{C}^{I_i}}^{(N)}, \mathfrak{s}_{I_i}^{(N)} \right)_\mathrm{disj} \to \left( \mathrm{Gr}_{G, \mathbb{C}^{ \sqcup_i I_i}}^{(N)}, \mathfrak{s}_{\sqcup_i I_i}^{(N)} \right).\] Restricting to \(\prod_i D_i^{I_i} = (\prod_i D_i^{I_i})_{\mathrm{disj}} \subset (\mathbb{C}^{I_1}\times\mathbb{C}^{I_2})_{\mathrm{disj}}\) on the LHS and to \((D')^{\sqcup_i I_i} \subseteq \mathbb{C}^{\sqcup_i I_i}\) on the RHS induces a map \[\prod_{i=1}^n \left( \mathrm{Gr}_{G,D_i^{I_i}}^{(N)}, \mathfrak{s}_{I_i}^{(N)} \right) \to \left( \mathrm{Gr}_{G, {D'}^{\sqcup_i I_i}}^{(N)}, \mathfrak{s}_{\sqcup_i I_i}^{(N)} \right).\] Thanks to 18, taking the colimit of these maps in \(N\) gives in turn a map \[\prod_{i=1}^n \left( \mathrm{Gr}_{G,D_i^{I_i}}, \mathfrak{s}_{I_i} \right) \to \left( \mathrm{Gr}_{G, {D'}^{\sqcup I_i}}, \mathfrak{s}_{\sqcup I_i} \right).\] Post-composing by the quotient map into \(\mathrm{Gr}_{G, \mathrm{Ran}_{\leq |\sqcup_i I_i|}(D')}\), we thus obtain a morphism \[\label{mapinran2n} \prod_{i=1}^n \left( \mathrm{Gr}_{G,D_i^{I_i}}, \mathfrak{s}_{I_i} \right) \to \left( \mathrm{Gr}_{G, \mathrm{Ran}_{\leq |\sqcup_i I_i|}(D')}, \mathfrak{s}_{\mathrm{Ran}} \right).\tag{19}\] Recall that the relation which defines the quotient map \(\left( \mathrm{Gr}_{G,D^{I}}, \mathfrak{s}_I \right) \to \left( \mathrm{Gr}_{G, \mathrm{Ran}_{\leq |I|}(D)}, \mathfrak{s}_{\mathrm{Ran}} \right)\) is \[(x_I, \mathcal{F},\alpha) \sim (x'_I, \mathcal{F}', \alpha') \iff \{x_1,\dots, x_{|I|}\}=\{x_1',\dots, x_{|I|}'\}, \mathcal{F}\simeq \mathcal{F}', \alpha\simeq \alpha'.\] Since also the product map \(\prod_{i=1}^n \mathrm{Gr}_{G, D_i^{I_i}} \rightarrow \prod_{i=1}^n \mathrm{Gr}_{G, \mathrm{Ran}_{\leq |I_i|}(D_i)}\) is a quotient map (by 48 and 43), the morphism 19 factors as \[\prod_{i=1}^n \left( \mathrm{Gr}_{G, \mathrm{Ran}_{\leq |I_i|}(D_i)}, \mathfrak{s}_{\mathrm{Ran}_{\leq |I_i|}} \right) \to \left( \mathrm{Gr}_{G, \mathrm{Ran}_{\leq |\sqcup_i I_i|}(D')}, \mathfrak{s}_{\mathrm{Ran}_{\leq |\sqcup_i I_i|}} \right).\] Note that this map is also stratified by the same argument at the end of the proof of 49. We can now use 18 again and obtain a continuous map at the level of \(\mathrm{Ran}\)’s: \[\prod_{i=1}^n \left( \mathrm{Gr}_{G, \mathrm{Ran}(D_i)}, \mathfrak{s}_\mathrm{Ran} \right) \to \left( \mathrm{Gr}_{G, \mathrm{Ran}(D')}, \mathfrak{s}_{\mathrm{Ran}} \right).\] Note also that this assignment on active morphisms respects composition, because the operation of gluing torsors via trivializations away from disjoint systems of points is associative (see the description in 95 and 96). Finally, note that any morphism in \(\mathrm{MDisk}(\mathbb{R}^2)^\otimes_\mathrm{nu}\) can be written uniquely as a product of inert morphisms followed by a product of active morphisms.

Let now \(\mathfrak{G}_{[W^{-1}]}:\mathrm{MDisk}(\mathbb{R}^2)^\otimes_\mathrm{nu}\to \mathrm{Str}{\mathrm{Top}}[W^{-1}]\) be the functor obtained by postcomposing \(\mathfrak{G}\) with the (1-categorical) localization at \(W\). Taking the nerve we get a functor of \(\infty\)-categories \[\mathrm{N}(\mathfrak{G}_{[W^{-1}]}): \mathrm{N}(\mathrm{MDisk}(\mathbb{R}^2)^\otimes_\mathrm{nu}) \to \mathrm{N}(\mathrm{Str}{\mathrm{Top}}[W^{-1}]).\] It turns out that \(\mathrm{N}(\mathfrak{G}_{[W^{-1}]})\) is lax [3]: for any object \((D_1,\dots, D_n) \in \mathrm{MDisk}(\mathbb{R}^2)^\otimes_\mathrm{nu})_{\langle n\rangle}\) the inert maps \(\mathfrak{G}(\rho_i): \mathfrak{G}(D_1,\dots, D_n) \to \mathfrak{G}(D_i)\) exhibit \(\mathfrak{G}(D_1,\dots, D_n)\) as a product \(\prod_i \mathfrak{G}(D_i)\). Localizing by a class \(W\) of maps closed under products preserves products, and so does taking the nerve. Hence \(\mathrm{N}(\mathfrak{G}_{[W^{-1}]})\) is lax. By [3] we then obtain a map of \(\infty\)-operads \[\mathrm{Gr}_{\mathrm{Ran}(-)}^\otimes: \mathrm{N}(\mathrm{MDisk}(\mathbb{R}^2)^\otimes_\mathrm{nu}) \to \mathrm{N}(\mathrm{Str}{\mathrm{Top}}[W^{-1}])^\times\] such that \(\pi\circ \mathrm{Gr}_{\mathrm{Ran}(-)}^\otimes\) is \(\mathrm{N}(\mathfrak{G}_{[W^{-1}]})\), where \(\pi\) is defined in [3].

Thanks to 77, in order to conclude the proof it remains to check that \(\mathrm{Gr}_{\mathrm{Ran}(-)}^\otimes\) is locally constant: this is a property at the level of the \(\langle 1 \rangle\)-fiber, over which the functor \(\pi|_{\langle 1 \rangle}\) is the identity (see its definition in [3]). Therefore it is enough to check that \[\mathrm{N}(\mathfrak{G}_{[W^{-1}]})_{\langle 1\rangle}: \mathrm{N}(\mathrm{MDisk}(\mathbb{R}^2)^\otimes_\mathrm{nu})_{\langle 1 \rangle} \to \mathrm{N}(\mathrm{Str}{\mathrm{Top}}[W^{-1}])_{\langle 1 \rangle}\] sends any morphism to an isomorphism of \(\mathrm{N}(\mathrm{Str}{\mathrm{Top}}[W^{-1}])\). This is precisely 59 which says that, for \(D'\subset D\) metric disks, the induced map \(\mathrm{Gr}_{\mathrm{Ran}(D')} \xhookrightarrow{i_\mathrm{Ran}} \mathrm{Gr}_{\mathrm{Ran}(D)}\) is a stratified homotopy equivalence. ◻