G}$ contains the whole group $\prod_{j\notin I}G_{j}$ rather than being a
subgroup of $G_{I}$. Nevertheless, by Lemma 3.3.8 (ii,iv), we know
$|{{\mathcal{Q}}}^{\times G}|=|{\widehat{{{\mathcal{Q}}}}}^{\times
G}|\simeq|{{\mathcal{Q}}}|/G$ while, because $G$ acts on
${\mathcal{X}}_{\mathcal{V}}$ by inner automorphisms, we have
$|{\mathcal{X}}_{\mathcal{V}}^{\times G}|\simeq|{\mathcal{X}}_{\mathcal{V}}|$
by Lemma 3.3.3 (iii). Since these categories all have the same object space,
$|{\iota}^{\times G}|:|{\widehat{{{\mathcal{Q}}}}}^{\times
G}|\to|{\mathcal{X}}_{\mathcal{V}}|$ is a homeomorphism. This proves (iii). ∎

######  Remark 4.2.6.

  * (i)

The groupoid ${\mathcal{X}}_{\mathcal{V}}$ is not a groupoid completion of
${\iota}({{\mathcal{Q}}}^{\times G})$ (cf. Definition 3.2.7) because, as we
saw at the very end of the above proof, the isotropy groups of elements in
${\iota}({{\mathcal{Q}}}^{\times G})$ are much too large.

  * (ii)

By Step 4 of the above proof, we may identify ${\widehat{{{\mathcal{Q}}}}}$
with the (nonfull!) subcategory
${\mathcal{X}}_{\mathcal{V}}^{{\smallsetminus}G}$ of
${\mathcal{X}}_{\mathcal{V}}$ that has the same object space $X_{\mathcal{V}}$
as ${\mathcal{X}}_{\mathcal{V}}$ and that is generated as a groupoid by the
projections $\rho_{IJ}:{\widetilde{V}}_{IJ}\to V_{IJ}\subset V_{I}$ that
relate the different components of $X_{\mathcal{V}}$. Thus we have removed the
morphisms that depend explicitly on the group actions, except for those needed
to form the groupoid completion. ${\Diamond}$

The following lemma shows that ${\mathcal{X}}_{\mathcal{V}}$ is the groupoid
completion434343See Definition 3.2.7 of its subcategory
${\mathcal{B}}_{\mathcal{V}}$ that has the same objects but morphisms between
$V_{I},V_{J}$ only for $I\subset J$. Indeed, it is the groupoid completion in
a very strong sense: any functor from ${\mathcal{B}}_{\mathcal{V}}$ to a
groupoid ${\mathcal{Z}}$ must extend to ${\mathcal{X}}_{\mathcal{V}}$.
(Contrast this with the example in Remark 3.2.10.) This lemma is an essential
ingredient of the proof of Theorem 5.2.6 below.

######  Lemma 4.2.7.

Let ${\mathcal{X}}_{\mathcal{V}}$ be as in Proposition 4.1.1, and denote by
${\mathcal{B}}_{\mathcal{V}}\subset{\mathcal{X}}_{\mathcal{V}}$ the
subcategory with the same object space but morphisms from $V_{I}$ to $V_{J}$
only if $I\subset J$. (Thus ${\mathcal{B}}_{\mathcal{V}}={\iota}^{\times
G}({{\mathcal{Q}}}^{\times G})$.) Then every functor
$\phi:{\mathcal{B}}_{\mathcal{V}}\to{\mathcal{Z}}$ from
${\mathcal{B}}_{\mathcal{V}}$ to an arbitrary groupoid has a unique extension
to a functor ${\mathcal{X}}_{\mathcal{V}}\to{\mathcal{Z}}$.

###### Proof.

Denote an arbitrary element in ${\rm
Mor}_{{\mathcal{X}}_{\mathcal{V}}}(V_{I},V_{J})$ by $m_{IJ}$ and for
$I\supsetneq J$ define

(4.2.24) $\displaystyle\phi(m_{IJ}):=\bigl{(}\phi(m_{IJ}^{-1})\bigr{)}^{-1}.$

Since $(s\times t)(m^{-1})=(t\times s)(m)$, it follows from the assumptions
that $\phi$ is compatible with the structure maps $s,t$ of
${\mathcal{X}}_{\mathcal{V}}$ as well as preserving identities and inverses.
Hence it remains to show that if $m_{IJ}\circ m_{JK}=m_{IK}$ in
${\mathcal{X}}_{\mathcal{V}}$ then

(4.2.25) $\displaystyle\phi(m_{IJ})\circ\phi(m_{JK})=\phi(m_{IK}).$

We discuss the various cases in turn.

  * $\bullet$

If $I\subset J\subset K$ then the identity $m_{IJ}\circ m_{JK}=m_{IK}$ is a
relation in ${\mathcal{B}}_{\mathcal{V}}$ so that (4.2.25) holds because
$\phi|_{{\mathcal{B}}_{\mathcal{V}}}$ is a functor.

  * $\bullet$

If $K\subset J\subset I$ then

$m^{\prime}_{KI}:=m_{IK}^{-1}=m_{JK}^{-1}\circ
m_{IJ}^{-1}=:m_{KJ}^{\prime}\circ m_{JI}^{\prime}$

is a relation in ${\mathcal{B}}_{\mathcal{V}}$ and so is preserved by $\phi$.
Therefore because $\phi(m^{-1})=\bigl{(}\phi(m)\bigr{)}^{-1}$ we have

$\bigl{(}\phi(m_{IK})\bigr{)}^{-1}=\bigl{(}\phi(m_{JK})\bigr{)}^{-1}\circ\bigl{(}\phi(m_{IJ})\bigr{)}^{-1}=\bigl{(}\phi(m_{IJ})\circ\phi(m_{JK})\bigr{)}^{-1},$

where the second equality is a relation in ${\mathcal{Z}}$, which implies
(4.2.25) in this case as well.

  * $\bullet$

If $I\subset K\subset J$ then we can rewrite the relation $m_{IJ}\circ
m_{JK}=m_{IK}$ as $m_{IJ}=m_{IK}\circ m_{KJ}^{\prime}$, where
$m_{KJ}^{\prime}=m_{JK}^{-1}$. Then, because the latter is a relation in
${\mathcal{B}}_{\mathcal{V}}$ we have

$\phi(m_{IJ})=\phi(m_{IK})\circ\phi(m_{KJ}^{\prime})=\phi(m_{IK})\circ\phi(m_{JK}^{-1})=\phi(m_{IK})\circ(\phi(m_{JK}))^{-1},$

which easily implies (4.2.25). By inverting this relation as in the second
bullet, we find that (4.2.25) also holds if $J\subset K\subset I$.

  * $\bullet$

Finally if $J\subset I\subset K$ we rewrite $m_{IJ}\circ m_{JK}=m_{IK}$ as
$m_{JK}=m_{JI}^{\prime}\circ m_{IK}$ where $m_{JI}^{\prime}=(m_{IJ})^{-1}$,
and then argue as above. By inverting, we may also obtain (4.2.25) in the case
$K\subset I\subset J$.

This completes the proof. ∎

######  Remark 4.2.8.

We saw in Proposition 4.2.4(iii) that the functor ${\iota}^{\times
G}:{{\mathcal{Q}}}^{\times G}\to{\mathcal{X}}_{\mathcal{V}}$ induces a
homeomorphism
$|{\widehat{{{\mathcal{Q}}}}}|/G\to|{\mathcal{X}}_{\mathcal{V}}|$. It follows
that each morphism $m\in{\mathcal{X}}_{\mathcal{V}}$ decomposes as a product
of a morphism in ${\widehat{{{\mathcal{Q}}}}}$ with a morphism given by the
action of $G$. This product can be written in either order since

$(I,J,x,g)=(I,I,\rho_{I}(x),g)\circ(I,J,x,{\rm id})=(I,J,g^{-1}*x,{\rm
id})\circ(J,J,\rho_{J}(x),g),\quad g\in G_{I\wedge J}.$

In terms of the map $\overline{\alpha}_{X_{\mathcal{V}}}:G\times
X_{\mathcal{V}}\to{\bf X}_{\mathcal{V}},(g,(I,x))\mapsto(I,I,g*x,g)$ in Lemma
3.3.3 (iii′) that defines the group action, the above identity can be written

(4.2.26) $\displaystyle\overline{\alpha}(g,g^{-1}*\rho_{I}(x))\circ(I,J,x,{\rm
id})=(I,J,g^{-1}*x,{\rm id})\circ\overline{\alpha}(g,g^{-1}*\rho_{J}(x)).$

${\Diamond}$

Finally we show that the category ${\mathcal{X}}_{\mathcal{V}}$ has a natural
system of good neighbourhoods.

######  Lemma 4.2.9.

The category ${\mathcal{X}}_{\mathcal{V}}$ carries a natural system of good
neighbourhoods as in Definition 3.4.6.

###### Proof.

Recall that a collection of open neighbourhoods $(U(x))_{x\in X}$ of the
points $x\in X$ of an étale proper category is called a good system of
neighbourhoods if

  * -

for every $x\in X$ the target map $t:s^{-1}({\rm cl}_{X}(U(x))\to X$ is
proper;

  * -

each $U(x)$ is equipped with the natural action of the isotropy group $G_{x}$
at $x$;

  * -

for any pair $x,x^{\prime}\in X$ and each connected component ${\Sigma}$ of
${\rm Mor}(U(x),U(x^{\prime}))$ the maps $s:{\Sigma}\to U(x)$ and
$t:{\Sigma}\to U(x^{\prime})$ are étale bijections onto their images.

Given $|y|\in|{\mathcal{X}}_{\mathcal{V}}|$, first note that if $U(|y|)$ is
any neighbourhood of $|y|\in|{\mathcal{X}}_{\mathcal{V}}|$ such that ${\rm
cl}(U(|y|))\subset\bigcap_{|y|\in|V_{J}|}\,|V_{J}|$, then the properness
condition holds by Lemma 3.2.5(iii). We also suppose that, with
$H_{|y|}=\bigcap_{|y|\in|V_{J}|}J$ as in Lemma 4.2.1 (i), for one (and hence
any) lift $x\in V_{H_{|y|}}$ of $y$ to $H_{|y|}$, the components of the
inverse image $\pi^{-1}(U(|y|)\cap V_{H_{|y|}}$ are freely permuted by the
group $G_{H_{|y|}}/G_{x}$, where $G_{x}$ is the isotropy group at $x$. It
follows that for any lift $y\in V_{J}$ of $|y|$ the components of
$\pi^{-1}(U(|y|)\cap V_{J}$ are freely permuted by the group $G_{J}/G_{x}$,
and we pick $U(y)$ to be the component of $\pi^{-1}(U(|y|)\cap V_{J}$ that
contains $y$. Then $U(y)$ is invariant under the action of the isotropy group
$G_{y}$. To check the last condition, suppose first that $U(x)\subset V_{I}$,
$U(x^{\prime})\subset V_{J}$ where $I\subset J$, and consider the component of
${\rm Mor}(U(x),U(x^{\prime}))$ containing $m=(I,J,y,g)$. Then $t(m)=y\in
U(x^{\prime})\subset{\widetilde{V}}_{IJ}$ and $s(m)=g*\rho_{I}(y)$, where
$\rho_{I}:{\widetilde{V}}_{IJ}\to V_{I}$ quotients by the free action of
$G_{J{\smallsetminus}I}$. But by assumption the images of $U(x^{\prime})$ by
$G_{J{\smallsetminus}I}$ are disjoint. Therefore given $s(m)$ and $g$ (both
uniquely determined by $m$), there is at most one possible choice for $y$.
Since the argument when $J\subset I$ is similar, this completes the proof. ∎

###  4.3. Bundles over ${\mathcal{X}}_{\mathcal{V}}^{{\smallsetminus}G}$ and
${\mathcal{X}}_{\mathcal{V}}$

In order to model the category
${\mathcal{X}}_{\mathcal{V}}^{{\smallsetminus}G}\times E$ in (2.1.5). we will
consider the product of the category ${{\mathcal{Q}}}$ as above with the
trivial category with objects $E$ and only identity morphisms. Here we suppose
that $E_{i}$ is a vector space on which $G_{i}$ acts, so that $G:=\prod_{i\in
A}G_{i}$ acts on $E:=\prod_{i\in A}E_{i}$ by the product action (where
$A:=\\{1,\dots,N\\}$). The statement in (ii) below should be contrasted with
the fact that, when the maps $\rho_{IJ}$ have closed graph, the space
$|{\widehat{{{\mathcal{Q}}}}}^{\times G}|\simeq|{\mathcal{X}}_{\mathcal{V}}|$
is Hausdorff by Proposition 4.1.1 (i). Again, we restate the relevant results
from §4.1 for the convenience of the reader.

######  Lemma 4.3.1.

Let ${\mathcal{X}}_{\mathcal{V}}$ and ${{\mathcal{Q}}}$ be as in Proposition
4.2.4, and consider the product category ${{\mathcal{Q}}}\times E=(Q\times
E,{{{\bf Q}}}\times{\bf E})$ as follows:

$Q\times E={\textstyle\bigsqcup_{I\subset A}V_{I}\times E,\quad{{{\bf
Q}}}\times{\bf E}=\bigsqcup_{I\subset J}{\widetilde{V}}_{IJ}\times E,}$

with groupoid completion ${\widehat{{{\mathcal{Q}}}}}\times
E=({\mathcal{X}}_{\mathcal{V}}^{{\smallsetminus}G})\times E$. Suppose further
that $G$ acts on $E$ as above. Then:

  * (i)

The category ${\widehat{{{\mathcal{Q}}}}}\times E$ is étale, and $G$ acts on
both ${{\mathcal{Q}}}\times E$ and its groupoid completion
${\widehat{{{\mathcal{Q}}}}}\times E$ by the product action on objects and
morphisms.

  * (ii)

If the action of $G$ in $E$ is trivial, then $|({{\mathcal{Q}}}\times
E)^{\times G}|=|({\widehat{{{\mathcal{Q}}}}}\times E)^{\times G}|$ is
Hausdorff. On the other hand, if $G_{j}$ acts nontrivially on $E_{j}$ for some
$j$, and for some $J$ with $j\in J$ and some
$H\subset(J{\smallsetminus}\\{j\\})$ there is a point $x\in{\rm
Fr}({\widetilde{V}}_{HJ})\subset V_{J}$ that does not lie in
${\widetilde{V}}_{FJ}$ for any $F\subset(J{\smallsetminus}\\{j\\})$, then the
realization $|({{\mathcal{Q}}}\times E)^{\times
G}|=|({\widehat{{{\mathcal{Q}}}}}\times E)^{\times G}|$ is not Hausdorff.

  * (iii)

There is a bundle ${\rm
pr}:{\mathcal{X}}_{\mathcal{V}}^{E}\to{\mathcal{X}}_{\mathcal{V}}$ with fiber
$E$ that supports a $G$-action (given on objects as in (i)) such that the
functorial inclusion
${\iota}:{\widehat{{{\mathcal{Q}}}}}={\mathcal{X}}_{\mathcal{V}}^{{\smallsetminus}G}\to{\mathcal{X}}_{\mathcal{V}}$
defined in Proposition 4.2.4 (iii) lifts to a $G$-equivariant functor
${\iota}_{E}:({\mathcal{X}}_{\mathcal{V}}^{{\smallsetminus}G})\times
E\to{\mathcal{X}}_{\mathcal{V}}^{E}$ that induces a homeomorphism

$|({\widehat{{{\mathcal{Q}}}}}\times E)^{\times
G}|=|({\mathcal{X}}_{\mathcal{V}}^{{\smallsetminus}G}\times E)^{\times
G}|\;=\;|{\mathcal{X}}_{\mathcal{V}}^{{\smallsetminus}G}\times
E|/G\simeq|{\mathcal{X}}_{\mathcal{V}}^{E}|/G.$

Further, under the conditions detailed in (ii) above,
$|{\mathcal{X}}_{\mathcal{V}}^{E}|/G$ is not Hausdorff.

###### Proof.

Part (i) follows from the fact that the projection maps $\rho_{IJ}$ are
equivariant with respect to the projection $G_{J}\to G_{I}$.

To prove (ii), first note that if the action of $G$ on $E$ is trivial then
$|({\widehat{{{\mathcal{Q}}}}}\times E)^{\times
G}|=(|{\widehat{{{\mathcal{Q}}}}}|/G)\times
E=|{\mathcal{X}}_{\mathcal{V}}|\times E$ by Proposition 4.2.4 (iii), and so is
Hausdorff. To prove the second claim in (ii), we show that under the stated
conditions there is a sequence $(x_{k},e)\in Q\times E$ whose image in
$|({\widehat{{{\mathcal{Q}}}}}\times E)^{\times G}|$ has two different limits.
Consider a point $x\in V_{J}$ that lies in the frontier ${\rm
Fr}({\widetilde{V}}_{HJ}):={\rm cl}({\widetilde{V}}_{HJ})\cap{\rm
cl}(V_{J}{\smallsetminus}{\widetilde{V}}_{HJ})$ of ${\widetilde{V}}_{HJ}$ but
does not lie in ${\widetilde{V}}_{FJ}$ for any
$F\subset(J{\smallsetminus}\\{j\\})$.444444 For example, we might take
$H=J{\smallsetminus}\\{j\\}$ and $x\in{\rm
Fr}({\widetilde{V}}_{HJ}){\smallsetminus}\bigcup_{F\subset
H}{\widetilde{V}}_{FJ}$ (assuming this set is nonempty). By taking $I=J$ in
(4.2.2), we see that for $F\subset J$, ${\widehat{{{\mathcal{Q}}}}}$ only
contains the partial actions of $G_{J{\smallsetminus}F}$ on
${\widetilde{V}}_{FJ}$. Therefore if $g\in G_{j}{\smallsetminus}\\{{\rm
id}\\}$ there is no morphism in ${\widehat{{{\mathcal{Q}}}}}$ with source
$(J,x)$ and target $(J,g*x)$. Thus
$(J,x)\not\sim_{{\widehat{{{\mathcal{Q}}}}}}(J,g*x)$. Now choose a convergent
sequence $x_{k}\to x$, where $x_{k}\in{\widetilde{V}}_{HJ}$. Then we have
$(J,x_{k})\sim_{{\widehat{{{\mathcal{Q}}}}}}(J,g*x_{k})$ since the action of
$G_{J{\smallsetminus}H}$ on ${\widetilde{V}}_{HJ}$ is included in the
morphisms of ${\widehat{{{\mathcal{Q}}}}}$. We then have

$(J,x_{k},e)\sim_{\mathcal{C}}(J,g*x_{k},e),\quad(J,x_{k},e)\sim_{\mathcal{C}}(J,g*x_{k},g*e).$

But if we choose $(g,e)\in G_{j}\times E_{j}$ so that $g*e\neq e$ we have

$(J,x,e)\not\sim_{\mathcal{C}}(J,g*x,e).$

Thus the sequence $|(x_{k},e)|$ has two limits $|(x,e)|$ and $|(g*x,e)|$ in
$|{\mathcal{C}}|$, as claimed.

To prove (iii), consider the category
${\mathcal{X}}_{\mathcal{V}}^{E}=(X_{\mathcal{V}}^{E},{\bf
X}_{\mathcal{V}}^{E})$ with

(4.3.1) $\displaystyle
X_{\mathcal{V}}^{E}:={\textstyle\bigsqcup_{I\in{\mathcal{I}}}\,V_{I}\times
E,\quad\ {\bf X}_{\mathcal{V}}^{E}:=\bigsqcup_{I,J\in{\mathcal{I}},I\sim
J}}\,{\widetilde{V}}_{(I\wedge J)(I\vee J)}\times G_{I\wedge J}\times E,$

where

(4.3.2) $\displaystyle(s\times t)\,(I,J,y,g,e)$
$\displaystyle=\bigl{(}\,(I,\,g^{-1}*\rho_{I}(y),g^{-1}*e),\;(J,\rho_{J}(y),e)\bigr{)}$
$\displaystyle{\rm id}_{(I,x,e)}:=(I,I,x,{\rm id},e),$
$\displaystyle\quad(I,J,x,g,e)^{-1}:=(J,I,g^{-1}*x,g^{-1},g^{-1}*e),$

and (where defined) composition is given by

(4.3.3)
$\displaystyle(I,J,x,g,e)\circ(J,K,y,h,e^{\prime}):=\bigl{(}I,K,\rho_{I\vee
K}(v),k(hg)|_{I\wedge K},e^{\prime}\bigr{)}$

where $v,k$ are defined in (4.1.4). Then ${\rm
pr}:{\mathcal{X}}_{\mathcal{V}}^{E}\to{\mathcal{X}}_{\mathcal{V}}$ is a bundle
with projection functor given by

$(I,x,e)\mapsto(I,x),\quad(I,J,y,g,e)\mapsto(I,J,y,g),$

and structural map

$\mu:{\bf X}\,{\vphantom{\times}}_{s}{\times}_{{\rm pr}}(X\times E)\to X\times
E,\quad\bigl{(}(I,J,y,g),(I,g^{-1}*\rho_{IJ}(y),e)\bigr{)}\mapsto(J,y,g*e).$

Then $G$ acts on the object space $X_{\mathcal{V}}^{E}=Q\times E$ by the
product action as in (i) above, and acts on the morphism space ${\bf
X}_{\mathcal{V}}^{E}$ by the following lift of the action on
${\mathcal{X}}_{\mathcal{V}}$ given in (4.2.23):

(4.3.4) $\displaystyle g*(I,y,e)$ $\displaystyle=(I,g|_{I}*y,g*e),$
$\displaystyle g*\bigl{(}I,J,y,h,e\bigr{)}$
$\displaystyle=\bigl{(}I,J,g|_{I\vee J}*y,g|_{I\wedge J}\,h\,g|_{I\wedge
J}^{-1},\,g*e\bigr{)}.$

We now define ${\iota}_{E}:{\widehat{{{\mathcal{Q}}}}}\times
E\to{\mathcal{X}}_{\mathcal{V}}^{E}$ to be the identity map on objects and the
inclusion on morphisms. This is equivariant, as claimed. Hence the final claim
in (iii) follows from (ii) and the fact that $|({{\mathcal{Q}}}\times
E)^{\times G}|\simeq|{{\mathcal{Q}}}\times E|/G|$ by Lemma 3.3.8 (iii). ∎

######  Remark 4.3.2.

Notice that the $G$-action on ${\mathcal{X}}_{\mathcal{V}}^{E}$ is not an
inner action, and hence is not lifted from the inner $G$-action on
${\mathcal{X}}_{\mathcal{V}}$ by the process of Lemma 3.3.6. For example,
because the group $G_{A{\smallsetminus}I}$ acts trivially on $V_{I}\subset
X_{\mathcal{V}}$, the lifted inner action is also trivial on $V_{I}\times
E_{A}$. But the action of $G_{A{\smallsetminus}I}$ described in (4.3.4) need
not be trivial on the $E_{A}$ factor in $V_{I}\times E_{A}$. This is the
reason for the conditions on the kernel of the functor $\tau$ in Proposition
4.3.3. ${\Diamond}$

We next explain the formal properties of the $G$-equivariant functor
$\tau:{\mathcal{X}}_{\mathcal{V}}^{{\smallsetminus}G}\times
E\to{\mathcal{W}}_{\mathcal{V}}$ whose existence is claimed in Theorem 2.1.2
(v). Recall from Proposition 4.1.2 (ii) that
${\mathcal{X}}_{\mathcal{V}}^{{\smallsetminus}G}=\widehat{{{\mathcal{Q}}}}$ is
the groupoid completion of the nonsingular étale category ${{\mathcal{Q}}}$.
It supports an action of $G$ by Lemma 4.2.3.

######  Proposition 4.3.3.

In the situation of Lemma 4.3.1, suppose given a bundle
${\mathcal{P}}:{\mathcal{W}}_{\mathcal{V}}\to{\mathcal{X}}_{\mathcal{V}}$ over
${\mathcal{X}}_{\mathcal{V}}$ with induced inner $G$-action as in Definition
3.3.4. Suppose further that there is a functor ${\tau}:{{\mathcal{Q}}}\times
E\to{\mathcal{W}}_{\mathcal{V}}$ such that

  * -

the induced map on object spaces is $G$-equivariant

  * -

its restriction to each fiber $(I,y)\times E$ is linear with kernel that
includes the subspace $E_{A{\smallsetminus}H_{y}}$, where $H_{y}=\min\\{H\
\big{|}\ y\in{\widetilde{Q}}_{HI}\\}$;

  * -

${\mathcal{P}}\circ{\tau}={\iota}\circ{\rm pr}:{{\mathcal{Q}}}\times
E\to{\mathcal{X}}_{\mathcal{V}}$,

where ${\iota}:{{\mathcal{Q}}}\to{\mathcal{X}}_{\mathcal{V}}$ is the
functorial inclusion from Proposition 4.1.3.

Then:

  * (i)

${\tau}$ extends uniquely to a functor
$\tau:{\mathcal{X}}_{\mathcal{V}}^{{\smallsetminus}G}\times
E\to{\mathcal{W}}_{\mathcal{V}}$.

  * (ii)

Moreover $\tau$ factors as the composite $\tau=\tau_{E}\circ{\iota}_{E}$ in
the following commutative diagram of functors

where ${\mathcal{X}}_{\mathcal{V}}^{E}$ is as in Lemma 4.3.1 (iii).

  * (iii)

Both functors $\tau_{E}$ and $\tau=\tau_{E}\circ{\iota}_{E}$ are
$G$-equivariant. In particular, $\tau(I,x,g*e)=g*\tau(I,g^{-1}*x,e).$

###### Proof.

We saw in Lemma 4.2.1 (i) that the category ${{{\mathcal{Q}}}}$ satisfies the
conditions in Lemma 3.2.9 (iv); namely it is a poset in which each equivalence
class has a unique minimum with morphisms to all other elements. Since the
product ${{{\mathcal{Q}}}}\times E$ also has this property, the functor
${\tau}$ extends by Lemma 3.2.9 (iv). This proves (i).

Towards (ii), consider the full subcategory
$({\mathcal{X}}_{\mathcal{V}}^{{\smallsetminus}G}\times E)^{\prime}$ of
${\mathcal{X}}_{\mathcal{V}}^{{\smallsetminus}G}\times E$ with the following
object space:

(4.3.5) $\displaystyle(X_{\mathcal{V}}\times E)^{\prime}$ $\displaystyle\
={\textstyle\bigsqcup_{I}}\bigl{\\{}(I,x,e)\in V_{I}\times E\ \big{|}\ e\in
E_{H_{x}}\bigr{\\}}$ $\displaystyle\
={\textstyle\bigsqcup_{I}\;\bigl{(}\bigcup_{K\subsetneq H\subset
I}}\bigl{(}{\widetilde{V}}_{HI}{\smallsetminus}{\widetilde{V}}_{KI}\bigr{)}\times
E_{H}\bigr{)}$

where $H_{x}$ is as in (4.2.2). Thus

(4.3.6) $\displaystyle H_{x}=\min\bigl{\\{}H\ |\
|x|\in|V_{H}|\subset|{\mathcal{X}}_{\mathcal{V}}^{{\smallsetminus}G}|\bigr{\\}}=\min\bigl{\\{}H\
|\
|x|\in|V_{H}|/G=|{\mathcal{X}}_{\mathcal{V}}^{{\smallsetminus}G}|/G\bigr{\\}}=:H_{|x|}$

where the above equality holds because the sets $V_{H}$ are $G_{H}$-invariant.
Below we denote by ${\rm pr}_{E}^{\prime}$ the (non continuous) projection map
$X_{\mathcal{V}}\times E\to(X_{\mathcal{V}}\times E)^{\prime}$ that maps the
fiber $(I,x)\times E$ at $(I,x)$ to $(I,x)\times E_{H_{x}}$ with kernel
$E_{A{\smallsetminus}H_{x}}$. Note that this extends to a $G$-equivariant
functor ${\rm pr}_{E}:{\mathcal{X}}_{\mathcal{V}}^{{\smallsetminus}G}\times
E\to({\mathcal{X}}_{\mathcal{V}}^{{\smallsetminus}G}\times E)^{\prime}$.

Denote by $({\mathcal{X}}_{\mathcal{V}}^{E})^{\prime}$ the corresponding full
subcategory of ${\mathcal{X}}_{\mathcal{V}}^{E}$ with objects
$(X_{\mathcal{V}}\times E)^{\prime}$. Since the group $G_{A{\smallsetminus}I}$
acts trivially on $\bigl{\\{}(I,x,e)\ \big{|}\ e\in E_{H_{x}}\bigr{\\}}\subset
V_{I}\times E_{I}$, the action of $G$ on
$({\mathcal{X}}_{\mathcal{V}}^{E})^{\prime}$ is an inner action given by
lifting the inner action on ${\mathcal{X}}_{\mathcal{V}}$ as follows:

(4.3.7) $\displaystyle\overline{\alpha}_{E^{\prime}}:(X_{\mathcal{V}}\times
E)^{\prime}\times G\to{\rm Mor}_{({\mathcal{X}}_{\mathcal{V}}^{E})^{\prime}},$
$\displaystyle\quad((I,x,e),g)\mapsto(I,I,g|_{I}*x,g|_{I},g|_{I}*e).$

Hence, because the only morphisms in
$({\mathcal{X}}_{\mathcal{V}}^{E})^{\prime}$ that are not in
$({\mathcal{X}}_{\mathcal{V}}^{{\smallsetminus}G}\times E)^{\prime}$ come from
the $G$-action, the functoriality of the lifting construction (see Remark
3.3.7) implies that any functor
$({\mathcal{X}}_{\mathcal{V}}^{{\smallsetminus}G}\times
E)^{\prime}\to{\mathcal{W}}$ that

  * (a)

covers the inclusion
${\mathcal{X}}_{\mathcal{V}}^{{\smallsetminus}G}\to{\mathcal{X}}_{\mathcal{V}}$
on the base, and

  * (b)

is given by a $G$-equivariant map $(X_{\mathcal{V}}\times E)^{\prime}\to W$ on
objects that is fiberwise linear,

extends to a functor
$({\mathcal{X}}_{\mathcal{V}}^{E})^{\prime}\to{\mathcal{W}}$.

Now consider the statement in the proposition. Since the categories
${\mathcal{X}}_{\mathcal{V}}^{{\smallsetminus}G}\times E$ and
${\mathcal{X}}_{\mathcal{V}}^{E}$ have the same object space, the only claim
that requires proof is that the map $X_{\mathcal{V}}^{E}\to W$ on object
spaces induced by $\tau$ does extend to a functor $\tau_{E}$ between these
categories. In other words we have to show that the fact that $\tau$ is
$G$-equivariant on objects implies that it respects the additional morphisms
in ${\mathcal{X}}_{\mathcal{V}}^{E}$ corresponding to the group action. But
the hypotheses on the functor $\tau$ imply that the induced map on objects
factors through the projection ${\rm pr}^{\prime}_{E}:X_{\mathcal{V}}\times
E\to(X_{\mathcal{V}}\times E)^{\prime}$. Thus $\tau$ induces a functor
$({\mathcal{X}}_{\mathcal{V}}^{{\smallsetminus}G}\times
E)^{\prime}\to{\mathcal{W}}$ that satisfies the conditions (a), (b) above and
hence extends to a $G$-equivariant functor
$\tau^{\prime}_{E}:({\mathcal{X}}_{\mathcal{V}}^{E})^{\prime}\to{\mathcal{W}}$.
We now define $\tau_{E}:=\tau_{E}^{\prime}\circ{\rm pr}_{{\mathcal{V}},E}$,
where ${\rm
pr}_{{\mathcal{V}},E}:{\mathcal{X}}_{\mathcal{V}}^{E}\to({\mathcal{X}}_{\mathcal{V}}^{E})^{\prime}$
is the extension of ${\rm
pr}_{E}:{\mathcal{X}}_{\mathcal{V}}^{{\smallsetminus}G}\times
E\to({\mathcal{X}}_{\mathcal{V}}^{{\smallsetminus}G}\times E)^{\prime}$ to
${\mathcal{X}}_{\mathcal{V}}^{E}$. This satisfies both (ii) and (iii). ∎

######  Remark 4.3.4.

The $G$-equivariance of $\tau$ implies that it extends to a functor

$\tau^{G}:({\mathcal{X}}_{\mathcal{V}}^{{\smallsetminus}G}\times E)^{\times
G}\to{\mathcal{W}}_{\mathcal{V}},$

that by Proposition 4.3.3 factors as

$\displaystyle({\mathcal{X}}_{\mathcal{V}}^{{\smallsetminus}G}\times
E)^{\times G}\longrightarrow({\mathcal{X}}_{\mathcal{V}}^{E})^{\times
G}\;\longrightarrow(({\mathcal{X}}_{\mathcal{V}}^{E})^{\prime})^{\times
G}\stackrel{{\scriptstyle{\iota}^{\times
G}}}{{\longrightarrow}}({\mathcal{X}}_{\mathcal{V}}^{E})^{\prime}\;\stackrel{{\scriptstyle\tau^{\prime}_{E}}}{{\longrightarrow}}\;{\mathcal{W}}_{\mathcal{V}},$

where $({\mathcal{X}}_{\mathcal{V}}^{E})^{\prime}$ has objects as in (4.3.5)
and ${\iota}^{\times G}$ is the functor described in Lemma 3.3.8 (v), which
exists because, as noted above, the $G$ action on
$({\mathcal{X}}_{\mathcal{V}}^{E})^{\prime}$ is an inner action. We saw in the
preceding proposition that
$\tau^{\prime}_{E}:({\mathcal{X}}_{\mathcal{V}}^{E})^{\prime}\to{\mathcal{W}}_{\mathcal{V}}$
covers the identity map on the base ${\mathcal{X}}_{\mathcal{V}}$ and that the
$G$ action on both spaces is the lift of the inner action on the base.
Therefore, there are enough morphisms $m$ in the base for the $G$-equivariance
of $\tau_{E}^{\prime}$ to be captured in the identity

(4.3.8)
$\displaystyle\mu_{W}\bigl{(}m,\tau^{\prime}_{E}(w)\bigr{)}=\tau^{\prime}_{E}\bigl{(}\mu_{E}(m,w)\bigr{)},\quad
m\in{\rm Mor}_{{\mathcal{X}}_{\mathcal{V}}},\;w\in{\rm
Obj}_{({\mathcal{X}}_{\mathcal{V}}^{E})^{\prime}},\;P(w)=s(m).$

where $\mu_{\bullet}$ is the structure function of the relevant bundle that
expresses how morphisms of the base lift to morphisms on the total space; see
Definition 3.4.1. ${\Diamond}$

We now use the results of Proposition 4.3.3 to construct multisections of the
bundle ${\mathcal{W}}_{\mathcal{V}}\to{\mathcal{X}}_{\mathcal{V}}$ with
properties as described in Proposition 4.1.6. As explained at the beginning of
§5.1, a multisection ${\Lambda}$ of a bundle
${\mathcal{P}}:{\mathcal{W}}\to{\mathcal{X}}$ is a functor
${\Lambda}:{\mathcal{W}}\to{\mathbb{Q}}^{\geq 0}$ with a local section
structure near each $x_{0}\in X$ as follows: for each $x_{0}\in X$ there is an
open set $U(x_{0})\subset X$ and finitely many sc+-sections
${\mathfrak{s}}_{1},\dots,{\mathfrak{s}}_{k}:U(x_{0})\to W$ with associated
positive rational numbers ${\sigma}_{1},\dots,{\sigma}_{k}$ such that
$\sum_{i}{\sigma}_{i}=1$ and

${\Lambda}(w)={\textstyle\sum_{\\{i\in\\{1,\dots
k\\}\,|\,{\mathfrak{s}}_{i}(P(w))=w\\}}{\sigma}_{i}.}$

The support ${\rm supp\,}({\Lambda})$ is given by

${\rm supp\,}({\Lambda})=\\{w\in W\ |\ {\Lambda}(w)>0\\}.$

If ${\Lambda}$ is a multisection of ${\mathcal{W}}\to{\mathcal{X}}$ and
$\psi:{\mathcal{X}}^{\prime}\to{\mathcal{X}}$ is any functor, then the formula
${\Lambda}^{\prime}:={\Lambda}\circ\Psi$ defines a pullback multisection of
the pullback bundle $\Psi:\psi^{*}{\mathcal{W}}\to{\mathcal{X}}^{\prime}$
defined in Lemma 3.4.4 (ii). The first part of the next lemma shows that there
is a notion of the pushforward of a multisection of
$\psi^{*}{\mathcal{W}}\to{\mathcal{X}}^{\prime}$, provided that the functor
$\psi:{\mathcal{X}}^{\prime}\to{\mathcal{X}}$ is sufficiently well behaved.
(Compare similar results in [HWZ, Thm.13.4.1]). The second part is exactly
what is needed to show that the functor $\tau$ in Proposition 4.3.3 can be
used to push forward multisections. In the applications in §5.3 and §5.4 we
will, of course, need to take into account the extra analytic conditions
required to control compactness and smoothness.

######  Lemma 4.3.5.

  * (i)

Let $\psi:{\mathcal{X}}^{\prime}\to{\mathcal{X}}$ be an equivalence, let
${\mathcal{W}}\to{\mathcal{X}}$ a bundle with pullback
$\psi^{*}{\mathcal{W}}\to{\mathcal{X}}^{\prime}$, and denote by
$\Psi:\psi^{*}{\mathcal{W}}\to{\mathcal{W}}$ the lift of
$\psi:{\mathcal{X}}^{\prime}\to{\mathcal{X}}$ from Lemma 3.4.4. If
${\Lambda}^{\prime}:\psi^{*}{\mathcal{W}}\to{\mathbb{Q}}^{\geq 0}$ is a
multisection, then the following defines a multisection of
${\mathcal{W}}\to{\mathcal{X}}$:

    * -

for each $w\in W$ there is $w^{\prime}\in W^{\prime}$ and a morphism in
${\mathcal{W}}$ from $w$ to $\Psi(w^{\prime})$, and we define

(4.3.9)
$\displaystyle\psi_{*}({\Lambda}^{\prime})(w):={\Lambda}^{\prime}(w^{\prime}).$

  * (ii)

Suppose that ${\mathcal{W}}^{\prime}\to{\mathcal{X}}^{\prime}$ and
${\mathcal{W}}\to{\mathcal{X}}$ are two bundles and that the functor
$\Psi:{\mathcal{W}}^{\prime}\to{\mathcal{W}}$ is the lift of a functor
$\psi:{\mathcal{X}}^{\prime}\to{\mathcal{X}}$ that is the identity map on
$X^{\prime}=X$ and is such that $\Psi({\mathcal{W}}^{\prime})$ is a full
subcategory of ${\mathcal{W}}$ that is invariant under the morphisms
$(m,w)\in{\bf W}$ in ${\mathcal{W}}$ in the sense that any morphism in
${\mathcal{W}}$ with source or target in the image $\Psi(W^{\prime})$ is in
fact contained in the image $\Psi({\bf W}^{\prime})$. Then, given a
multisection ${\Lambda}^{\prime}:{\mathcal{W}}^{\prime}\to{\mathbb{Q}}^{\geq
0}$, the following formula defines a multisection of
${\mathcal{W}}\to{\mathcal{X}}$:

(4.3.10) $\displaystyle\Psi_{*}({\Lambda}^{\prime})(w):$
$\displaystyle=\sum_{w^{\prime}\in\Psi^{-1}(w)}{\Lambda}^{\prime}(w^{\prime}),$

where the sum is defined to be $0$ when $\Psi^{-1}(w)=\emptyset$.

###### Proof.

Consider (i). To see that $\psi_{*}({\Lambda}^{\prime}):W\to{\mathbb{Q}}^{\geq
0}$ is well defined, we note first that every $w\in W$ is equivalent w.r.t.
$\sim_{{\mathcal{W}}}$ to some element $\Psi(w^{\prime})$ since
$|\Psi|:|{\mathcal{W}}^{\prime}|\to|{\mathcal{W}}|$ is surjective, and second
that the injectivity of $|\Psi|$ and the fact that ${\Lambda}^{\prime}$ is
constant on the equivalence classes of $\sim_{{\mathcal{W}}^{\prime}}$ imply
that ${\Lambda}(w)$ does not depend on the choice of $w^{\prime}$ such that
$\Psi(w^{\prime})\sim_{{\mathcal{W}}}w.$ Further, $\psi_{*}({\Lambda})$ is a
functor because, by construction, it is constant on the equivalence classes of
$\sim_{{\mathcal{W}}}$. Finally, because $\psi_{*}({\Lambda})$ is a functor,
every morphism $m:x_{0}\mapsto x_{1}$ in ${\mathcal{X}}$ extends to a locally
invertible étale map and has a family of fiberwise lifts that take a local
section structure for ${\Lambda}$ at $x_{0}$ to one at $x_{1}$. Hence, because
$|\psi|:|{\mathcal{X}}^{\prime}|\to|{\mathcal{X}}|$ is surjective, it suffices
to check that there is a local section structure near every point
$x_{0}\in\psi(X^{\prime})$. To this end, choose
$x_{0}^{\prime}\in\psi^{-1}(x_{0})$ and $U(x_{0}^{\prime})$ so small that
$\psi^{\prime}|_{U(x_{0}^{\prime})}$ is injective. Then, because
$\psi^{\prime}$ is étale, $U(x_{0})=\psi^{\prime}(U(x_{0}^{\prime}))$ is a
neighbourhood of $x_{0}$ and the local section structure
$\bigl{(}{\mathfrak{s}}_{i}^{\prime}:U(x_{0}^{\prime})\to
W^{\prime},{\sigma}_{i}^{\prime}\bigr{)}$ at $x_{0}^{\prime}$ pushes forward
to the local section structure
$\bigl{(}{\mathfrak{s}}_{i}=\Psi\circ{\mathfrak{s}}_{i}\circ\psi^{-1}:U(x_{0})\to
W,{\sigma}_{i}^{\prime}\bigr{)}$ at $x_{0}$. This completes the proof of (i).

In (ii), the map on the base is the identity, while the map on the fiber need
be neither injective not surjective. Nevertheless, the given formula for
$\Psi_{*}({\Lambda}^{\prime})$ defines a functor on ${\mathcal{W}}$ because
the image $\Psi({\rm Mor}_{{\mathcal{W}}^{\prime}})$ contains all morphisms in
${\mathcal{W}}$ with source or target in $\Psi(W)$. Note also that the numbers
$\Psi_{*}({\Lambda}^{\prime})(w)$ sum to $1$ over each fiber of
${\mathcal{W}}\to{\mathcal{X}}$, and that because $\psi$ is the identity on
objects, each family $({\mathfrak{s}}^{\prime}_{i})$ of local sections at $x$
for ${\Lambda}^{\prime}$ gives rise to a family
$({\mathfrak{s}}_{i}:=\Psi\circ{\mathfrak{s}}_{i}^{\prime})$ of local sections
for ${\Lambda}$. ∎

It remains to prove Proposition 4.1.6 that we now restate for the convenience
of the reader.

######  Proposition 4.3.6.

Let the bundle ${\mathcal{W}}_{\mathcal{V}}\to{\mathcal{X}}_{\mathcal{V}}$,
group $G$, and functor $\breve{\tau}:{{\mathcal{Q}}}\times
E\to{\mathcal{W}}_{\mathcal{V}}$ with extension
$\tau:{\mathcal{X}}_{\mathcal{V}}^{{\smallsetminus}{\Gamma}}\times
E\to{\mathcal{W}}_{\mathcal{V}}$ be as in Proposition 4.1.5.

  * (i)

Each $e\in E$ induces a multisection functor
${\Lambda}_{{\mathcal{V}},e}:{\mathcal{W}}_{\mathcal{V}}\to{\mathbb{Q}}^{\geq
0}$ given by

(4.3.11) $\displaystyle\textstyle{\Lambda}_{{\mathcal{V}},e}(w)$
$\displaystyle=\tfrac{1}{|G|}\ \\#\\{h\in
G\,|\,\tau(P_{\mathcal{V}}(w),h*e)=w\\},$

where $\\#$ denotes the number of elements in a finite set.

  * (ii)

This multisection ${\Lambda}_{{\mathcal{V}},e}$ is globally structured in the
sense of Definition 3.4.5 by the sections $({\mathfrak{s}}_{g})_{g\in G}$ and
correspondence ${\kappa}:{\rm Mor}_{{\mathcal{X}}_{\mathcal{V}}}\to\\{b:G\to
G,{\rm bijection}\\}$ given by

(4.3.12) $\displaystyle{\mathfrak{s}}_{g}(I,x)$ $\displaystyle=\tau(I,x,g*e),\
\quad{\kappa}(I,J,y,g)(h)=g*h.$

  * (iii)

The triple
$({\Lambda}_{{\mathcal{V}},e},({\mathfrak{s}}_{i})_{i\in{\mathcal{I}}},{\kappa})$
is structurable in the sense of [HWZ, Def.13.3.6], and determines a unique
structured multisection
$[{\Lambda}_{{\mathcal{V}},e},{\mathcal{U}},{\mathcal{S}},\tau]$ in the sense
of [HWZ, Def.13.3.8], where $(U(x)_{x\in{\mathcal{X}}_{\mathcal{V}}}$ is the
system of good neighbourhoods in ${\mathcal{X}}_{\mathcal{V}}$ defined in
Lemma 4.2.9.

###### Proof.

To prove (i) we must show that if the bundle
${\mathcal{W}}_{\mathcal{V}}\to{\mathcal{X}}_{\mathcal{V}}$, group $G$, and
functor $\tau:{\mathcal{X}}_{\mathcal{V}}^{{\smallsetminus}G}\times
E\to{\mathcal{W}}_{\mathcal{V}}$ are as in Proposition 4.3.3, then each $e\in
E$ determines a multisection functor
${\Lambda}_{{\mathcal{V}},e}:{\mathcal{W}}_{\mathcal{V}}\to{\mathbb{Q}}^{\geq
0}$ given by

(4.3.13) $\displaystyle\textstyle{\Lambda}_{{\mathcal{V}},e}(w)$
$\displaystyle=\tfrac{1}{|G|}\\#\\{h\in
G\,|\,\tau(P_{\mathcal{V}}(w),h*e)=w\\},$

where $|G|$ denotes the order of the group $G$.

For fixed $e\in E$, consider the functor
${\Lambda}^{{\smallsetminus}G}_{e}:{\mathcal{X}}_{\mathcal{V}}^{{\smallsetminus}G}\times
E\to{\mathbb{Q}}^{\geq 0}$ defined by setting

$\displaystyle{\Lambda}^{{\smallsetminus}G}_{e}(x,e^{\prime})$
$\displaystyle=\begin{cases}1/|G|\;\mbox{ if }e^{\prime}\in G*e,\\\ 0\;\mbox{
otherwise}.\end{cases}$

This is a $G$-equivariant functor with support
${\mathcal{X}}_{\mathcal{V}}^{{\smallsetminus}G}\times\\{G*e\\}$, and hence
clearly is a multisection of the bundle
${\mathcal{X}}_{\mathcal{V}}^{{\smallsetminus}G}\times
E\to{\mathcal{X}}_{\mathcal{V}}^{{\smallsetminus}G}$. Because
${\Lambda}_{e}^{{\smallsetminus}G}$ is $G$-invariant, we may also consider it
as a multisection of the bundle
$({\mathcal{X}}_{\mathcal{V}}^{{\smallsetminus}G}\times E)^{\times
G}\to({\mathcal{X}}_{\mathcal{V}}^{{\smallsetminus}G})^{\times G}$ (see Lemma
3.3.8 and Proposition 4.2.4 (iii)). Next note that because it is
$G$-equivariant, $\tau$ extends to a functor
$\tau^{G}:({\mathcal{X}}_{\mathcal{V}}^{{\smallsetminus}G}\times E)^{\times
G}\to{\mathcal{W}}_{\mathcal{V}}$, and it is immediate that its image is
invariant under the morphisms of ${\mathcal{W}}_{\mathcal{V}}$ in the sense
defined in Lemma 4.3.5 (ii). Hence there is a well defined pushforward
multisection $(\tau^{G})_{*}({\Lambda}^{{\smallsetminus}G}_{e})$, and it
follows by comparing (4.3.10) with (4.3.13) that

$(\tau^{G})_{*}({\Lambda}^{{\smallsetminus}G}_{e})={\Lambda}_{{\mathcal{V}},e}.$

This completes the proof of (i).

To prove (ii), note that because ${\kappa}$ is locally constant, it suffices
to check that (4.3.12) holds; namely we require that that for all morphisms
$m$ in ${\mathcal{X}}_{\mathcal{V}}$

${\mathfrak{s}}_{{\kappa}(m)(h)}(t(m))=\mu\bigl{(}m,{\mathfrak{s}}_{h}(s(m))\bigr{)}\in
W_{t(m)}.$

With $m=(I,J,y,g)$ we therefore require that
${\mathfrak{s}}_{gh}(J,\rho_{J}(y))=\mu\bigl{(}m,{\mathfrak{s}}_{h}(I,g^{-1}*\rho_{I}(y))$.
Equivalently, since
${\mathfrak{s}}_{h}(I,g^{-1}*\rho_{I}(y))=\tau(I,g^{-1}*\rho_{I}(y),h*e)$, we
require that

$\tau(J,\rho_{J}(y),gh*e)=\mu\bigl{(}(I,J,y,g),\tau(I,g^{-1}*\rho_{I}(y),h*e)\bigr{)}$

Since every morphism in ${\mathcal{X}}_{\mathcal{V}}$ is the composite of a
morphism $(I,J,y,{\rm id})$ in
${\mathcal{X}}_{\mathcal{V}}^{{\smallsetminus}G}$ with a morphism $(J,J,y,g)$
given by the $G$ action in ${\mathcal{X}}_{\mathcal{V}}$, it suffices to check
the above identity for these two types of morphisms. But this identity holds
for the morphisms in ${\mathcal{X}}_{\mathcal{V}}^{{\smallsetminus}G}$ since
$\tau$ is a functor ${\mathcal{X}}_{\mathcal{V}}^{{\smallsetminus}G}\times
G\to{\mathcal{W}}_{\mathcal{V}}$ by Proposition 4.3.3 (i), and it holds for
the morphisms that give the $G$ action since $\tau$ is $G$-equivariant by
Proposition 4.3.3 (iii).

Finally the claims in (iii) hold by Proposition 3.4.11. ∎

##  5\. Proofs of Main Theorems

This section proves Theorem 2.1.2 and Corollary 2.1.3. Thus we consider a sc-
Fredholm section functor $f:{\mathcal{X}}\to{\mathcal{W}}$ of a strong bundle
${{\mathcal{P}}:{\mathcal{W}}\to{\mathcal{X}}}$ with compact solution set
$S:=|f^{-1}(0)|\subset|{\mathcal{X}}|$. We begin in §5.2 by constructing
suitable étale data as described in §4.1 from a family of local uniformizers
for the ep-groupoid ${\mathcal{X}}$ that covers a neighbourhood of the compact
zero set of the polyfold-Fredholm section $f:{\mathcal{X}}\to{\mathcal{W}}$.
We then apply the constructions in §4 to obtain the bottom row of the diagram
(2.1.5). This is the content of Theorem 5.2.6. In Theorem 5.3.1 we construct
the pullback bundle
${\mathcal{W}}_{\mathcal{V}}\to{\mathcal{X}}_{\mathcal{V}}$ of the strong
bundle ${\mathcal{W}}\to{\mathcal{X}}$, and establish the existence and
properties of the right hand square in (2.1.5). Finally, under the assumption
that the local uniformizers are chosen so that they extend to local
stabilizations as in Lemma 5.4.1 that are compatible with suitable compactness
controlling data, Theorem 5.4.2 constructs the left hand square in (2.1.5).
The proof of Theorem 2.1.2 is given in §5.4. Corollary 2.1.3 then follows by
applying Proposition 4.1.6.

###  5.1. Fundamental classes induced by Fredholm sections over polyfolds

In this subsection we consider a compact moduli space
${\overline{{\mathcal{M}}}}$ without formal boundary and of expected dimension
$d$ (given by the Fredholm index of local models). For example,
${\overline{{\mathcal{M}}}}$ might be the Gromov-Witten moduli space
${\overline{{\mathcal{M}}}}_{g,k}(M,J,A)$ of $J$-holomorphic curves in
homology class $A\in H_{*}(M)$ in a closed symplectic manifold $M$, of fixed
genus $g\geq 0$ and with $k\in{\mathbb{N}}_{0}$ marked points, with $d=\dim
M+2c_{1}(A)+2k+6g-6$. In that setting, a Kuranishi atlas ${\mathcal{K}}$ for
${\overline{{\mathcal{M}}}}$ consists of an interconnected family of local
models without boundary, and we showed in [MW3] how to construct a fundamental
class
$[{\overline{{\mathcal{M}}}}]_{\mathcal{K}}\in\check{H}_{d}({\overline{{\mathcal{M}}}};{\mathbb{Q}})$
by compatibly perturbing the local models. The main purpose of this section is
to construct an analogous rational Čech homology class454545See the Appendix
for a brief discussion of rational Čech homology.
$[{\overline{{\mathcal{M}}}}]_{f}\in\check{H}_{d}({\overline{{\mathcal{M}}}};{\mathbb{Q}})$
for polyfold descriptions of ${\overline{{\mathcal{M}}}}$ as in Theorem 5.1.4.
We also briefly discuss the situation when ${\overline{{\mathcal{M}}}}$ has
nonempty formal boundary.

######  Remark 5.1.1.

(Čech homology and Fundamental classes) We use rational Čech homology here due
to its identification with singular homology for finite simplicial complexes
and its tautness property explained in [MW2, Remark 8.2.4] and Appendix A.
This allows for a perturbative construction of $[{\overline{{\mathcal{M}}}}]$
as follows: Suppose the moduli space is identified
${\overline{{\mathcal{M}}}}\simeq S\subset Y$ with a compact subset $S$ of a
metric space $Y$. Then we can construct a fundamental class
$[S]\in\check{H}_{d}(S;{\mathbb{Q}})$ and push it forward to
$[{\overline{{\mathcal{M}}}}]\in\check{H}_{d}({\overline{{\mathcal{M}}}};{\mathbb{Q}})$
via $\phi:S\stackrel{{\scriptstyle\simeq}}{{\to}}{\overline{{\mathcal{M}}}}$.
To construct $[S]$ we choose a nested sequence of open subsets $S\subset
B_{k}\subset B_{k-1}\subset Y$ such that $\bigcap_{k\in{\mathbb{N}}}B_{k}=S$.
For example, the balls $B_{k}:=\\{p\in Y\,|\,\inf_{s\in S}d(p,s)<1/k\\}$ of
radius $1/k$ provide such nested neighbourhoods of $S$. Then these inclusions
induce a system of maps
$\check{H}_{d}(S;{\mathbb{Q}})\to\check{H}_{d}(B_{k+1};{\mathbb{Q}})\to\check{H}_{d}(B_{k};{\mathbb{Q}})$,
and the meaning of tautness is that their inverse limit equals the Čech
homology
$\check{H}_{d}(S;{\mathbb{Q}})\;\simeq\;\underset{\leftarrow}{\lim}\,\check{H}_{d}(B_{k};{\mathbb{Q}})$.

We can use this limit to define a fundamental class

$[S]:=\underset{\leftarrow}{\lim}\,(\phi_{k})_{*}[Z_{k}]\;\in\;\check{H}_{d}(S;{\mathbb{Q}})$

by constructing continuous maps $\phi_{k}:Z_{k}\to B_{k}$ from sets $Z_{k}$
(such as those in Lemma 5.1.3) that carry natural fundamental classes
$[Z_{k}]\in\check{H}_{d}(Z_{k};{\mathbb{Q}})$. Existence of this limit follows
if we can ensure compatibility
$(\phi_{k})_{*}[Z_{k}]=(\iota_{k\ell})_{*}\bigl{(}(\phi_{\ell})_{*}[Z_{\ell}]\bigr{)}\in\check{H}_{d}(B_{k};{\mathbb{Q}})$
for $k<\ell$, where $\iota_{k\ell}:B_{\ell}\to B_{k}$ is the inclusion map.
This in turn can be achieved by constructing cobordisms $Z_{k\ell}$ with
boundary $\partial Z_{k\ell}\simeq Z^{-}_{k}\sqcup Z_{\ell}$ and continuous
maps $\phi_{k\ell}:Z_{k\ell}\to B_{k}$ that restrict to $\phi_{k}$ resp.
$\iota_{k\ell}\circ\phi_{\ell}$ on the boundary. ${\Diamond}$

######  Definition 5.1.2.

A polyfold model without boundary
${\mathcal{E}}=\bigl{(}f:{\mathcal{X}}\to{\mathcal{W}},\psi\bigr{)}$ of
Fredholm index $d\in{\mathbb{Z}}$ for a compact Hausdorff space
${\overline{{\mathcal{M}}}}$ consists of the following data:

  * $\bullet$

${{\mathcal{P}}:{\mathcal{W}}\to{\mathcal{X}}}$ is a strong bundle over an ep-
groupoid ${\mathcal{X}}$ with empty boundary $\partial X=\emptyset$.

  * $\bullet$

$f:{\mathcal{X}}\to{\mathcal{W}}$ is an oriented sc-Fredholm section functor
of constant Fredholm index $d\in{\mathbb{Z}}$.

  * $\bullet$

$\psi:|f^{-1}(0)|\to{\overline{{\mathcal{M}}}}$ is a homeomorphism.

Here the homeomorphism $|f^{-1}(0)|\to{\overline{{\mathcal{M}}}}$ guarantees
that the zero set $|f^{-1}(0)|\subset|{\mathcal{X}}|$ is compact. On the other
hand, the homeomorphism $|f^{-1}(0)|\to{\overline{{\mathcal{M}}}}$ then
induces a metrizable topology on ${\overline{{\mathcal{M}}}}$ since
$|f^{-1}(0)|\subset|{\mathcal{X}}|$ is a subset of the metrizable space
$|{\mathcal{X}}|$ by [HWZ, Thm.7.3.1]. Thus rational Čech homology
$\check{H}_{d}({\overline{{\mathcal{M}}}};{\mathbb{Q}})$ is well defined, and
we will apply its tautness properties as explained in [MW2, Remark 8.2.4] and
Appendix A to construct the fundamental class
$[{\overline{{\mathcal{M}}}}]\in\check{H}_{d}({\overline{{\mathcal{M}}}};{\mathbb{Q}})$.

######  Lemma 5.1.3.

Suppose that ${\partial}{\mathcal{X}}=0$ and $f:{\mathcal{X}}\to{\mathcal{W}}$
is an oriented sc-Fredholm section functor of constant Fredholm index
$d\in{\mathbb{Z}}$ with compact zero set $|f^{-1}(0)|$ and
$\Lambda:{\mathcal{W}}\to{\mathcal{X}}$ is a sc+-multisection such that
$\Lambda\circ f:X\to{\mathbb{Q}}^{\geq 0}$ is a compact, oriented, tame
branched, ep+-subgroupoid as in [HWZ, Thms.15.3.7, 15.4.1]. Then the solution
set $|S_{\Lambda\circ f}|:=|{\rm supp\,}({\Lambda}\circ
f)|\subset|{\mathcal{X}}|$ carries a natural fundamental class
$[{\Lambda}\circ f]\in\check{H}_{d}(|S_{\Lambda\circ f}|;{\mathbb{Q}})$. In
particular, if $M:=|S_{\Lambda\circ f}|$ is an oriented manifold, and
$\Lambda\circ f\equiv\theta\in{\mathbb{Q}}^{\geq 0}$ is constant, then the
fundamental class is $[{\Lambda}\circ f]=\theta[M]$.

Moreover, consider two such sc+-multisections
$\Lambda^{0},\Lambda^{1}:{\mathcal{W}}\to{\mathcal{X}}$ whose solution sets
$|S_{\Lambda^{0}\circ f}|,|S_{\Lambda^{1}\circ f}|\subset B$ are contained in
a common open subset $B\subset|{\mathcal{X}}|$. Suppose that
$\Theta:[0,1]\times{\mathcal{X}}\to{\mathbb{Q}}^{\geq 0}$ is a compact,
oriented, tame branched ep+-subgroupoid with support $|{\rm
supp\,}\Theta|\subset[0,1]\times B$ and restrictions
$\Theta|_{[0,{\varepsilon})\times{\mathcal{X}}}(t,x)={\Lambda}^{0}(f(x))$,
$\Theta|_{(1-{\varepsilon},1]\times{\mathcal{X}}}(t,x)={\Lambda}^{1}(f(x))$
for some ${\varepsilon}>0$. If the orientation of ${\Lambda}^{1}\circ
f=\Theta|_{\\{1\\}\times{\mathcal{X}}}$ given by boundary restriction of
$\Theta$ as in [HWZ, Def.9.3.8] agrees with the orientation induced by $f$ via
the solution set convention in §2.3 (generalized to multisections in [HWZ,
§9]), and the orientation of ${\Lambda}^{0}\circ
f=\Theta|_{\\{0\\}\times{\mathcal{X}}}$ given by restriction of $\Theta$ is
opposite to the orientation induced by $f$, then

$(\phi^{0})_{*}[{\Lambda}^{0}\circ f]=(\phi^{1})_{*}[{\Lambda}^{1}\circ
f]\in\check{H}_{d}(B;{\mathbb{Q}}),$

where $\phi^{i}:|S_{\Lambda^{i}\circ f}|\to B$ denotes the inclusion map for
$i=0,1$.

###### Proof.

To explain what is meant by a “natural fundamental class” we first describe
the structure of the support ${\rm supp\,}(\theta)=\\{x\in X\ |\
\theta(x)>0\\}$ of the functor $\theta:={\Lambda}\circ f$. By [HWZ,
Def.9.1.2], each point $x\in{\rm supp\,}(\theta)$ has a local branching
structure. This means that there is an open neighbourhood $U(x)$ of $x\in X$
and a finite collection of open smooth $n$-dimensional submanifolds
$M_{x,i}\ni x,i\in I_{x}$ of $X$, (called local branches) together with
weights ${\sigma}_{x,i}\in{\mathbb{Q}}^{>0}$ such that each inclusion
$M_{x,i}\to U(x)$ is proper, and

(5.1.1) $\displaystyle\theta(y)={\textstyle\sum_{i\in I_{x}:y\in
M_{x,i}}{\sigma}_{x,i}\quad\mbox{ for all }\ y\in U(x).}$

In particular $U(x)=\bigcup_{i\in I_{x}}M_{x,i}$. We define the branch locus
${\rm Br}$ of $S:=|{\rm supp\,}(\theta)|\subset|{\mathcal{X}}|$ to consist of
all $z\in S\subset|{\mathcal{X}}|$ that do not have a neighbourhood in $S$
that is entirely contained in one of the sets $|M_{x,i_{0}}|$. By [HWZ,
Def.9.2.2,Lem.9.2.3], the complement $S{\smallsetminus}{\rm Br}$ is the image
in $|{\rm supp\,}(\theta)|$ of the set of good points in ${\rm
supp\,}(\theta)$. Note that by definition, given a local branching structure
$(M_{x,i})_{i\in I_{x}}$ each good point $y$ has a neighbourhood
$V(y)\subset{\rm supp\,}(\theta)$ such that $V(y)\cap M_{x,i}=V(y)\cap
M_{x,j}$ for all $i,j\in I_{x}$ for which $y\in M_{x,i}\cap M_{x,j}$. Further
${\rm Br}$ is closed and nowhere dense in $S$ by [HWZ, Lem.9.2.7].

With this understood, we require that the fundamental class
$[\theta,{\mathfrak{o}}]$ has the following property: for each local branching
structure $(M_{x,i})_{i\in I_{x}}$ over $U(x)$ and good point $y\in U(x)$, the
restriction of $[\theta,{\mathfrak{o}}]$ to $|V(y)|$ is given by

(5.1.2)
$\displaystyle{\textstyle\rho_{S,|V(y)|}([\theta,{\mathfrak{o}}])=\sum_{i\in
I_{x},y\in
M_{x,i}}{\sigma}_{x,i}\,\mu_{|V(y)|}\in\check{H}^{\infty}_{n}\bigl{(}|V(y)|\bigr{)}}$

where $\mu_{|V(y)|}$ denotes the pushforward by
$\pi_{\mathcal{X}}:V(y)\to|V(y)|\subset|{\mathcal{X}}|$ of the fundamental
class $\mu_{V(y)}$ of the oriented $n$-manifold $V(y)$ described in (A.3). To
see that the condition in (5.1.2) makes sense, notice firstly that the map
$\pi_{\mathcal{X}}:V(y)\to|V(y)|$ is proper because (as in [HWZ, Prop.9.1.12])
we can choose $V(y)$ to be a subset of a local uniformizer for
${\mathcal{X}}$, and secondly that the pushforward class $\mu_{|V(y)|}$ is
defined for proper maps by the discussion before (A.5). Another important
point about equation (5.1.1) is that it uses the Çech homology theory
$\check{H}^{\infty}_{n}(\cdot)$ that is analogous to locally finite singular
homology and is dual to compactly supported rational Çech cohomology. Thus it
equals the standard Çech homology $\check{H}_{n}(\cdot)$ on compact spaces,
but has very different behavior on noncompact spaces.

We next claim that for each local branching structure $(M_{x,i})_{i\in I_{x}}$
over $U(x)$ the condition (5.1.2) determines the restriction of
$[\theta,{\mathfrak{o}}]$ to $|U(x)|{\smallsetminus}{\rm Br}$. Indeed,
$|U(x)|{\smallsetminus}{\rm Br}$ is a (possibly infinite) union of connected
components $|C|$, where $\pi_{\mathcal{X}}^{-1}(|C|)\cap U(x)=:C$ is the union
of the submanifolds $V(y)$ with $|y|\in|C|$. Thus $C$ can be identified with
an open subset of an oriented $n$-manifold. The set $C$ may have a finite
number of connected components $(C_{i})_{i\in I_{x,|C|}}$ (where $I_{x,|C|}$
indexes the relevant local branches), but, because $|C|$ is disjoint from the
branch locus, each such component satisfies $|C_{i}|=|C|$. This allows us to
show that the restriction of $[\theta,{\mathfrak{o}}]$ to $|C|$ is simply a
multiple of $\mu_{|C|}$. Indeed, because restricting to an open subset
$V^{\prime}\subset V$ takes the fundamental class of $V$ to that of
$V^{\prime}$, this restriction is $\sum_{i\in
I_{x,|C|}}{\sigma}_{x,i}\,\mu_{|C|}$.

Next observe that because $S$ is compact and the map
$\pi_{\mathcal{X}}:X\to|{\mathcal{X}}|$ is open, $S$ has a finite open cover
by sets $|U_{i}|$, where each $U_{i}\subset X$ supports a local branching
structure. Assume for some $k\geq 0$ that we have constructed the fundamental
class $[\theta]_{k}$ over $W_{k}:=\bigcup_{j=1}^{k}|U_{j}|$ so that (5.1.2)
holds for each $y$ with $|V(y)|\subset W_{k}$, and for $W:=|U_{k+1}|$ define
$[\theta]_{k,W}\in\check{H}_{n}(W_{k}\cap W)$ by restriction,
$[\theta]_{k,W}:=\rho_{W_{k},W_{k}\cap W}([\theta]_{k})$. As noted above, the
requirement (5.1.2) specifies the restriction of $[\theta]$ to
$W{\smallsetminus}{\rm Br}$. We will prove the following:

  * (a)

there is a unique class $[\theta]_{W}\in\check{H}^{\infty}_{n}(W)$ such that
$\rho_{W,W{\smallsetminus}Br}([\theta])=[\theta]_{W{\smallsetminus}{\rm Br}}$.

  * (b)

$\check{H}^{\infty}_{n+1}(W_{k}\cap W)=0$, and

  * (c)

$\rho_{W_{k},W_{k}\cap W}([\theta]_{k})=\rho_{W,W_{k}\cap
W}([\theta]_{W})\in\check{H}^{\infty}_{n}(W_{k}\cap W)$.

Granted this, the Mayer–Vietoris sequence in (A.6) implies that there is a
unique class $[\theta]_{k+1}\in\check{H}^{\infty}_{n}(W_{k+1})$ that restricts
to
$[\theta]_{k}\oplus[\theta]_{W}\in\check{H}^{\infty}_{n}(W_{k})\oplus\check{H}^{\infty}_{n}(W)$.
Then $[\theta]_{k+1}$ satisfies (5.1.2) for each $y$ with $|y|\in|W_{k+1}|$,
so that the construction of $[\theta]$ is completed by a finite number of
these steps.

As a preliminary step towards (a), we prove that the homology of the branch
locus ${\rm Br}$ of $S$ vanishes for $q\geq n$; indeed

(5.1.3) $\displaystyle\check{H}^{\infty}_{q}({\rm Br}\cap
W_{k})=0,\quad\forall q\geq n,\ k.$

To see this, we first claim that $\check{H}^{\infty}_{q}(A)=0,q\geq n$ for
every closed, proper (i.e. $A\neq Y$) subset $A$ of a connected $n$-manifold
$Y$. This holds because, by the universal coefficient theorem [Ma, Cor.4.18],
we may identify $\check{H}^{\infty}_{q}(A)$ with the dual ${\rm
Hom}(\check{H}_{c}^{q}(A),{\mathbb{Q}})$ of the corresponding cohomology
group, and $\check{H}_{c}^{q}(A)=0$ for these $A$ by [Ma, Ex.1,p78]. But each
intersection ${\rm Br}\cap W_{k}$ is a finite union of such subsets $A$, since
${\rm Br}$ is closed and nowhere dense in $S$ by [HWZ, Lem.9.2.7], and each
$W_{k}$ is a finite union of open subsets of ${\mathbb{R}}^{n}$. Therefore,
(5.1.3) holds by an inductive argument using the Mayer-Vietoris sequence in
(iv).

Claim (a) now follows by applying the exact sequence (A.5) with $i=n$, $A={\rm
Br}\cap W$ and $Y=W$. Further (b) holds because of the exactness of

$\check{H}^{\infty}_{n+1}\bigl{(}{\rm Br}\cap(W_{k}\cap
W)\bigr{)}\to\check{H}^{\infty}_{n+1}(W_{k}\cap
W)\to\check{H}^{\infty}_{n+1}\bigl{(}(W_{k}\cap W){\smallsetminus}{\rm
Br}\bigr{)},$

and the fact that $(W_{k}\cap W){\smallsetminus}{\rm Br}$ is a disjoint union
of connected $n$-manifolds so that its homology vanishes in dimension $>n$.
Similarly, to prove (c) it suffices to check that the two classes
$\rho_{W_{k},W_{k}\cap W}([\theta]_{k}),\ \rho_{W,W_{k}\cap W}([\theta]_{W})$
in $\check{H}^{\infty}_{n}(W_{k}\cap W)$ have the same restriction to
$(W_{k}\cap W){\smallsetminus}{\rm Br}$. But, just as in the first paragraph
of this proof, it follows from (5.1.2) that for each connected component $|C|$
of this space the class $[\theta]_{W}$ has the form ${\sigma}_{C}|\mu_{C}|$,
where ${\sigma}_{C}$ is the sum of the weights ${\sigma}_{x,i}$ of the local
branches that map to $|C|$. A similar claim holds for $[\theta]_{k}$. Thus the
restriction to $|C|$ of each of these two classes is ${\sigma}_{C}|\mu_{C}|$.
This completes the definition of $[\theta]$ and hence of $[{\Lambda}\circ f]$.

To prove the identity $(\phi^{0})_{*}[{\Lambda}^{0}\circ
f]=(\phi^{1})_{*}[{\Lambda}^{1}\circ f]\in\check{H}_{d}(B;{\mathbb{Q}})$ in
the second paragraph, note that by (A.5) there is a commutative diagram with
exact rows

where the vertical maps are induced by the inclusion $\phi:|{\rm
supp\,}(\Theta)|\to[0,1]\times B$ and $\theta_{i}:=\Lambda^{i}\circ f$. The
result will follow if we show that the boundary map

${\delta}:\check{H}^{\infty}_{d+1}\bigl{(}|{\rm
supp\,}(\Theta)|\cap\bigr{(}(0,1)\times
B\bigr{)}\bigr{)}\to\check{H}^{\infty}_{d}\bigl{(}|{\rm
supp\,}(\theta_{0})|\sqcup|{\rm supp\,}(\theta_{1})|\bigr{)}$

takes the fundamental class of $|{\rm supp\,}(\Theta)|\cap\bigr{(}(0,1)\times
B$ to the pair $(-[\theta_{0}],[\theta_{1}])$. But by the naturality property
(5.1.2) and condition (a) above, it suffices to check this on each component
of the set of good points; and it holds in this case because, for each open
subset $U\subset{\mathbb{R}}^{d}$, the map
${\delta}:\check{H}_{d+1}^{\infty}\bigl{(}(0,1)\times
U\bigr{)}\to\check{H}^{\infty}_{d}\bigl{(}\\{0,1\\}\times U\bigr{)}$ takes
$\mu_{(0,1)\times U}$ to $(-\mu_{\\{0\\}\times U},\mu_{\\{0\\}\times U}).$
Since $\check{H}^{\infty}_{d}(Y)=\check{H}_{d}(Y)$ for compact spaces $Y$,
this completes the proof. ∎

######  Theorem 5.1.4.

Every polyfold model without boundary
$\bigl{(}f:{\mathcal{X}}\to{\mathcal{W}},\psi\bigr{)}$ for a compact space
${\overline{{\mathcal{M}}}}$ induces a well defined fundamental class

(5.1.4)
$[{\overline{{\mathcal{M}}}}]_{f}=\psi_{*}\bigl{(}\underset{\leftarrow}{\lim}\,(\phi_{k})_{*}[\Lambda_{k}\circ
f]\bigr{)}\;\in\;\check{H}_{d}({\overline{{\mathcal{M}}}};{\mathbb{Q}}).$

In particular, this limit exists and is independent of the choice of
compactness controlling data $(N,{\mathcal{U}})$, nested sequence of open
subsets $B_{k}\subset B_{k-1}\subset|{\mathcal{U}}|\subset|{\mathcal{X}}|$
such that $\bigcap_{k\in{\mathbb{N}}}B_{k}=|f^{-1}(0)|$, and sc+-multisections
$\Lambda_{k}$ with the following properties.

  * (i)

Each $\Lambda_{k}:{\mathcal{W}}\to{\mathbb{Q}}^{\geq 0}$ is a sc+-multisection
as in [HWZ, Def.13.2.1] that is transverse, supported in
${\mathcal{U}}\cap\pi_{\mathcal{X}}^{-1}(B_{k})$, and bounded
$N(\Lambda_{k})<1$ in the sense of [HWZ, Thm.15.3.7].

  * (ii)

$\Lambda_{k}\circ f:X\to{\mathbb{Q}}^{\geq 0}$ is a compact, oriented, tame
branched ep+-subgroupoid of ${\mathcal{X}}$ in the sense of [HWZ, §9]. Its
solution set $S_{k}:=|{\rm supp\,}({\Lambda}_{k}\circ
f)|\subset|{\mathcal{X}}|$ carries a fundamental class $[\Lambda_{k}\circ
f]\in\check{H}_{d}(S_{k};{\mathbb{Q}})$, that is pushed forward to
$(\phi_{k})_{*}[\Lambda_{k}\circ f]\in\check{H}_{d}(B_{k};{\mathbb{Q}})$ by
the inclusion map $\phi_{k}:S_{k}\to B_{k}\subset|{\mathcal{X}}|$.

###### Proof.

A nested sequence $B_{k}\subset B_{k-1}\subset|{\mathcal{U}}|$ with
$\bigcap_{k\in{\mathbb{N}}}B_{k}=|f^{-1}(0)|$ exists since $|{\mathcal{X}}|$
is metrizable, so we can use balls around $|f^{-1}(0)|$ intersected with
$|{\mathcal{U}}|$. For any choice of nested sequence, denote
${\mathcal{U}}_{k}:=\pi_{\mathcal{X}}^{-1}(B_{k})\subset{\mathcal{U}}\subset
X$. Then Remark 3.5.2 guarantees that $(N,{\mathcal{U}}_{k})$ controls
compactness of $f$. Now [HWZ, Thm.15.3.7] guarantees the existence of
$(N,{\mathcal{U}}_{k})$-regular sc+-multisections
$\Lambda_{k}:{\mathcal{W}}\to{\mathbb{Q}}^{\geq 0}$. Their domain support in
${\mathcal{U}}_{k}$ [HWZ, Def.13.2.2.] guarantees that $\Lambda_{k}(f(x))=0$
for all $x\in X{\smallsetminus}{\mathcal{U}}_{k}$, so that the realization
$C_{k}$ of the “perturbed solution set” $|S_{\Lambda_{k}\circ f}|=|\\{x\in
X\,|\,\Lambda_{k}(f(x))>0\\}|$ is compact: indeed, it is closed and contained
in $\bigl{|}\\{x\in{\mathcal{U}}\,|\,f(x)\in
W[1],N(f(x))<1\\}\bigr{|}\bigr{)}$ by the bound $N(\Lambda_{k})<1$, which is
shorthand for $N(\Lambda_{k}(w))<1$ whenever $\Lambda_{k}(w)\neq 0$. Further
$|S_{\Lambda_{k}\circ f}|\subset B_{k}$.

Next, $|S_{\Lambda_{k}\circ f}|$ is given a smooth structure by the fact that
$\Lambda_{k}\circ f:X\to{\mathbb{Q}}^{\geq 0}$ is a compact, tame branched
ep+-subgroupoid; see [HWZ, Thm.15.3.7]. In addition, [HWZ, Thm.15.4.1]
transfers the orientation of $f$ to a canonical orientation of this branched
ep+-subgroupoid (by applying the solution set convention in §2.3 to each
branch). Thus $|S_{\Lambda_{k}\circ f}|$ has a natural fundamental class
$[{\Lambda}_{k}\circ f]\in\check{H}_{d}(|S_{\Lambda_{k}\circ
f}|;{\mathbb{Q}})$ by Lemma 5.1.3, which can be pushed forward by the
continuous inclusion map $\phi_{k}:S_{k}\to B_{k}\subset|{\mathcal{X}}|$.

It remains to prove that the limit
$\underset{\leftarrow}{\lim}\,(\phi_{k})_{*}[\Lambda_{k}\circ
f]\;\in\;\check{H}_{d}(|f^{-1}(0)|;{\mathbb{Q}})$ exists and is independent of
the choices involved. Here the main technical work is to construct a cobordism
given any choice of compactness controlling data
$(N^{\prime},{\mathcal{U}}^{\prime})$ and two
$(N^{\prime},{\mathcal{U}}^{\prime})$-regular sc+-multisections
$\Lambda^{i}:{\mathcal{W}}\to{\mathbb{Q}}^{\geq 0}$, following the arguments
of [HWZ, Cor.15.3.10].

As a first step, notice that $\widetilde{f}(t,x):=(t,f(x))$ is a sc-Fredholm
section of the strong bundle
$[0,1]\times{\mathcal{W}}\to[0,1]\times{\mathcal{X}}$ with compact solution
set $|\widetilde{f}^{-1}(0)|=[0,1]\times|f^{-1}(0)|$, and
$\widetilde{N}(t,w):=N^{\prime}(w)$ and
$\widetilde{\mathcal{U}}:=[0,1]\times{\mathcal{U}}^{\prime}$ control
compactness of $\widetilde{f}$. Moreover, the orientation of $f$ and natural
orientation of $[0,1]$ induce an orientation of $\widetilde{f}$ by the product
convention in §2.3. Next, instead of the general multisection extensions in
[HWZ, §14], we can directly construct a sc+-multisection
$\Lambda^{\prime}:[0,1]\times{\mathcal{W}}\to{\mathbb{Q}}^{\geq 0}$ which
extends $\widetilde{\Lambda}|_{\\{i\\}\times{\mathcal{W}}}=\Lambda^{i}$ given
on the boundary
$\partial|[0,1]\times{\mathcal{X}}|=\\{0,1\\}\times|{\mathcal{X}}|$ and
satisfies properties (1) and (2) of
$(\widetilde{N},\widetilde{\mathcal{U}})$-regularity. For that purpose let
$\beta:[0,1]\to[0,1]$ be a smooth function with
$\beta|_{[0,\frac{1}{4}]}\equiv 1$, $1>\beta|_{(\frac{1}{4},\frac{1}{3})}>0$,
$\beta|_{[\frac{1}{3},1]}\equiv 0$. Then we obtain a sc+-multisection
$\Lambda^{\prime}:[0,1]\times{\mathcal{W}}\to{\mathbb{Q}}^{\geq 0}$ by
extending $\Lambda^{\prime}(t,w):=\Lambda^{0}(\beta(t)^{-1}w)$ for
$t<\frac{1}{3}$ and $\Lambda^{\prime}(t,w):=\Lambda^{1}(\beta(1-t)^{-1}w)$ for
$t>\frac{2}{3}$ with the trivial multisection for $\frac{1}{3}\leq
t\leq\frac{2}{3}$, that is $\Lambda^{\prime}(t,0_{x})=1$ and
$\Lambda^{\prime}(t,w)=0$ for all $w\neq 0_{P(w)}$. To check the
sc+-smoothness and $(\widetilde{N},\widetilde{\mathcal{U}})$-regularity, let
$({\mathfrak{s}}_{i},\sigma_{i})_{i\in I}$ be local sections and weights
representing $\Lambda^{0}$. Then
$\Lambda^{\prime}|_{[0,\frac{2}{3}]\times{\mathcal{W}}}$ is locally
represented by $(\beta{\mathfrak{s}}_{i},\sigma_{i})_{i\in I}$ and we can
check that (1) $N^{\prime}(\beta{\mathfrak{s}}_{i})<1$ and (2) ${\rm
supp\,}(\beta{\mathfrak{s}}_{i})\subset[0,1]\times{\mathcal{U}}^{\prime}$.
Analogously, $\Lambda^{\prime}|_{[\frac{1}{3},0]\times{\mathcal{W}}}$ inherits
local section structures from $\Lambda^{1}$, and both of these section
structures for $\Lambda^{\prime}$ match with a trivial section structure
${\mathfrak{s}}_{i}(x)=0_{x}$ with weights $\sum\sigma_{i}=1$ representing
$\Lambda^{\prime}$ on $[\frac{1}{3},\frac{2}{3}]\times{\mathcal{W}}$.

Moreover, $\Lambda^{\prime}$ already satisfies the transversality conditions
(3),(4) in [HWZ, Thm.15.3.7] over the boundary of $[0,1]\times{\mathcal{X}}$.
Indeed, for a local section structure $({\mathfrak{s}}_{i},\sigma_{i})_{i\in
I}$ representing $\Lambda^{0}$, we have $\Lambda^{\prime}$ for $t<\frac{1}{4}$
represented by the local sections
${\mathfrak{s}}^{\prime}_{i}(t,x)={\mathfrak{s}}_{i}(x)$ and weights
$\sigma_{i}$. Then ${\rm T}_{(\widetilde{f},\Lambda^{\prime})}(t,x)$ is
shorthand for the collection of linearized operators ${\rm
D}(\widetilde{f}-{\mathfrak{s}}^{\prime}_{i})(t,x):{\rm T}_{(t,x)}[0,1]\times
X\to W_{x}$ for each $i\in I$ with
$\widetilde{f}(t,x)={\mathfrak{s}}^{\prime}_{i}(t,x)$. For $0\leq
t\leq\frac{1}{4}$ those are the operators ${\mathbb{R}}\times{\rm T}_{x}X\to
W_{x},(\delta t,\delta x)\mapsto{\rm D}(f-{\mathfrak{s}}_{i})(x)\delta x$ for
each $i\in I$ with $f(x)={\mathfrak{s}}_{i}(x)$. So surjectivity follows from
transversality of $\Lambda^{0}\circ f$ and the kernels are $\ker{\rm
D}(\widetilde{f}-{\mathfrak{s}}^{\prime}_{i})(t,x)={\mathbb{R}}\times\ker{\rm
D}(f-{\mathfrak{s}}_{i})(x)$. The latter satisfy the general position
requirement [HWZ, Def.5.3.9] since the reduced tangent space464646For readers
tracing references in [HWZ] it might help to note that $E_{x}\subsetneq E$ in
[HWZ, Def.2.4.7] is defined on the prior page. In our setting, the partial
cone is $C=[0,\infty)\oplus W$ for which $E_{(0,w)}=\\{0\\}\oplus W$. is ${\rm
T}_{(0,x)}([0,1]\times X)=\\{0\\}\times{\rm T}_{x}X$. Hence suitable
complements of $\ker{\rm D}(\widetilde{f}-{\mathfrak{s}}^{\prime}_{i})$ are
given by complements of $\ker{\rm D}(f-{\mathfrak{s}}_{i})(x)$. Analogously,
transversality and general position for $t\geq\frac{3}{4}$ follows from the
properties of $\Lambda^{1}$.

Now the proof of [HWZ, Thm.15.3.9] provides an
$(\widetilde{N},\widetilde{\mathcal{U}})$-regular sc+-multisection
$\widetilde{\Lambda}:[0,1]\times{\mathcal{W}}\to{\mathbb{Q}}^{\geq 0}$ which
extends $\widetilde{\Lambda}|_{\\{i\\}\times{\mathcal{W}}}=\Lambda^{i}$ by
adding to $\Lambda^{\prime}$ finitely many contributions supported in the
interior. In fact, the additional contributions can be chosen with support
over $(\frac{1}{4},\frac{3}{4})\times{\mathcal{X}}$ so that we have
$\Theta|_{[0,{\varepsilon})\times{\mathcal{X}}}(t,x)={\Lambda}^{0}(f(x))$,
$\Theta|_{(1-{\varepsilon},1]\times{\mathcal{X}}}(t,x)={\Lambda}^{1}(f(x))$
for ${\varepsilon}=\frac{1}{4}$. The resulting functor
$\Theta:=\widetilde{\Lambda}\circ\widetilde{f}:[0,1]\times
X\to{\mathbb{Q}}^{\geq 0}$ is a compact, tame branched ep+-subgroupoid, and
[HWZ, Thm.15.4.1] transfers the orientation of $\widetilde{f}$ to a canonical
orientation of this branched ep+-subgroupoid by applying the solution set
convention in §2.3 to each branch. And as in Remark 2.3.2 the orientation of
$\widetilde{f}|_{\\{1\\}\times X}$ induced by boundary restriction of the
product orientation of $\widetilde{f}$ agrees with the orientation of $f$, and
the orientation of $\widetilde{f}|_{\\{0\\}\times X}$ is opposite to the
orientation of $f$. Thus with the solution set orientation convention we
obtain oriented branched ep+-subgroupoid identifications ${\Lambda}^{1}\circ
f=\Theta|_{\\{1\\}\times{\mathcal{X}}}$ and $({\Lambda}^{0}\circ
f)^{-}=\Theta|_{\\{0\\}\times{\mathcal{X}}}$, so that Lemma 5.1.3 identifies
the resulting fundamental classes

(5.1.5) $(\phi^{0})_{*}[\Lambda^{0}\circ f]=(\phi^{1})_{*}[\Lambda^{1}\circ
f]\;\in\;\check{H}_{d}(|{\mathcal{U}}^{\prime}|;{\mathbb{Q}}).$

Here $\phi^{i}:|S_{\Lambda^{i}\circ f}|\to B$ denotes the inclusion maps for
$i=0,1$.

We will use this to prove existence of the limit
$\underset{\leftarrow}{\lim}\,(\phi_{k})_{*}[\Lambda_{k}\circ f]$ for the
system of homeomorphisms
$(\iota_{k\ell})_{*}:\check{H}_{d}(B_{\ell};{\mathbb{Q}})\to\check{H}_{d}(B_{k};{\mathbb{Q}})$
induced by the inclusions $\iota_{k\ell}:B_{\ell}\to
B_{k}\subset|{\mathcal{U}}|$ for $k<\ell$ and any choice of
$(N,{\mathcal{U}}_{k})$-regular sc+-multisections $\Lambda_{k}$. The inclusion
${\mathcal{U}}_{\ell}=\pi_{\mathcal{X}}^{-1}(B_{\ell})\subset{\mathcal{U}}_{k}$
guarantees that $\Lambda_{\ell}$ is also $(N,{\mathcal{U}}_{k})$-regular.
Comparing it to $\Lambda_{k}$ in (5.1.5) for
$(N^{\prime},{\mathcal{U}}^{\prime})=(N,{\mathcal{U}}_{k})$ yields
$(\phi_{\ell,k})_{*}[\Lambda_{\ell}\circ f]=(\phi_{k})_{*}[\Lambda_{k}\circ
f]\;\in\;\check{H}_{d}(B_{k};{\mathbb{Q}})$, where the continuous map
$\phi_{\ell,k}:|S_{\Lambda_{\ell}\circ f}|\to B_{k}$ is given by composing
$\phi_{\ell}:|S_{\Lambda_{\ell}\circ f}|\to B_{\ell}$ with $\iota_{k\ell}$. So
this identity becomes
$(\iota_{k\ell})_{*}(\phi_{\ell})_{*}[\Lambda_{\ell}\circ
f]=(\phi_{k})_{*}[\Lambda_{k}\circ f]$, as required for the limit.

To prove independence of choices we need to identify the inverse limits
arising from different choices, indexed by $i=0,1$, of compactness controlling
data $(N^{i},{\mathcal{U}}^{i})$, nested sequence $B^{i}_{k}\subset
B^{i}_{k-1}\subset|{\mathcal{U}}^{i}|$ with
$\bigcap_{k\in{\mathbb{N}}}B^{i}_{k}=|f^{-1}(0)|$, and
$(N^{i},{\mathcal{U}}^{i}_{k})$-regular multisections $\Lambda^{i}_{k}$ for
${\mathcal{U}}^{i}_{k}:=\pi_{\mathcal{X}}^{-1}(B^{i}_{k})$. We can compare
both limits to one arising from the compactness controlling data
$(\max\\{N^{0},N^{1}\\},{\mathcal{U}}^{0}_{k}\cap{\mathcal{U}}^{1}_{k})$, the
nested sequence $B^{0}_{k}\cap
B^{1}_{k}\subset|{\mathcal{U}}^{0}_{k}\cap{\mathcal{U}}^{1}_{k}|$, and
corresponding multisections. Thus it suffices to prove equality of the induced
fundamental class for choices as above where either just the multisections
differ, or we can (after shifting notation) assume $N^{0}\leq N^{1}$,
${\mathcal{U}}^{1}\subset{\mathcal{U}}^{0}$, and $B^{1}_{k}\subset B^{0}_{k}$.
In both situations we can identify the limits by ensuring that the inclusion
maps $\iota^{01}_{k}:B^{1}_{k}\hookrightarrow B^{0}_{k}$ yield

(5.1.6) $(\phi^{0}_{k})_{*}[\Lambda^{0}_{k}\circ
f]=(\iota^{01}_{k})_{*}(\phi^{1}_{k})_{*}[\Lambda^{1}_{k}\circ
f]\;\in\;\check{H}_{d}(B^{0}_{k};{\mathbb{Q}})\qquad\text{for sufficiently
large}\;k.$

To do so first note that for fixed $k$, the multisection $\Lambda^{1}_{k}$ is
also $(N^{0},{\mathcal{U}}^{0}_{k})$-regular, so that just the maps
$\phi^{1}_{k}:|\Lambda^{1}_{k}\circ f|\to|{\mathcal{U}}^{1}_{k}|$ and
$\phi^{1,0}_{k}:|\Lambda^{1}_{k}\circ f|\to|{\mathcal{U}}^{0}_{k}|$ differ by
$\phi^{1,0}_{k}=\iota^{10}_{k}\circ\phi^{0}_{k}$. Thus we obtain

$(\phi^{1,0}_{k})_{*}[\Lambda^{1}_{k}\circ
f]=(\iota^{10}_{k})_{*}(\phi^{1}_{k})_{*}[\Lambda^{1}_{k}\circ
f]\;\in\;\check{H}_{d}(|{\mathcal{U}}^{0}_{k}|;{\mathbb{Q}}).$

Now it remains to compare the classes arising from the two
$(N^{0},{\mathcal{U}}^{0})$-regular multisections
$\Lambda^{1}_{k},\Lambda^{0}_{k}$. Here the prior cobordism construction
applies, so that (5.1.5) yields

$(\phi^{0}_{k})_{*}[\Lambda^{0}_{k}\circ
f]=(\phi^{1,0}_{k})_{*}[\Lambda^{1}\circ
f]\;\in\;\check{H}_{d}(|{\mathcal{U}}^{0}_{k}|;{\mathbb{Q}}).$

Combining both identities proves (5.1.6) and thus establishes independence of
the fundamental class $[{\overline{{\mathcal{M}}}}]_{f}$ from the choices in
this construction. ∎

######  Remark 5.1.5.

The above construction can be adapted to the more general case when the base
polyfold ${\mathcal{X}}$ has formal boundary, so that the zero set
${\mathcal{M}}:=|f^{-1}(0)|$ of a polyfold-Fredholm section
$f:{\mathcal{X}}\to{\mathcal{W}}$ has formal boundary
${\partial}{\overline{{\mathcal{M}}}}=|f^{-1}(0)|\cap|{\partial}{\mathcal{X}}|$.
In this case, as stated in Corollary 2.1.3, there is a relative fundamental
class $[{\mathcal{M}}]_{f}$ constructed as above that is an element of the
Çech homology group
$\check{H}^{\infty}_{d}({\overline{{\mathcal{M}}}}{\smallsetminus}{\partial}{\overline{{\mathcal{M}}}})$.
Further details are left to the interested reader.

###  5.2. The construction of ${\mathcal{X}}_{\mathcal{V}}$ from
${\mathcal{X}}$

Consider an ep-groupoid ${\mathcal{X}}=(X,{\bf X})$ as in Definition 2.1.1.
This subsection constructs the ep-groupoid ${\mathcal{X}}_{\mathcal{V}}$ of
Theorem 2.1.2 which is covered by local uniformizers
$X_{\mathcal{V}}=\bigsqcup_{J}V_{J}$ and ${\rm
Mor}_{{\mathcal{X}}_{\mathcal{V}}}(V_{J},V_{J})=G_{J}\times V_{J}$, along with
a functor $\psi:{\mathcal{X}}_{\mathcal{V}}\to{\mathcal{X}}|_{V}$ which
induces a homeomorphism
$|\psi|:|{\mathcal{X}}_{\mathcal{V}}|\stackrel{{\scriptstyle\simeq}}{{\to}}|V|\subset|{\mathcal{X}}|$
to a neighbourhood of a given compact solution set $S\subset|{\mathcal{X}}|$.
The construction proceeds in the following steps:

  * $\bullet$

In Lemma 5.2.1 we find a finite collection of local uniformizers
$(U_{i},G_{i})_{i=1,\dots,N}$ for ${\mathcal{X}}$ (see Definition 3.2.1) whose
realizations $\bigcup_{i}|U_{i}|\subset|{\mathcal{X}}|$ cover the compact
solution set $S\subset|{\mathcal{X}}|$.

  * $\bullet$

Lemma 5.2.2 compiles these local uniformizers into data
$(U_{J},G_{J},\rho_{IJ})_{J\subset A}$ which satisfies the conditions (a)-(c)
of §4.1 for étale data, except for the (separation) property. Such data
consists of translation groupoids $(U_{J},G_{J}\times U_{J})$ for each
$J\subset A:=\\{1,\ldots,N\\}$ and étale maps $\rho_{IJ}:U_{J}\to U_{I}$ for
$I\subset J$. It is related to the original ep-groupoid ${\mathcal{X}}$ by so-
called footprint maps $\psi_{J}:U_{J}\to|{\mathcal{X}}|$ which lift
homeomorphisms $\text{\raise 3.44444pt\hbox{$\scriptstyle U_{J}\\!$}/\lower
3.44444pt\hbox{$\\!\scriptstyle G_{J}$}}\simeq\bigcap_{i\in J}|U_{i}|=:F_{J}$.
The $\rho_{IJ}$ can then be thought of as lifts of the inclusions
$F_{J}\subset F_{I}$.

  * $\bullet$

Lemma 5.2.5 constructs subsets $V_{J}\subset U_{J}$ from a cover reduction
$F^{\prime}_{J}\subset F_{J}$ of $S\subset\bigcup_{J\in A}F^{\prime}_{J}$ such
that the restrictions
$(V_{J},G_{J},\rho_{IJ}|_{{\widetilde{V}}_{IJ}})_{J\subset A}$ with
${\widetilde{V}}_{IJ}=\rho_{IJ}^{-1}(V_{I})\subset V_{J}$ form étale data as
in §4.1. This cover reduction is done to meet the separation properties: ${\rm
cl}(V_{HI})\cap{\rm cl}(V_{HJ})=\emptyset$ unless $I\subset J$ or $J\subset
I$, and ${\rm cl}({\widetilde{V}}_{IJ})\cap{\rm
cl}({\widetilde{V}}_{HJ})=\emptyset$ unless $I\subset H$ or $H\subset I$.

  * $\bullet$

Theorem 5.2.6 then uses Proposition 4.1.1 to compile the ep-groupoid
${\mathcal{X}}_{\mathcal{V}}$ from the étale data $(V_{J},G_{J},\rho_{IJ})$.
Here each $(V_{J},G_{J})$ will turn out to be a local uniformizer for
${\mathcal{X}}_{\mathcal{V}}$, and the footprint maps
$\psi_{J}:V_{J}\to|{\mathcal{X}}|$ are lifted to construct the functor
$\psi:{\mathcal{X}}_{\mathcal{V}}\to{\mathcal{X}}$.

Although the main application is to the polyfold case, all constructions in
this subsection are valid when ${\mathcal{X}}$ is a general étale proper
groupoid in the sense of Definition 3.1.2 with a given compact subset
$S\subset|{\mathcal{X}}|$ that is covered by a finite family of local
uniformizers. The proof is a more detailed version of the arguments in [Mc1].

######  Lemma 5.2.1.

Let ${\mathcal{X}}=(X,{\bf X})$ be an ep-groupoid as in Definition 2.1.1, and
fix a compact subset $S\subset|{\mathcal{X}}|$. Then there exists a finite
collection of local uniformizers $(U_{i},G_{i})_{i=1,\dots,N}$ as in
Definition 3.2.1, whose realizations $(|U_{i}|)_{i=1,\dots,N}$ cover
$S\subset\bigcup_{i=1}^{N}|U_{i}|\subset|{\mathcal{X}}|$.

###### Proof.

Given any object $x\in X$, the isotropy group $G_{x}=\\{g\in{\bf
X}\,|\,s(g)=t(g)=x\\}$ is finite by [HWZ, Proposition 7.1.12], and [HWZ,
Proposition 7.1.19] guarantees the existence of a local uniformizer
$(U_{x},G_{x},{\Gamma}_{x})$ around $x$ as follows:

  * $\bullet$

$U_{x}\subset X$ is an open neighbourhood of $x$, equipped with the natural
$G_{x}$-action from [HWZ, Theorem 7.1.13].

  * $\bullet$

${\Gamma}_{x}:(U_{x},G_{x}\times U_{x})\to(X,{\bf X})$ is a functor consisting
of the inclusion $U_{x}\to X$ and a bijection $G_{x}\times U_{x}\to{\rm
Mor}_{\mathcal{X}}(U_{x},U_{x})$, which induces a homeomorphism $\text{\raise
3.44444pt\hbox{$\scriptstyle U_{x}\\!$}/\lower 3.44444pt\hbox{$\\!\scriptstyle
G_{x}$}}\to|U_{x}|\subset|{\mathcal{X}}|$.

This data also forms a local uniformizer in the sense of Definition 3.2.1
since these two notions are the same by Remark 3.2.2 (ii).

Since $S$ is compact and $S\subset\bigcup_{x\in\pi^{-1}(S)}|U_{x}|$, we can
choose a finite subcover $S\subset\bigcup_{i=1}^{N}|U_{x_{i}}|$ indexed by
finitely many $x_{1},\ldots,x_{N}\in X$. This proves the Lemma, where we
rename $U_{i}:=U_{x_{i}}$ and $G_{i}:=G_{x_{i}}$. ∎

The following constructions are formulated for general étale proper groupoids.
The special case of an ep-groupoid (which represents a polyfold) is obtained
by working in the étale category whose spaces are M-polyfolds, and whose maps
are local sc-diffeomorphisms (that is, sc-smooth maps that have local sc-
smooth inverses).

######  Lemma 5.2.2.

Let $(U_{i},G_{i})_{i\in A}$ be a finite collection of local uniformizers of
an étale proper groupoid ${\mathcal{X}}$ with footprints
$(F_{i}:=|U_{i}|)_{i\in A}$, and for every $I\subset A$ define
$F_{I}:=\cap_{i\in I}F_{i}$. Choose an identification $A=\\{1,\dots,N\\}$,
which induces an order $I=\\{i_{1}<\dots<i_{k}\\}$ on each $I\subset A$. Then
the following holds.

  * (i)

For every nonempty $I=\\{i_{1}<\dots<i_{k}\\}\subset A$ there is a translation
groupoid $(U_{I},G_{I}\times U_{I})$ with footprint map $\psi_{I}:U_{I}\to
F_{I}\subset|{\mathcal{X}}|$ as follows:

    * (a)

The domain

$\displaystyle U_{I}:=U_{i_{1}\dots i_{k}}$
$\displaystyle:=\bigl{\\{}{\underline{x}}=\bigl{(}x_{i_{1}},m_{i_{1}i_{2}},x_{i_{2}},\ldots,m_{i_{k-1}i_{k}},x_{i_{k}}\bigr{)}\
\big{|}$ $\displaystyle\quad\qquad x_{i_{\ell}}\in
U_{i_{\ell}}\,,\,m_{i_{j}i_{\ell}}\in{\rm
Mor}_{\mathcal{X}}(x_{i_{j}},x_{i_{\ell}}),\ |x_{i_{1}}|\in F_{I}\bigr{\\}}$

carries a natural étale structure so that $s_{I}:U_{I}\to
U_{i_{1}},(x_{i_{1}},m_{i_{1}i_{2}},\ldots,x_{i_{k}})\mapsto x_{i_{1}}$ and
$t_{I}:U_{I}\to U_{i_{k}},(x_{i_{1}},m_{i_{1}i_{2}},\ldots,x_{i_{k}})\mapsto
x_{i_{k}}$ are étale.

    * (b)

The finite group $G_{I}:=\prod_{i_{j}\in I}G_{i_{j}}$ acts by étale maps on
$U_{I}$, with the action determined by $g_{i_{\ell}}\in G_{i_{\ell}}$ acting
as474747 In case $\ell=1$ resp. $\ell=k$ this is to be read as
$\bigl{(}g_{i_{1}}*x_{i_{1}}\,,\,g_{i_{1}}^{-1}\circ
m_{i_{1}i_{2}}\,\ldots\bigr{)}$ resp. $\bigl{(}\ldots\,m_{i_{k-1}i_{k}}\circ
g_{i_{k}}\,,\,g_{i_{k}}*x_{i_{k}}\bigr{)}$.

(5.2.1) $\displaystyle
g_{i_{\ell}}*(x_{i_{1}},m_{i_{1}i_{2}},\ldots,x_{i_{k}})=(\ldots\,m_{i_{\ell-1}i_{\ell}}\circ
g_{i_{\ell}}\,,\,g_{i_{\ell}}*x_{i_{\ell}}\,,\,g_{i_{\ell}}^{-1}\circ
m_{i_{\ell}i_{\ell+1}}\,\ldots).$

Moreover, for any $i\in I$ the action of the subgroup484848 For one element
sets $I=\\{i\\}$ we interpret
$G_{I{\smallsetminus}\\{i\\}}=G_{\emptyset}=\\{{\rm id}\\}$ to be the trivial
group. $G_{I{\smallsetminus}\\{i\\}}\subset G_{I}$ on $U_{I}$ is free.

    * (c)

The footprint map

$\psi_{I}\;:\;U_{I}\;\to\;|{\mathcal{X}}|,\qquad(x_{i_{1}},m_{i_{1}i_{2}},x_{i_{2}},\ldots,m_{i_{k-1}i_{k}},x_{i_{k}})\;\mapsto\;|x_{i_{1}}|=\ldots=|x_{i_{k}}|$

induces a homeomorphism $|\psi_{I}|:\text{\raise 3.44444pt\hbox{$\scriptstyle
U_{I}\\!$}/\lower 3.44444pt\hbox{$\\!\scriptstyle G_{I}$}}\to F_{I}$.

  * (ii)

For $I\subset J$, define $U_{IJ}:=\psi_{I}^{-1}(F_{J})$. Then there is an
étale map $\rho_{IJ}:U_{J}\to U_{I}$ with image $U_{IJ}$ given by forgetting
the $x_{j_{\ell}}$ for $j_{\ell}\in J{\smallsetminus}I$ and suitably composing
or forgetting morphisms. That is, for $I=J{\smallsetminus}\\{j_{\ell}\\}$ we
define $\rho_{IJ}$ which forgets the object $x_{j_{\ell}}$ by

$\rho_{(J{\smallsetminus}\\{j_{\ell}\\})J}(\ldots,x_{j_{\ell-1}},m,x_{j_{\ell}},m^{\prime},x_{j_{\ell+1}},\ldots)\,:=\,(\ldots,x_{j_{\ell-1}},m\circ
m^{\prime},x_{j_{\ell+1}},\ldots)$

for $1<\ell<|J|$, and for $\ell=1$ resp. $\ell=|J|$ by

$\displaystyle\rho_{(J{\smallsetminus}\\{j_{1}\\})J}(x_{j_{1}},m,x_{j_{2}},\ldots):=(x_{j_{2}},\ldots),$
$\displaystyle\rho_{(J{\smallsetminus}\\{j_{|J|}\\})J}(\ldots,x_{j_{|J|-1}},m,x_{j_{|J|}}):=(\ldots,x_{j_{|J|-1}}).$

Then for $J{\smallsetminus}I=(j_{\ell_{1}}<\dots<j_{\ell_{k}})$ we construct
the map $\rho_{IJ}:=\rho_{J_{k}J_{k-1}}\circ\ldots\circ\rho_{J_{1}J_{0}}$ by
denoting $J_{0}:=J$ and $J_{n}:=J_{n-1}{\smallsetminus}\\{j_{\ell_{n}}\\}$.

These maps $\rho_{IJ}$ satisfy the following conditions for all $I\subset
J\subset K$

    * (a)

$\rho_{II}={\rm id}_{U_{I}}$;

    * (b)

$\rho_{IJ}:U_{J}\to U_{I}$ is an étale map with closed graph that is the
composite of the free quotient $U_{J}\to\text{\raise
3.44444pt\hbox{$\scriptstyle U_{J}\\!$}/\lower 3.44444pt\hbox{$\\!\scriptstyle
G_{J{\smallsetminus}I}$}}$ with a $G_{I}$-equivariant étale homeomorphism
$\text{\raise 3.44444pt\hbox{$\scriptstyle U_{J}\\!$}/\lower
3.44444pt\hbox{$\\!\scriptstyle
G_{J{\smallsetminus}I}$}}\stackrel{{\scriptstyle\simeq}}{{\to}}U_{IJ}\subset
U_{I}$; in particular it is equivariant with respect to the projection
$G_{J}\to G_{I}$ and induces a proper map $\rho_{IJ}:U_{J}\to U_{IJ}\subset
U_{J}$;

    * (c)

$\psi_{J}=\psi_{I}\circ\rho_{IJ}$ for all $I\subset J$;

    * (d)

for $H\subset I\subset J$ we have $\rho_{HJ}=\rho_{HI}\circ\rho_{IJ}$.

######  Remark 5.2.3.

In the following argument, we will frequently use the fact that the embeddings
${\Gamma}_{i}(g,\cdot):U_{i}\to{\rm Mor}_{\mathcal{X}}(U_{i},U_{i})$ that are
part of the local uniformizers $(U_{i},G_{i})$ yield actions of $G_{i}$ and
$G_{j}$ on the morphism space ${\rm Mor}_{\mathcal{X}}(U_{i},U_{j})$,

$\displaystyle G_{i}\times{\rm Mor}_{\mathcal{X}}(U_{i},U_{j})\;\to\;{\rm
Mor}_{\mathcal{X}}(U_{i},U_{j}),\qquad$ $\displaystyle(g,m)\;\mapsto\;g\circ
m:={\Gamma}_{i}(g,g^{-1}*s(m))\circ m,$ $\displaystyle G_{j}\times{\rm
Mor}_{\mathcal{X}}(U_{i},U_{j})\;\to\;{\rm
Mor}_{\mathcal{X}}(U_{i},U_{j}),\qquad$ $\displaystyle(g,m)\;\mapsto\;m\circ
g:=m\circ{\Gamma}_{j}(g,t(m)).$

The case $i=j$ results from the natural inner action of $G_{i}$ on the local
uniformizer $(U_{i},G_{i},{\Gamma}_{i})$ in Remark 3.3.5 (ii). ${\Diamond}$

###### Proof of Lemma 5.2.2.

Definition of the $\mathbf{U}_{I}$: To understand the construction of the sets
$U_{I}$ in (i), note that for one element sets $I=\\{i\\}$ the domain
$U_{\\{i\\}}=U_{i}$ is naturally an étale space since it is an open subset of
$X$. For a two element set $I=\\{i<j\\}$ the domain

$U_{\\{i,j\\}}=:U_{ij}=\bigl{\\{}\bigl{(}s(m),m,t(m)\bigr{)}\ \big{|}\
m\in{\rm Mor}_{\mathcal{X}}(U_{i},U_{j}),\ |s(m)|\in F_{ij}\bigr{\\}}$

is naturally identified with the open subset of morphisms ${\rm
Mor}_{\mathcal{X}}(U_{i},U_{j})\subset{\bf X}$, and thus also carries an étale
structure. This identifies the maps $s_{I},t_{I}$ with the source and target
maps on ${\rm Mor}_{\mathcal{X}}(U_{i},U_{j})$, which are étale by the étale
property of the groupoid ${\mathcal{X}}$.

For general $I=\\{i_{1}<\dots<i_{k}\\}$ we can identify the space
$U_{i_{1}\dots i_{k}}:=U_{I}$ with the set of $(k-1)$-tuples of composable
morphisms $U_{i_{1}}\to U_{i_{2}}\to\dots\to U_{i_{k}}$ in ${\mathcal{X}}$.
This in particular implies that the footprint map
$\psi_{I}(x_{i_{1}},m_{i_{1}i_{2}},\ldots,x_{i_{k}})=|x_{i_{1}}|=\dots=|x_{i_{k}}|\in|{\mathcal{X}}|$
is well defined with image $F_{I}:=\bigcap_{i\in I}|U_{i}|$.

Proof of (i): For $k=1$ the properties of local uniformizers assert that
$U_{i_{1}}$ is a $G_{i_{1}}$-invariant étale space with
$s_{i_{1}}=t_{i_{1}}:U_{i_{1}}\to U_{i_{1}}$ the identity map, and
$|\psi_{i_{1}}|:\text{\raise 3.44444pt\hbox{$\scriptstyle
U_{i_{1}}\\!$}/\lower 3.44444pt\hbox{$\\!\scriptstyle
G_{i_{1}}$}}\to|U_{i_{1}}|$ is a homeomorphism. Moreover,
$G_{I{\smallsetminus}\\{i_{1}\\}}=G_{\emptyset}$ is the trivial group, which
does act freely.

To prove the regularity assertions in (i)(a) for a general
$U_{J}=U_{j_{1}\dots j_{k}}$ we may assume by induction that
$U_{J^{\prime}}:=U_{j_{1}\dots j_{k-1}}$ carries an étale structure such that
both $s_{J^{\prime}}$ and

$t_{J^{\prime}}:U_{J^{\prime}}\to
U_{j_{k-1}},\quad(x_{j_{1}},m_{j_{1}j_{2}},\ldots,x_{j_{k-1}})\mapsto
x_{j_{k-1}}$

are étale. Then we identify

(5.2.2) $U_{J}=\bigl{\\{}\bigl{(}{\underline{x}},m,t(m)\bigr{)}\ \big{|}\
{\underline{x}}\in U_{J^{\prime}}\,,\,m\in{\rm
Mor}_{\mathcal{X}}(U_{j_{k-1}},U_{j_{k}})\,,\,t_{J^{\prime}}({\underline{x}})=s(m)\bigr{\\}}$

with the fiber product
$U_{J^{\prime}}\,{\vphantom{\times_{s}}}_{t_{J^{\prime}}}{\times_{s}}{\rm
Mor}_{\mathcal{X}}(U_{j_{k-1}},U_{j_{k}})$. This naturally carries an étale
structure since both $t_{J^{\prime}}:U_{J^{\prime}}\to X$ and $s:{\bf X}\to X$
are étale. A proof in the polyfold case is given in [HWZ, Prop.5.2.20] and
generalizes to other étale categories.

In particular, we obtain charts from sufficiently small subsets of ${\rm
Mor}_{\mathcal{X}}(U_{j_{k-1}},U_{j_{k}})\subset{\bf X}$. On those charts, the
map $t_{J}:U_{J}\to U_{j_{k}}$ is identified with a restriction of the target
map $t:{\bf X}\to X$, which is étale hence also induces an étale structure.
Note for later that the forgetful map

$\rho_{J^{\prime}J}:U_{J}\to
U_{J^{\prime}},\quad({\underline{x}},m,t(m))\mapsto{\underline{x}}$

is étale, and that it is $G_{J^{\prime}}$-equivariant and constant on
$G_{j_{k}}$-orbits in $U_{J}$, so also equivariant w.r.t. the projection
$\rho_{J^{\prime}J}^{G}:G_{J}\to G_{J^{\prime}}$ that forgets $g_{j_{k}}$. In
these charts, $s_{J}:U_{J}\to U_{j_{1}}$ is identified with
$s_{J^{\prime}}\circ\rho_{J^{\prime}J}$ and thus also is étale.

To prove the claims in (i)(b) about the action of $G_{J}$ on $U_{J}$ we may
assume by induction that $G_{J^{\prime}}:=\prod_{\ell=1}^{k-1}G_{j_{\ell}}$
acts as claimed on $U_{J^{\prime}}$. Then the action of
$({\underline{g}},g_{j_{k}})\in G_{J^{\prime}}\times G_{j_{k}}=G_{J}$ is

$({\underline{g}}^{\prime},g_{j_{k}})*(x_{j_{1}},m_{j_{1}j_{2}},\ldots,x_{j_{k}})=\bigl{(}{\underline{g}}^{\prime}*(x_{j_{1}},m_{j_{1}j_{2}},\ldots,x_{j_{k-1}})\,,\,g_{j_{k-1}}^{-1}\circ
m_{j_{k-1}j_{k}}\circ g_{j_{k}}\,,\,g_{j_{k}}*x_{j_{k}}\bigr{)}.$

Since $G_{J}$ is finite, it suffices to check that this action is étale by
fixing ${\underline{g}}=({\underline{g}}^{\prime},g_{j_{k}})$ and considering
the map ${\underline{x}}\mapsto{\underline{g}}*{\underline{x}}$ in a local
chart given by the map $t_{J}({\underline{x}})=x_{k}$. In these coordinates,
the action $x_{j_{k}}\mapsto g_{j_{k}}*x_{j_{k}}$ is étale by the properties
of the local uniformizer. Moreover, this determines an action of $G_{J}$ on
$U_{J}$ because

$\displaystyle({\underline{g}},g_{j_{k}})*({\underline{h}},h_{j_{k}})*({\underline{x}}^{\prime},m,x_{j_{k}})$
$\displaystyle=({\underline{g}},g_{j_{k}})*\bigl{(}{\underline{h}}*{\underline{x}}^{\prime}\,,\,h_{j_{k-1}}^{-1}\circ
m\circ h_{j_{k}}\,,\,h_{j_{k}}*x_{j_{k}}\bigr{)}$
$\displaystyle=\bigl{(}{\underline{g}}*{\underline{h}}*{\underline{x}}^{\prime}\,,\,g_{j_{k-1}}^{-1}\circ\bigl{(}h_{j_{k-1}}^{-1}\circ
m\circ h_{j_{k}}\bigr{)}\circ
g_{j_{k}}\,,\,g_{j_{k}}*h_{j_{k}}*x_{j_{k}}\bigr{)}$
$\displaystyle=\bigl{(}({\underline{g}}{\underline{h}})*{\underline{x}}^{\prime}\,,\,(g_{j_{k-1}}h_{j_{k-1}})^{-1}\circ
m\circ(g_{j_{k}}h_{j_{k}})\,,\,(g_{j_{k}}h_{j_{k}})*x_{j_{k}}\bigr{)}$
$\displaystyle=\bigl{(}({\underline{g}},g_{j_{k}})({\underline{h}},h_{j_{k}})\bigr{)}*({\underline{x}}^{\prime},m,x_{j_{k}}).$

Next, freedom of the action of $G_{J{\smallsetminus}\\{j\\}}$ on $U_{J}$ for
$j\neq j_{k}$ follows directly from freedom of
$G_{J^{\prime}{\smallsetminus}\\{j\\}}$ acting on $U_{J^{\prime}}$. For
$j=j_{k}$ the action of ${\underline{g}}=(g_{j_{1}},\ldots,g_{j_{k-1}})\in
G_{J{\smallsetminus}\\{j_{k}\\}}$ on $U_{J}$ is

$({\underline{g}},{\rm
id}_{G_{j_{k}}})*({\underline{x}}^{\prime},m,x_{j_{k}})=\bigl{(}{\underline{g}}*{\underline{x}}^{\prime}\,,\,g_{j_{k-1}}^{-1}\circ
m\,,\,x_{j_{k}}\bigr{)}.$

Here we use that fact that the action $m\mapsto g_{j_{k-1}}^{-1}\circ m$ of
$G_{j_{k-1}}$ on ${\rm Mor}_{\mathcal{X}}(U_{j_{k-1}},U_{j_{k}})$ is free. To
see this, notice that otherwise there is $g\in G_{j_{k-1}},g\neq{\rm id},$ and
$m\in{\rm Mor}_{\mathcal{X}}(U_{j_{k-1}},U_{j_{k}})$ such that $g\circ m=m$.
In the notation of Definition 3.2.1(iii) we therefore have
${\Gamma}(g,s(x))={\rm id}_{s(x)}$. But by assumption ${\Gamma}:G\times
U\to{\rm Mor}(U,U)$ is injective, so that this is possible only if $g={\rm
id}$. So given any

$({\underline{g}},{\rm
id}_{G_{j_{k}}})*({\underline{x}}^{\prime},m,x_{j_{k}})=({\underline{x}}^{\prime},m,x_{j_{k}})$

we deduce $g_{j_{k-1}}={\rm id}$ and hence

${\underline{g}}*{\underline{x}}^{\prime}=(g_{j_{1}},\ldots,g_{j_{k-2}},{\rm
id})*{\underline{x}}^{\prime}={\underline{x}}^{\prime}.$

Now freedom of the action of $G_{J^{\prime}{\smallsetminus}\\{j_{k-1}\\}}$ on
$U_{J^{\prime}}$ implies ${\underline{g}}={\rm id}$, which confirms the
desired freedom of $G_{J{\smallsetminus}\\{j_{k}\\}}$ acting on $U_{J}$.

To prove that $|\psi_{J}|:\text{\raise 3.44444pt\hbox{$\scriptstyle
U_{J}\\!$}/\lower 3.44444pt\hbox{$\\!\scriptstyle G_{J}$}}\to
F_{J}\subset|{\mathcal{X}}|$ is a homeomorphism we may assume by induction
that $\psi_{J^{\prime}}({\underline{x}}):=|x_{j_{1}}|=\ldots|x_{j_{k-1}}|$
induces a homeomorphism $\text{\raise 3.44444pt\hbox{$\scriptstyle
U_{J^{\prime}}\\!$}/\lower 3.44444pt\hbox{$\\!\scriptstyle
G_{J^{\prime}}$}}\to F_{J^{\prime}}$. Then using (5.2.2) we have

$\psi_{J}({\underline{x}},m,t(m)):=|x_{j_{1}}|=\ldots=|x_{j_{k-1}}|=|s(m)|=|t(m)|=|x_{j_{k}}|.$

This implies
$\psi_{J}({\underline{x}},m,t(m))=\psi_{J^{\prime}}({\underline{x}})$ so we
can factor

$\psi_{J}=\psi_{J^{\prime}}\circ|\rho_{J^{\prime}J}|,$

where $|\rho_{J^{\prime}J}|:\text{\raise 3.44444pt\hbox{$\scriptstyle
U_{J}\\!$}/\lower 3.44444pt\hbox{$\\!\scriptstyle G_{J}$}}\to\text{\raise
3.44444pt\hbox{$\scriptstyle U_{J^{\prime}}\\!$}/\lower
3.44444pt\hbox{$\\!\scriptstyle G_{J^{\prime}}$}}$ is induced by the forgetful
map

$\rho_{J^{\prime}J}:U_{J}\to
U_{J}^{\prime},({\underline{x}},m,t(m))\mapsto{\underline{x}}.$

As we saw in (a), $\rho_{J^{\prime}J}$ is étale, thus an open and continuous
map. Since it is equivariant w.r.t. the projection
$\rho_{J^{\prime}J}^{G}:G_{J}\to G_{J^{\prime}}$, it induces an open and
continuous map $|\rho_{J^{\prime}J}|$. The latter is also injective since for
${\underline{x}}\in U_{J^{\prime}}$ the preimage

$\rho_{J^{\prime}J}^{-1}({\underline{x}})=\\{({\underline{x}},m,t(m))\in
U_{J}\\}\simeq\\{m\in{\rm
Mor}_{\mathcal{X}}(t^{\prime}({\underline{x}}),U_{j_{k}})\\}$

is a $G_{j_{k}}$-orbit, and we have the splitting $G_{J}=G_{J^{\prime}}\times
G_{j_{k}}$. Finally, the image of $|\psi_{J}|$ is as claimed:

$\displaystyle|\psi_{J}|\bigl{(}\text{\raise 3.44444pt\hbox{$\scriptstyle
U_{J}\\!$}/\lower 3.44444pt\hbox{$\\!\scriptstyle
G_{J}$}}\bigr{)}=\psi_{J}(U_{J})$
$\displaystyle=\\{\psi_{J^{\prime}}({\underline{x}})\,|\,{\underline{x}}\in
U_{J^{\prime}},{\rm
Mor}_{\mathcal{X}}(t^{\prime}({\underline{x}}),U_{j_{k}})\neq\emptyset\\}$
$\displaystyle=\psi_{J^{\prime}}(U_{J^{\prime}})\cap\psi_{j_{k}}(U_{j_{k}})$
$\displaystyle=F_{J^{\prime}}\cap F_{j_{k}}\;=\;F_{J}.$

Proof of (ii): Next, we check the properties of the projections $\rho_{IJ}$.
The identity $\rho_{II}={\rm id}_{U_{I}}$, footprint compatibility
$\psi_{I}\circ\rho_{IJ}=\psi_{J}:{\underline{x}}\mapsto|x_{i}|$ for any $i\in
I\subset J$, and cocycle condition $\rho_{HI}\circ\rho_{IJ}=\rho_{HJ}$ hold
directly by construction. Equivariance of $\rho_{IJ}$ w.r.t. $G$ requires
$\rho_{IJ}$ to be constant on the orbits of $G_{J{\smallsetminus}I}$. This
holds since the action of $g_{j_{\ell}}\in G_{j_{\ell}}$ for $j_{\ell}\in
J{\smallsetminus}I$ in (5.2.1) yields

$\rho_{IJ}(g_{j_{\ell}}*{\underline{x}})=(\ldots\,m_{j_{\ell-1}j_{\ell}}\circ
g_{j_{\ell}}^{-1}\circ g_{j_{\ell}}\circ
m_{j_{\ell}j_{\ell+1}}\,\ldots)=\rho_{IJ}({\underline{x}}).$

This also implies $G_{I}$-equivariance of the induced map $\text{\raise
3.44444pt\hbox{$\scriptstyle U_{J}\\!$}/\lower 3.44444pt\hbox{$\\!\scriptstyle
G_{J{\smallsetminus}I}$}}\to U_{IJ}$.

To prove that $\rho_{IJ}$ for $I\subset J$ is étale and induces an étale
homeomorphism $\text{\raise 3.44444pt\hbox{$\scriptstyle U_{J}\\!$}/\lower
3.44444pt\hbox{$\\!\scriptstyle G_{J{\smallsetminus}I}$}}\to U_{IJ}$, we work
by induction over $|J|=k$. The statement holds for $k=1$ since $\rho_{II}={\rm
id}_{U_{I}}$ is the identity map on $U_{I}=U_{II}$ and
$G_{I{\smallsetminus}I}=\\{\rm id\\}$. Given $|J|\geq 2$ we have again the
identity $\rho_{JJ}={\rm id}_{U_{J}}$ with $G_{J{\smallsetminus}J}=\\{\rm
id\\}$ for $I=J$, and for $I\subsetneq J$ can factor
$\rho_{IJ}=\rho_{IJ^{\prime}}\circ\rho_{J^{\prime}J}$ with
$J^{\prime}=J{\smallsetminus}\\{j_{i}\\}$ for some choice of $j_{i}\in
J{\smallsetminus}I$. By the induction hypothesis,
$\rho_{IJ^{\prime}}:U_{J^{\prime}}\to U_{I}$ is étale inducing an étale
homeomorphism $\text{\raise 3.44444pt\hbox{$\scriptstyle
U_{J^{\prime}}\\!$}/\lower 3.44444pt\hbox{$\\!\scriptstyle
G_{J^{\prime}{\smallsetminus}I}$}}\to
U_{IJ^{\prime}}=\psi_{I}^{-1}(|U_{J^{\prime}}|)$, so it remains to consider
the map

$\rho_{J^{\prime}J}\,:\;U_{J}\;\to\;U_{J^{\prime}},\qquad(\ldots,x_{j_{i-1}},m,x_{j_{i}},m^{\prime},x_{j_{i+1}},\ldots)\;\mapsto\;(\ldots,x_{j_{i-1}},m\circ
m^{\prime},x_{j_{i+1}},\ldots).$

For $j_{i}=j_{k}$ this is the $G_{J^{\prime}}$-equivariant étale map from
(i)(a) that is constant on $G_{j_{i}}$-orbits. For $j_{i}\neq j_{k}$ it is
also étale because it intertwines the maps $t_{J}:U_{J}\to U_{j_{k}}$ and
$t_{J^{\prime}}:U_{J^{\prime}}\to U_{j_{k}}$ which are étale by (i)(a); and it
is also $G_{J^{\prime}}$-equivariant and constant on $G_{j_{i}}$-orbits. This
already implies that $\rho_{IJ}=\rho_{IJ^{\prime}}\circ\rho_{J^{\prime}J}$ is
étale. Moreover, $\rho_{J^{\prime}J}$ induces a $G_{J^{\prime}}$-equivariant
étale map $\text{\raise 3.44444pt\hbox{$\scriptstyle U_{J}\\!$}/\lower
3.44444pt\hbox{$\\!\scriptstyle G_{j_{i}}$}}\to U_{J^{\prime}}$, which we
claim is injective and thus an étale homeomorphism to its image. Once this is
established, we may quotient by the action of
$G_{J^{\prime}{\smallsetminus}I}$ – which is free by (i)(b) – to see that the
map

$\text{\raise 3.44444pt\hbox{$\scriptstyle U_{J}\\!$}/\lower
3.44444pt\hbox{$\\!\scriptstyle G_{J{\smallsetminus}I}$}}=\bigl{(}\text{\raise
3.44444pt\hbox{$\scriptstyle U_{J}\\!$}/\lower 3.44444pt\hbox{$\\!\scriptstyle
G_{j_{i}}$}}\bigr{)}/\,G_{J^{\prime}{\smallsetminus}I}\to\text{\raise
3.44444pt\hbox{$\scriptstyle U_{J^{\prime}}\\!$}/\lower
3.44444pt\hbox{$\\!\scriptstyle G_{J^{\prime}{\smallsetminus}I}$}}$

induced by $\rho_{J^{\prime}J}$ is an étale homeomorphism to its image.
Composition with the map $\text{\raise 3.44444pt\hbox{$\scriptstyle
U_{J}^{\prime}\\!$}/\lower 3.44444pt\hbox{$\\!\scriptstyle
G_{J^{\prime}{\smallsetminus}I}$}}\to\psi_{I}^{-1}(F_{J^{\prime}})$ induced by
$\rho_{IJ^{\prime}}$, which is an étale homeomorphism, shows that the map
$\text{\raise 3.44444pt\hbox{$\scriptstyle U_{J}\\!$}/\lower
3.44444pt\hbox{$\\!\scriptstyle G_{J{\smallsetminus}I}$}}\to U_{I}$ induced by
$\rho_{IJ}$ is also an étale homeomorphism to its image. Properness of the
surjective map $\rho_{IJ}:U_{J}\to\rho_{IJ}(U_{J})\subset U_{I}$ then follows
since it is the composition of a free finite quotient map and a homeomorphism.

To check injectivity of the map $\text{\raise 3.44444pt\hbox{$\scriptstyle
U_{J}\\!$}/\lower 3.44444pt\hbox{$\\!\scriptstyle G_{j_{i}}$}}\to
U_{J^{\prime}}$ induced by $\rho_{J^{\prime}J}$, note that the preimage of
$(\ldots,x_{j_{i-1}},m^{\prime\prime},x_{j_{i+1}},\ldots)$ is given by tuples
$(\ldots,x_{j_{i-1}},m,x_{j_{i}},m^{\prime},x_{j_{i+1}},\ldots)$ with $m\circ
m^{\prime}=m^{\prime\prime}$ and $x_{j_{i}}=t(m)=s(m^{\prime})\in U_{j_{i}}$.
Due to the groupoid properties, $m^{\prime\prime}\in{\rm
Mor}_{\mathcal{X}}(U_{j_{i-1}},U_{j_{i+1}})$ determines $m,m^{\prime}\in{\rm
Mor}_{\mathcal{X}}$ up to $m\mapsto
m\circ\gamma,m^{\prime}\mapsto\gamma^{-1}\circ m^{\prime}$, and the ambiguity
is exactly $\gamma\in{\rm Mor}_{\mathcal{X}}(t(m),U_{j_{i}})\simeq G_{j_{i}}$
by the properties of local uniformizers. Finally, the image of
$\rho_{J^{\prime}J}$ is

$\displaystyle\bigl{\\{}(\ldots,x_{j_{i-1}},m^{\prime\prime},x_{j_{i+1}},\ldots)\in
U_{J^{\prime}}\,\big{|}\,\exists\,m,m^{\prime}\in{\rm
Mor}_{\mathcal{X}}:m^{\prime\prime}=m\circ m^{\prime},t(m)=s(m^{\prime})\in
U_{j_{i}}\bigr{\\}}$
$\displaystyle=\bigl{\\{}(\ldots,x_{j_{i-1}},m^{\prime\prime},x_{j_{i+1}},\ldots)\in
U_{J^{\prime}}\,\big{|}\,|x_{j_{i-1}}|=|x_{j_{i+1}}|\in
F_{j_{i}}\bigr{\\}}\;=\;\psi_{J^{\prime}}^{-1}(F_{j_{i}}).$

Here the inclusion $\subset$ follows from $m\in{\rm
Mor}_{\mathcal{X}}(x_{i-1},U_{j_{i}})$, $m^{\prime}\in{\rm
Mor}_{\mathcal{X}}(U_{j_{i}},x_{i+1})$, and towards the inclusion $\supset$ we
know that $|x_{j_{i-1}}|\in|U_{j_{i}}|$ implies existence of a morphism
$m\in{\rm Mor}_{\mathcal{X}}(x_{j_{i-1}},U_{j_{i}})$; so then
$m^{\prime}:=m^{-1}\circ m^{\prime\prime}$ gives the required decomposition. ∎

The next steps in the construction of ${\mathcal{X}}_{\mathcal{V}}$ are to
choose a cover reduction of the covering $S\subset\bigcup_{i=1}^{N}F_{i}$ by
the open subsets $F_{i}:=|U_{i}|\subset|{\mathcal{X}}|$ and construct the
corresponding reduction of the data constructed in Lemma 5.2.2. In the context
of Theorem 2.1.2, this covering reduction will be chosen compatible with the
supports $C_{i}:={\rm supp\,}\beta_{i}\subset F_{i}$ of a partition of unity
$\beta_{i}:X\to[0,1]$ used in the constructions of §5.4. We state all cover
reduction properties here, though only (i)-(iii) are needed for the
construction of ${\mathcal{X}}_{\mathcal{V}}$. The following result applies in
our setting since the realization $Y:=|{\mathcal{X}}|$ is metrizable by [HWZ,
Thm.7.3.1], and metric spaces are normal by [Mu, Thm. 32.2].

######  Lemma 5.2.4.

Let $S\subset Y$ be a closed subset of a normal topological space $Y$ with a
finite cover by open subsets $S\subset\bigcup_{i\in A}F_{i}$. Then there is a
cover reduction $\bigl{(}F^{\prime}_{I}\bigr{)}_{I\subset A}$ by open subsets
$F^{\prime}_{I}\subset Y$ in the following sense:

  * (i)

The closures are contained in the given intersections:
$\overline{F^{\prime}_{I}}\,\subset\,F_{I}:=\bigcap_{i\in I}F_{i}$.

  * (ii)

The closures intersect, i.e.
$\overline{F^{\prime}_{I}}\cap\overline{F^{\prime}_{J}}\neq\emptyset$, only if
$I\subset J$ or $J\subset I$.

  * (iii)

The open sets cover the given set $S\,\subset\,\bigcup_{I\subset
A}F^{\prime}_{I}$.

If, in addition, $S\subset\bigcup_{i\in A}C_{i}$ is another cover by closed
subsets $C_{i}\subset F_{i}$, then the cover reduction
$\bigl{(}F^{\prime}_{I}\bigr{)}_{I\subset A}$ can be chosen such that

  * (iv)

$F^{\prime}_{I}\cap C_{j}=\emptyset\quad\forall j\in A{\smallsetminus}I$.

###### Proof.

Since $Y$ is normal, we may choose open subsets $F_{i}^{0}\sqsubset F_{i}$
that still cover $S$, where the notation $A\sqsubset B$ means that the closure
of $A$ is contained in $B$. If closed subsets $C_{i}\subset F_{i}$ are given,
we more over choose the $F_{i}^{0}$ so that $C_{i}\subset F_{i}^{0}$. Now we
can follow the proof of [MW1, Lemma 5.3.1] by choosing nested open sets

(5.2.3) $\displaystyle
F_{i}^{0}\,\sqsubset\,G_{i}^{1}\,\sqsubset\,F_{i}^{1}\,\sqsubset\,G_{i}^{2}\,\sqsubset\,\ldots\,\sqsubset\,F_{i}^{N}=F_{i}.$

Then [MW1, Lemma 5.3.1] shows that the required cover reduction is achieved by

(5.2.4) $\displaystyle F^{\prime}_{I}\,:=\;\Bigl{(}{\textstyle\bigcap_{i\in
I}}G_{i}^{|I|}\Bigr{)}\;{\smallsetminus}\;\Bigl{(}\textstyle{\bigcup_{j\notin
I}}\overline{F^{|I|}_{j}}\Bigr{)}.$

If closed subsets $C_{i}\subset F_{i}$ were given, then for $j\notin I$ this
construction satisfies $C_{j}\subset F_{j}^{0}\subset F^{|I|}_{j}$ and thus
$F^{\prime}_{I}\cap C_{j}=\emptyset$. ∎

######  Lemma 5.2.5.

Let $(U_{i},G_{i})_{i\in A}$ be a finite set of local uniformizers as in Lemma
5.2.2 whose footprints $(F_{i}:=|U_{i}|)_{i\in A}$ cover a compact subset
$S\subset|{\mathcal{X}}|$, and let $\bigl{(}F^{\prime}_{I}\subset\bigcap_{i\in
I}F_{i}\bigr{)}_{I\subset A}$ be a cover reduction as in Lemma 5.2.4. Choose
an order of the set $A$ and for each $I\subset A$ define
$V_{I}:=U_{I}\cap\psi_{I}^{-1}(F_{I}^{\prime})$, where $U_{I}$ is as in Lemma
5.2.2. Then $(V_{I},G_{I},\rho_{IJ})_{I\subset A}$ forms étale data of type
$V$ in the sense of §4.1 that is compatible with the footprint maps $\psi_{I}$
as follows:

  * $\bullet$

For each $I\subset A$ the map $\psi_{I}|_{V_{I}}:V_{I}\to|{\mathcal{X}}|$
induces a homeomorphism $|\psi_{I}|:\text{\raise 3.44444pt\hbox{$\scriptstyle
V_{I}\\!$}/\lower 3.44444pt\hbox{$\\!\scriptstyle
G_{I}$}}\stackrel{{\scriptstyle\simeq}}{{\to}}F_{I}^{\prime}$.

  * $\bullet$

$\psi_{J}\big{|}_{{\widetilde{V}}_{IJ}}=\psi_{I}\circ\rho_{IJ}$.

###### Proof.

We must check that the properties (a), (b) and (c) listed in §4.1 are
satisfied. Note that (a) and (b) hold by construction. To check (c), for
$I\subset J$ we define

${\widetilde{V}}_{IJ}:=\psi_{J}^{-1}(F^{\prime}_{I}\cap F^{\prime}_{J})\subset
V_{J},\quad V_{IJ}:=\psi_{I}^{-1}(F^{\prime}_{I}\cap F^{\prime}_{J})\subset
V_{I}.$

Then, $\rho_{IJ}$ restricts to give a map ${\widetilde{V}}_{IJ}\to V_{IJ}$
because $\psi_{J}=\psi_{I}\circ\rho_{IJ}:U_{J}\to|{\mathcal{X}}|$. Further,
the other properties listed in Lemma 5.2.2 (ii) readily imply that, for each
$I\subset J$, ${\widetilde{V}}_{IJ}\subset V_{J}$ is a $G_{J}$-invariant open
subset on which $G_{J{\smallsetminus}I}$ acts freely, and
$\rho_{IJ}:{\widetilde{V}}_{IJ}\to V_{I}$ is a composite
${\widetilde{V}}_{IJ}\to\text{\raise
3.44444pt\hbox{$\scriptstyle{\widetilde{V}}_{IJ}\\!$}/\lower
3.44444pt\hbox{$\\!\scriptstyle G_{J{\smallsetminus}I}$}}\to V_{I}$ of the
quotient map with an injective étale map with open image
$V_{IJ}:=\rho_{IJ}({\widetilde{V}}_{IJ})\subset V_{I}$. Thus the identity and
equivariance properties in §4.1 (c) hold. To check the closed graph property,
consider a sequence
$(x_{k},y_{k})=(x_{k},\rho_{IJ}(x_{k}))\in{\widetilde{V}}_{IJ}\times
V_{IJ}\subset V_{J}\times V_{I}$ with limit $(x_{\infty},y_{\infty})\in
V_{J}\times V_{I}$. Then $|x_{k}|=|y_{k}|\in F_{J}^{\prime}\cap
F_{I}^{\prime}$ so that $|x_{\infty}|=|y_{\infty}|$. But
$|x_{\infty}|\in|V_{J}|=F_{J}^{\prime}$ and
$|y_{\infty}|\in|V_{I}|=F_{I}^{\prime}$. Therefore $|x_{\infty}|,|y_{\infty}|$
both lie in $F_{J}^{\prime}\cap F_{I}^{\prime}$ which implies that
$x_{\infty}\in{\widetilde{V}}_{IJ},y_{\infty}\in V_{IJ}$ and
$(x_{\infty},y_{\infty})\in{\rm graph}\rho_{IJ}$, as required.

The next step is to check that
${\widetilde{V}}_{HJ}\cap{\widetilde{V}}_{IJ}=\rho_{IJ}^{-1}({\widetilde{V}}_{HI}\cap
V_{IJ})$ when $H\subset I\subset J$. To prove this, we argue by induction as
in the proof of Lemma 5.2.2 (i), assuming this holds when $J$ is replaced by
$J^{\prime}=J-\\{j_{i}\\}$, where $j_{i}\in J{\smallsetminus}I$. We first
compute the image $V_{IJ}$ in $V_{I}$ of the étale homeomorphism induced by
$\rho_{IJ}=\rho_{IJ^{\prime}}\circ\rho_{J^{\prime}J}$ on $\scriptstyle
V_{J}\\!$/$\\!\scriptstyle G_{J{\smallsetminus}I}$ by combining the identities
$\rho_{J^{\prime}J}(V_{J})=\psi_{J^{\prime}}^{-1}(F^{\prime}_{j_{i}})$ with
${\rm im\,}\rho_{IJ^{\prime}}=\psi_{I}^{-1}(F^{\prime}_{J^{\prime}})$ and
$\psi_{J^{\prime}}^{-1}=\rho_{I{J^{\prime}}}^{-1}\circ\psi_{I}^{-1}$ to obtain

$\displaystyle
V_{IJ}:\;=\;\rho_{IJ^{\prime}}\bigl{(}\rho_{J^{\prime}J}(V_{J})\bigr{)}$
$\displaystyle=\;\rho_{IJ^{\prime}}\bigl{(}\psi_{J^{\prime}}^{-1}(F^{\prime}_{j_{i}})\bigr{)}\;=\;{\rm
im\,}\rho_{IJ^{\prime}}\cap\psi_{I}^{-1}(F^{\prime}_{j_{i}})\bigr{)}$
$\displaystyle=\;\psi_{I}^{-1}\bigl{(}F^{\prime}_{J^{\prime}}\cap
F^{\prime}_{j_{i}}\bigr{)}\;=\;\psi_{I}^{-1}(F^{\prime}_{J})\;=\;\psi_{I}^{-1}(F^{\prime}_{J}\cap
F^{\prime}_{I}).$

Therefore

$\displaystyle\rho_{IJ}^{-1}({\widetilde{V}}_{HI}\cap V_{IJ})$
$\displaystyle=\rho_{IJ}^{-1}(\psi_{I}^{-1}(F^{\prime}_{H})\cap\psi_{I}^{-1}(F^{\prime}_{J}))$
$\displaystyle=\psi_{J}^{-1}(F^{\prime}_{H}\cap F^{\prime}_{J}\cap
F^{\prime}_{I})={\widetilde{V}}_{HJ}\cap{\widetilde{V}}_{IJ}.$

It follows that the domain of the composite $\rho_{HI}\circ\rho_{IJ}$, when
restricted to the sets $V_{\bullet}\subset U_{\bullet}$, is precisely
${\widetilde{V}}_{HJ}\cap{\widetilde{V}}_{IJ}$. Hence the identity

$\rho_{HJ}|_{{\widetilde{V}}_{HJ}\cap{\widetilde{V}}_{IJ}}=\rho_{HI}\circ\rho_{IJ}$

holds since $\rho_{HI}\circ\rho_{IJ}=\rho_{HJ}:U_{J}\to U_{H}$. Thus the
composition property holds.

Further, the separation conditions follow from property (ii) in Lemma 5.2.4
because

  * -

if $H\subset I,J$ then ${\rm cl}(V_{HI}),{\rm cl}(V_{HJ})\subset V_{H}$ and
$\psi_{H}({\rm cl}(V_{HI})\cap{\rm
cl}(V_{HJ}))=\overline{F^{\prime}_{H}}\cap\overline{F^{\prime}_{I}}\cap\overline{F^{\prime}_{J}}$,

  * -

if $H,I\subset J$, then ${\rm cl}({\widetilde{V}}_{HI}),{\rm
cl}({\widetilde{V}}_{HJ})\subset V_{J}$ and $\psi_{J}({\rm
cl}({\widetilde{V}}_{HI})\cap{\rm
cl}({\widetilde{V}}_{HJ}))=\overline{F^{\prime}_{H}}\cap\overline{F^{\prime}_{I}}\cap\overline{F_{J}^{\prime}}$.

This completes the proof that the tuples $(V_{I},G_{I},\rho_{IJ})_{I\subset
A}$ form étale data of type $V$.

Finally, the compatibility of this data with the footprint maps follows from
the properties (i)(c) and (ii)(c) in Lemma 5.2.2. ∎

We are now ready to prove part (ii) of Theorem 2.1.2.

######  Theorem 5.2.6.

Let $(U_{i},G_{i})_{i\in A}$ be a finite set of local uniformizers as in Lemma
5.2.2 whose footprints cover a given compact subset $S\subset\bigcup_{i\in
A}|U_{i}|$, let $\bigl{(}F^{\prime}_{I}\subset\bigcap_{i\in
I}|U_{i}|\bigr{)}_{I\subset A}$ be a cover reduction as in Lemma 5.2.4, and
choose an order $A=\\{1,\ldots,N\\}$. Then the étale groupoid
${\mathcal{X}}_{\mathcal{V}}$ constructed by Proposition 4.1.1 from the étale
data in Lemma 5.2.5 has the following properties with
$V:=\pi_{\mathcal{X}}^{-1}\bigl{(}\bigcup_{I\subset
A}\psi_{I}(V_{I})\bigr{)}=\pi_{\mathcal{X}}^{-1}\bigl{(}\bigcup_{I\subset
A}F^{\prime}_{I}\bigr{)}\subset X$.

  * (i)

The étale category ${\mathcal{X}}_{\mathcal{V}}=(X_{\mathcal{V}},{\bf
X}_{\mathcal{V}})$ is covered by the local uniformizers
$(V_{I},G_{I})_{I\subset A}$, and is proper. In detail, it is given by

$\displaystyle X_{\mathcal{V}}$ $\displaystyle=\textstyle\bigsqcup_{I\subset
A}V_{I}=\textstyle\bigsqcup_{I\subset
A}\bigl{\\{}(I,{\underline{x}})\,\big{|}\,{\underline{x}}=(x_{i_{1}},\ldots)\in
V_{I}\bigr{\\}},$ $\displaystyle{\rm
Mor}_{{\mathcal{X}}_{\mathcal{V}}}(V_{I},V_{J})$
$\displaystyle=\begin{cases}\bigl{\\{}\bigl{(}I,J,{\underline{x}},{\underline{g}}\bigr{)}\phantom{{}^{-1}}\,\big{|}\,{\underline{x}}\in{\widetilde{V}}_{IJ},{\underline{g}}\in
G_{I}\bigr{\\}}&;I\subset J\\\
\bigl{\\{}\bigl{(}J,I,{\underline{x}},{\underline{g}}\bigr{)}^{-1}\,\big{|}\,{\underline{x}}\in{\widetilde{V}}_{JI},{\underline{g}}\in
G_{J}\bigr{\\}}&;J\subset I\\\ \ \emptyset&;\text{otherwise}\end{cases}$

with the structure maps

$\displaystyle
s(I,J,{\underline{x}},{\underline{g}})=\bigl{(}I,{\underline{g}}^{-1}*\rho_{IJ}({\underline{x}})\bigr{)},\quad
t(I,J,{\underline{x}},{\underline{g}})=\bigl{(}J,{\underline{x}}\bigr{)},\qquad\mbox{
for }I\subseteq J$ $\displaystyle{\rm
id}_{I,{\underline{x}}}=(I,I,{\underline{x}},\underline{{\rm
id}}),\qquad(I,J,{\underline{x}},{\underline{g}})^{-1}=(J,I,\,{\underline{g}}^{-1}*{\underline{x}},\,{\underline{g}}^{-1}),\qquad\mbox{
for }I\supseteq J$

and composition given as in Proposition 4.1.1.

  * (ii)

There is an étale functor $\psi:{\mathcal{X}}_{\mathcal{V}}\to{\mathcal{X}}$
given by

$\displaystyle X_{\mathcal{V}}\to
X,\qquad\bigl{(}I,(x_{i_{1}},m_{i_{1}i_{2}},\ldots,x_{i_{k}})\bigr{)}\mapsto
x_{i_{1}},$ $\displaystyle{\bf X}_{\mathcal{V}}\to{\bf
X},\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad$
$\bigl{(}I=\\{i_{1}<\cdot\cdot\\},J=\\{j_{1}<\cdot\cdot\\},{\underline{x}},{\underline{g}}\bigr{)}\;\mapsto\;\begin{cases}m_{i_{1}i_{2}}\circ\ldots\circ
m_{i_{\ell-1}i_{\ell}}&;\mbox{if }i_{1}<j_{1}=i_{\ell},\\\ g_{i_{1}}\circ{\rm
id}_{x_{i_{1}}}&;\mbox{if }i_{1}=j_{1},\\\
g_{j_{\ell}}\circ\bigl{(}m_{j_{1}j_{2}}\circ\ldots\circ
m_{j_{\ell-1}j_{\ell}}\bigr{)}^{-1}&;\mbox{if
}j_{1}<i_{1}=j_{\ell}.\end{cases}$

This is globally finite-to-one on both objects and morphisms, and lifts the
footprint maps $\psi_{I}|_{V_{I}}$ of Lemma 5.2.5 in the sense that

(5.2.5) $\pi_{\mathcal{X}}\circ\psi|_{V_{I}}=\psi_{I}|_{V_{I}}:V_{I}\to
F^{\prime}_{I}\subset|{\mathcal{X}}|\qquad\text{is surjective for all}\quad
I\subset A.$

Restricting the target to be the full subcategory ${\mathcal{X}}|_{V}$ of
${\mathcal{X}}$ with objects $V$, the functor
$\psi:{\mathcal{X}}_{\mathcal{V}}\to{\mathcal{X}}|_{V}$ is an equivalence in
the sense of [HWZ, Def.10.1.1], that is $\psi$ restricts to bijections ${\rm
Mor}_{{\mathcal{X}}_{\mathcal{V}}}({\underline{x}},{\underline{x}}^{\prime})\to{\rm
Mor}_{\mathcal{X}}(\psi({\underline{x}}),\psi({\underline{x}}^{\prime}))$ for
all ${\underline{x}},{\underline{x}}^{\prime}\in X_{\mathcal{V}}$, and induces
a homeomorphism

$|\psi|:|{\mathcal{X}}_{\mathcal{V}}|\to|V|\subset|{\mathcal{X}}|.$

  * (iii)

${\mathcal{X}}_{\mathcal{V}}$ is equipped with an inner action of
$G:=\prod_{i\in A}G_{i}$ given by
${\underline{g}}*(I,{\underline{x}})=(I,{\underline{g}}|_{I}*{\underline{x}})$
and ${\alpha}:{\rm Obj}_{{\mathcal{X}}_{\mathcal{V}}}\times G\to{\rm
Mor}_{{\mathcal{X}}_{\mathcal{V}}}$,
$((I,{\underline{x}}),{\underline{g}})\mapsto(I,I,{\underline{x}},{\underline{g}}|_{I})$
with $(g_{1},\ldots,g_{N})|_{I}:=(g_{i_{1}},\ldots,g_{i_{|I|}})$ as in
Proposition 4.1.1 (i). Moreover, $\psi$ is locally $G$-equivariant in the
following sense: For $I=\\{i_{1}<\cdot\cdot\\}$ we have $\psi(V_{I})\subset
U_{i_{1}}\subset{\rm Obj}_{{\mathcal{X}}}$ and $\psi:V_{I}\to U_{i_{1}}$ is
equivariant with respect to the projection $G\to G_{i_{1}}$ to the isotropy
group acting on the local uniformizer $U_{i_{1}}$.

  * (iv)

${\mathcal{X}}_{\mathcal{V}}^{{\smallsetminus}G}$ is the étale (but usually
not proper) groupoid given by Proposition 4.1.2 (iii). In detail, this
groupoid is given by the structure maps of ${\mathcal{X}}_{\mathcal{V}}$ as in
Theorem 5.2.6 restricted to

$\displaystyle{\rm Obj}_{{\mathcal{X}}_{\mathcal{V}}^{{\smallsetminus}G}}$
$\displaystyle=\ \textstyle{\rm Obj}_{{\mathcal{X}}_{\mathcal{V}}}\ =\
\bigsqcup_{I\subset A}V_{I},$ $\displaystyle{\rm
Mor}_{{\mathcal{X}}_{\mathcal{V}}^{{\smallsetminus}G}}(V_{I},V_{J})$
$\displaystyle=\
\begin{cases}\bigl{\\{}(I,J,{\underline{x}},{\underline{g}})\phantom{{}^{-1}}\,\big{|}\,{\underline{x}}\in{\widetilde{V}}_{IJ},{\underline{g}}\in
G_{I{\smallsetminus}H_{{\underline{x}}}}\bigr{\\}}&;I\subset J\\\
\bigl{\\{}(J,I,{\underline{x}},{\underline{g}})^{-1}\,\big{|}\,{\underline{x}}\in{\widetilde{V}}_{JI},{\underline{g}}\in
G_{J{\smallsetminus}H_{{\underline{x}}}}\bigr{\\}}&;J\subset I\\\ \
\emptyset&;\text{otherwise}\end{cases}$

where we denote494949See Remark 4.2.2 (ii).
$H_{\underline{x}}:=\min\\{H\,|\,{\underline{x}}\in{\widetilde{V}}_{HJ}\\}$.
Further, ${\mathcal{X}}_{\mathcal{V}}^{{\smallsetminus}G}$ is equipped with an
action of $G$ so that the functor
${\iota}:{\mathcal{X}}_{\mathcal{V}}^{{\smallsetminus}G}\to{\mathcal{X}}_{\mathcal{V}}$
given by inclusion is $G$-equivariant in the sense of Definition 3.3.1 and
induces a homeomorphism
$|{\iota}|:|{\mathcal{X}}_{\mathcal{V}}^{{\smallsetminus}G}|\,/G\stackrel{{\scriptstyle\simeq}}{{\to}}|{\mathcal{X}}_{\mathcal{V}}|$.

###### Proof.

We first prove (ii). The definition of ${\mathcal{X}}_{\mathcal{V}}$ implies
that each object and morphism in ${\mathcal{X}}$ has a finite number of
preimages. Hence $\psi$ is finite-to-one. In order to establish the other
properties of $\psi$, it is useful to consider the subcategory
${\mathcal{B}}_{\mathcal{V}}$ of ${\mathcal{X}}_{\mathcal{V}}$ that has the
same objects as ${\mathcal{X}}_{\mathcal{V}}$, but has morphisms from $V_{I}$
to $V_{J}$ only if $I\subset J$. Thus

$\displaystyle{\rm Mor}_{{\mathcal{B}}_{\mathcal{V}}}(V_{I},V_{J})$
$\displaystyle={\rm
Mor}_{{\mathcal{X}}_{\mathcal{V}}}(V_{I},V_{J}),\quad\mbox{ if }I\subset J,$
$\displaystyle=\emptyset\quad\mbox{ otherwise}$

To see that $\psi:{\mathcal{X}}_{\mathcal{V}}\to{\mathcal{X}}$ is a well
defined functor, it will suffice to show that the natural lifts of the
footprint maps $\psi_{J}:V_{J}\to F^{\prime}_{J}\subset|{\mathcal{X}}|$ fit
together to give maps $\psi:X_{\mathcal{V}}\to X$ and $\psi:{\bf
X}_{\mathcal{V}}\to{\bf X}$ whose restriction to the subcategory
${\mathcal{B}}_{\mathcal{V}}$ is a functor that is compatible with inverses.
For then we may apply Lemma 4.2.7 to conclude that $\psi$ must coincide with
the unique functorial extension of $\psi|_{{\mathcal{B}}_{\mathcal{V}}}$.

Consider the restriction of $\psi$ to ${\mathcal{B}}_{\mathcal{V}}$. Let us
denote

$I=\\{i_{1}<i_{2}<\cdots\\},\;\;J=\\{j_{1}<j_{2}<\cdot\\},\;\;{\underline{g}}=(g_{i_{1}},g_{i_{2}},\cdots)\in
G_{I}.$

To prove that $\psi:{\mathcal{B}}_{\mathcal{V}}\to{\mathcal{X}}$ is a functor
we must first check that the given formulas are compatible with source and
target map. Suppose first that $j_{1}:=i_{\ell}$ for some $\ell>1$ so that
$J\subset I$. Then $(I,J,{\underline{x}},{\underline{g}})\in{\rm
Mor}(V_{I},V_{J})$ has ${\underline{g}}=(g_{j_{1}},\cdots)\in G_{J}$ so that
$g_{i_{1}}={\rm id}$. Hence

$\psi\bigl{(}s(I,J,{\underline{x}},{\underline{g}})\bigr{)}=\psi(I,{\underline{g}}^{-1}*{\underline{x}})=x_{i_{1}},\quad\psi\bigl{(}t(I,J,{\underline{x}},{\underline{g}})\bigr{)}=\psi(J,{\underline{x}})=x_{i_{\ell}}=x_{j_{1}}.$

Thus,
$\psi((I,J,{\underline{x}},{\underline{g}}))=m_{i_{1}i_{2}}\circ\cdots\circ
m_{i_{\ell-1}i_{\ell}}\in{\rm Mor}_{{\mathcal{X}}}(x_{i_{1}},x_{i_{\ell}})$
has the same source and target. If $i_{1}=j_{1}$ this is the source and target
of
$\psi\bigl{(}(I,J,{\underline{x}},{\underline{g}})\bigr{)}=g_{i_{1}}\circ{\rm
id}_{x_{1}}$. Therefore the formulas are compatible with source and target
maps if $J\subset I$ or if $i_{1}=j_{1}$. Finally, note that if
$i_{1}=j_{\ell}$ for $\ell>1$ then $I\subset J$ so that ${\underline{x}}\in
V_{J}$, and we have

$\displaystyle(s\times
t)\,\bigl{(}\psi(I,J,{\underline{x}},{\underline{g}})\bigr{)}$
$\displaystyle=(s\times
t)\,\bigl{(}g_{j_{\ell}}\circ(m_{j_{\ell-1}j_{\ell}})^{-1}\cdots\circ(m_{j_{1}j_{2}})^{-1}\bigr{)}$
$\displaystyle=\bigl{(}g_{j_{\ell}}^{-1}*x_{j_{\ell}},\
x_{j_{1}}\bigr{)}=\bigl{(}g_{i_{1}}^{-1}*x_{i_{1}},\ x_{j_{1}}\bigr{)}$

as required.

Now consider the composition rule for $I\subset J\subset K$, where we write
$K:=\\{k_{1}<k_{2}<\cdots\\}$ so that $k_{1}\leq j_{1}:=k_{r}\leq
i_{1}=j_{\ell}=k_{s}$. We must show that

$\psi(I,J,{\underline{x}},{\underline{g}})\circ\psi(J,K,{\underline{y}},{\underline{h}})\
=\ \psi(I,K,{\underline{y}},{\underline{h}}|_{I}{\underline{g}}),$

when ${\underline{x}}\in V_{J},\ {\underline{y}}\in V_{K},{\underline{g}}\in
G_{I},\ {\underline{h}}\in G_{J}$. Since this is true if either $k_{1}=j_{1}$
or $j_{1}=i_{1}$, we will assume that
$k_{1}<j_{1}:=k_{r}<i_{1}=j_{\ell}=k_{s}$. Since $J\subset K$, for any
$j_{a}\in J$ there is $p<q$ so that $j_{a}=k_{p}<j_{a+1}=k_{q}$ and we may
write505050 Here the superscripts $J,K$ indicate that the entries of $m,n$ are
indexed by pairs of elements from the sets $J,K$ respectively. Thus
$m^{J}_{ab}$ is the composite $m_{j_{p}j_{p+1}}\circ
m_{j_{p+1}j_{p+2}}\circ\dots\circ m_{j_{q-1}j_{q}}$ where $a=j_{p}<b=j_{q}$.

(5.2.6) $\displaystyle{\underline{x}}=(x_{j_{1}},m_{j_{1}j_{2}},\dots)\in
V_{J},\quad{\underline{y}}:=(y_{k_{1}},n_{k_{1}k_{2}},\dots)\in V_{K},$
$\displaystyle m^{J}_{j_{1}j_{\ell}}:=m_{j_{1}j_{2}}\circ\dots\circ
m_{j_{\ell-1}j_{\ell}},\quad
n^{K}_{j_{a}j_{a+1}}:=n^{K}_{k_{p}k_{q}}=n_{k_{p}k_{p+1}}\circ\dots\circ
n_{k_{q-1}k_{q}}.$

Then the identity
${\underline{x}}={\underline{h}}^{-1}*\rho_{JK}({\underline{y}})$ gives

$(x_{j_{1}},m_{j_{1}j_{2}},x_{j_{2}}\dots)=(h_{j_{1}}^{-1}*y_{j_{1}},h_{j_{1}}\circ
n^{K}_{j_{1}j_{2}}\circ h_{j_{2}}^{-1},h_{j_{2}}^{-1}*y_{j_{2}},\ \cdots),$

and more generally for each $a$ and $p<t$ we have

(5.2.7) $\displaystyle m_{j_{a}j_{a+1}}=h_{j_{a}}\circ
n^{K}_{j_{a}j_{a+1}}\circ h_{j_{a+1}}^{-1},\quad
m^{J}_{k_{p}k_{t}}=h_{k_{p}}\circ n^{K}_{k_{p}k_{t}}\circ h_{k_{t}}^{-1}$

Further, since $k_{1}<j_{1}:=k_{r}<i_{1}=j_{\ell}=k_{s}$,

$\displaystyle\psi\bigl{(}(I,J,{\underline{x}},{\underline{g}})\bigr{)}=g_{j_{\ell}}\circ(m^{J}_{j_{1}j_{\ell}})^{-1}=g_{k_{s}}\circ(m^{J}_{k_{r}k_{s}})^{-1},$
$\displaystyle\psi\bigl{(}(J,K,{\underline{y}},{\underline{h}})\bigr{)}=h_{k_{r}}\circ(n^{K}_{k_{1}k_{r}})^{-1},\quad\psi\bigl{(}(I,K,{\underline{y}},{\underline{h}}|_{I}{\underline{g}})\bigr{)}=h_{k_{s}}g_{k_{s}}\circ(n^{K}_{k_{1}k_{s}})^{-1}$

Therefore, we need to show

(5.2.8) $\displaystyle g_{k_{s}}\circ(m^{J}_{k_{r}k_{s}})^{-1}\circ
h_{k_{r}}\circ(n^{K}_{k_{1}k_{r}})^{-1}=h_{k_{s}}g_{k_{s}}\circ(n^{K}_{k_{1}k_{s}})^{-1}.$

But $h_{k_{s}}g_{k_{s}}=g_{k_{s}}\circ h_{k_{s}}$ because of the categorical
composition convention explained in (3.2.1). Therefore (5.2.8) follows from
(5.2.7).

Hence the restriction of $\psi$ to ${\mathcal{B}}_{\mathcal{V}}$ preserves its
structure maps and so defines a functor
${\mathcal{B}}_{\mathcal{V}}\to{\mathcal{X}}$. Now let us check the
compatibility of $\psi$ with the inverse operation

$(I,J,{\underline{x}},{\underline{g}})\mapsto(I,J,{\underline{x}},{\underline{g}})^{-1}:=(J,I,{\underline{g}}^{-1}*{\underline{x}},{\underline{g}}^{-1}).$

If $I\subset J$ with $i_{1}=j_{\ell}$, then ${\underline{x}}\in
V_{J},{\underline{g}}\in G_{I}$ and $g_{j_{a}}={\rm id}$ for $a<\ell$ so that
we have

$\displaystyle\psi((I,J,{\underline{x}},{\underline{g}}))$
$\displaystyle=g_{j_{\ell}}\circ(m_{j_{\ell-1}j_{\ell}})^{-1}\cdots\circ(m_{j_{1}j_{2}})^{-1}=g_{j_{\ell}}\circ(m^{J}_{j_{1}j_{\ell}})^{-1},$
$\displaystyle\psi(J,I,{\underline{g}}^{-1}*{\underline{x}},{\underline{g}}^{-1})$
$\displaystyle=m_{j_{1}j_{2}}\cdots\circ m_{j_{\ell-1}j_{\ell}}\circ
g_{j_{\ell}}^{-1}=m^{J}_{j_{1}j_{\ell}}\circ g_{j_{\ell}}^{-1}.$

Hence $\psi$ preserves the inverse operation. Therefore $\psi$ is a functor
${\mathcal{X}}_{\mathcal{V}}\to{\mathcal{X}}$ by Lemma 4.2.7.

Next, note that the functor $\psi:{\mathcal{X}}_{\mathcal{V}}\to{\mathcal{X}}$
is étale on the object space since on each component $V_{I}\subset{\rm
Obj}_{{\mathcal{X}}_{\mathcal{V}}}$ it is given by the restriction of the
étale source map $s_{I}$ from Lemma 5.2.2 (i)(a). On morphisms, the functor is
étale because it consists of étale actions of the finite isotropy groups by
Remark 5.2.3, and the identity map as well as composition of morphisms in
${\mathcal{X}}$ are étale since ${\mathcal{X}}$ is. Moreover, given any
$(I,{\underline{x}}),(I^{\prime},{\underline{y}})\in{\rm
Obj}_{{\mathcal{X}}_{\mathcal{V}}}$, we need to check that $\psi$ restricts to
a bijection ${\rm
Mor}_{{\mathcal{X}}_{\mathcal{V}}}\bigl{(}(I,{\underline{x}}),(I^{\prime},{\underline{y}})\bigr{)}\to{\rm
Mor}_{\mathcal{X}}(x_{1},y_{1})$. In case $I\subset I^{\prime}$ this is
exactly the content of Lemma 5.2.7 below, and composition with the (bijective
and $\psi$-compatible) inverse maps in ${\mathcal{X}}_{\mathcal{V}}$ and
${\mathcal{X}}$ establishes the bijection for the case $I^{\prime}\subset I$.

We now show that the induced map

(5.2.9)
$\displaystyle|\psi|:|{\mathcal{X}}_{\mathcal{V}}|\overset{\simeq}{\to}F^{\prime}:={\textstyle\bigcup_{I\subset
A}F^{\prime}_{I}\subset|{\mathcal{X}}|}$

is a homeomorphism between the realizations. This map is surjective because
$|\psi(V_{I})|$ is the image $\psi_{I}(V_{I})=F^{\prime}_{I}$ of the footprint
map in Lemma 5.2.5. To check that it is injective recall that
$\psi_{I},\psi_{I^{\prime}}$ have disjoint images unless $I\subset I^{\prime}$
or $I^{\prime}\subset I$. Now for w.l.o.g. $I\subset I^{\prime}$ with
$|x_{i_{1}}|=|\psi(I,{\underline{x}})|=|\psi(I^{\prime},{\underline{x}}^{\prime})|=|x^{\prime}_{i^{\prime}_{1}}|$
the equivalence $x_{i_{1}}\sim_{\mathcal{X}}x^{\prime}_{i^{\prime}_{1}}$
implies ${\rm
Mor}_{\mathcal{X}}(x_{i_{1}},x^{\prime}_{i^{\prime}_{1}})\neq\emptyset$, which
transfers to ${\rm
Mor}_{{\mathcal{B}}_{\mathcal{V}}}((I,{\underline{x}}),(I^{\prime},{\underline{x}}^{\prime}))\neq\emptyset$
by Lemma 5.2.7. This proves
$(I,{\underline{x}})\sim_{{\mathcal{X}}_{\mathcal{V}}}(I^{\prime},{\underline{x}}^{\prime})$,
as required for injectivity of $|\psi|$. This map and its inverse are
continuous since its lift $\psi$ on objects is continuous with continuous
local inverses. This completes the proof of (ii).

To see that $\psi$ has the local equivariance stated in (iii), note that since
$\psi:{\rm Mor}_{{\mathcal{X}}_{\mathcal{V}}}(V_{I},V_{I})\to{\rm
Mor}_{\mathcal{X}}(V_{i_{1}},V_{i_{1}})$ is given by the formula
$(I,I,{\underline{x}},{\underline{g}})\mapsto g_{i_{1}}\circ{\rm id}_{x_{1}}$,
the $G$ action on $V_{I}$ given by
${\underline{x}}\mapsto{\underline{g}}*{\underline{x}}$ maps to the
$G_{i_{1}}$ action on $V_{i_{1}}\subset U_{i_{1}}$.

It follows that each pair $(V_{I},G_{I})$ is a local uniformizer for
${\mathcal{X}}_{\mathcal{V}}$. Thus $|{\mathcal{X}}_{\mathcal{V}}|$ is covered
by finitely many uniformizers and hence, by Corollary 3.2.6, is proper because
its realization $|{\mathcal{X}}_{\mathcal{V}}|\simeq F^{\prime}$ is Hausdorff.
Thus (i) also holds.

Finally, the claims in (iv) about the relation between the groupoids
${\mathcal{X}}_{\mathcal{V}}^{{\smallsetminus}G}$ and
${\mathcal{X}}_{\mathcal{V}}$ are proved in Proposition 4.2.4 (iii). ∎

######  Lemma 5.2.7.

In the setting of Lemma 5.2.5, given

$\displaystyle\qquad
I=(k_{\ell_{1}}=i_{1}<\dots)\;\subset\;K=(k_{1}<\ldots),\quad\mbox{\; and }$
$\displaystyle{\underline{x}}=(x_{i_{1}},m_{i_{1}i_{2}},\ldots)\in U_{I},\
\;{\underline{y}}=(y_{k_{1}},n_{k_{1}k_{2}},\ldots)\in U_{K}\mbox{\; with
}\psi_{I}({\underline{x}})=\psi_{K}({\underline{y}})$

there is a bijection ${\rm
Mor}_{{\mathcal{B}}_{\mathcal{V}}}((I,{\underline{x}}),(K,{\underline{y}}))\to{\rm
Mor}_{\mathcal{X}}(x_{i_{1}},y_{k_{1}})$ given by

$U_{K}\times
G_{I}\;\ni\;\bigl{(}{\underline{y}},(g_{i_{1}},\ldots,g_{i_{|I|}})\bigr{)}\mapsto
g_{i_{1}}\circ n_{k_{\ell_{1}-1}k_{\ell_{1}}}^{-1}\circ\ldots\circ
n_{k_{1}k_{2}}^{-1},$

where $i_{s}=k_{\ell_{s}},1\leq s\leq|I|$.

###### Proof.

The morphism space with fixed source and target in
${\mathcal{B}}_{\mathcal{V}}$ is

${\rm
Mor}_{{\mathcal{B}}_{\mathcal{V}}}((I,{\underline{x}}),(K,{\underline{y}}))=\bigl{\\{}({\underline{y}},{\underline{g}})\,\big{|}\,{\underline{g}}\in
G_{I},{\underline{g}}*{\underline{x}}=\rho_{IK}({\underline{y}})\bigr{\\}},$

where
$\rho_{IK}({\underline{y}})=(y_{k_{\ell_{1}}},n^{K}_{i_{1}i_{2}},\ldots,n^{K}_{i_{|I|-1}i_{|I|}},y_{k_{\ell_{|I|}}})\in
U_{I}$, in the notation of (5.2.6). This does not capture the morphisms
$n_{k_{1}k_{2}},\ldots,n_{k_{\ell_{1}-1}k_{\ell_{1}}}$ in ${\underline{y}}\in
U_{K}$, which compose to

$n^{K}_{k_{1}k_{\ell_{1}}}:=n_{k_{1}k_{2}}\circ\ldots\circ
n_{k_{\ell_{1}-1}k_{\ell_{1}}}\in{\rm
Mor}_{\mathcal{X}}(y_{k_{1}},y_{k_{\ell_{1}}}).$

Here by assumption we have515151Recall from Lemma 5.2.5 that
$\psi_{K}:V_{K}\to|{\mathcal{X}}|$ is the footprint map, not to be confused
with the functor $\psi:{\mathcal{X}}_{\mathcal{V}}\to{\mathcal{X}}$.

$\psi_{k_{\ell_{1}}}(y_{k_{\ell_{1}}})=\psi_{K}({\underline{y}})=\psi_{I}({\underline{x}})=\psi_{i_{1}}(x_{i_{1}})\;\in\;|{\mathcal{X}}|,$

and since $|\psi_{k_{\ell_{1}}}|:V_{k_{\ell_{1}}}/G_{k_{\ell_{1}}}\to
F^{\prime}_{k_{\ell_{1}}}\subset|{\mathcal{X}}|$ is a homeomorphism, this
implies $y_{k_{\ell_{1}}}=g_{i_{1}}*x_{i_{1}}$ for some $g_{i_{1}}\in
G_{i_{1}}$ that is unique up to $G_{x_{i_{1}}}={\rm
Mor}_{\mathcal{X}}(x_{i_{1}},x_{i_{1}})\subset G_{i_{1}}$. Any such choice
yields $n^{K}_{k_{1}k_{\ell_{1}}}\circ g_{i_{1}}^{-1}\in{\rm
Mor}_{\mathcal{X}}(y_{k_{1}},x_{i_{1}})$, and taking the inverse in the
groupoid ${\mathcal{X}}$ yields
$g_{i_{1}}\circ(n^{K}_{k_{1}k_{\ell_{1}}})^{-1}\in{\rm
Mor}_{\mathcal{X}}(x_{i_{1}},y_{k_{1}})$, which shows that the given map is
well defined.

To show that the map is a bijection, we will show that any $m\in{\rm
Mor}_{\mathcal{X}}(x_{i_{1}},y_{k_{1}})$ has exactly one preimage. More
precisely, given $m,{\underline{x}},{\underline{y}}$ we will iteratively
determine the entries in ${\underline{g}}=(g_{i_{1}},g_{i_{2}},\ldots)\in
G_{I}$ from the conditions
${\underline{g}}*{\underline{x}}=\rho_{IK}({\underline{y}})$ and
$g_{i_{1}}\circ(n^{K}_{k_{1}k_{\ell_{1}}})^{-1}=m$. The iteration starts with
$g_{i_{1}}\in G_{i_{1}}$ being determined uniquely by

${\rm id}_{x_{i_{1}}}\circ g_{i_{1}}=m\circ n^{K}_{k_{1}k_{\ell_{1}}}\in{\rm
Mor}_{\mathcal{X}}(x_{i_{1}},y_{k_{\ell_{1}}}),\quad\mbox{where
}k_{\ell_{1}}=i_{1},$

using notation from Remark 5.2.3. The properties of the local uniformizer in
Definition 3.2.1 then also guarantees the first object in
${\underline{g}}^{-1}*\rho_{IK}({\underline{y}})$ to be
$g_{i_{1}}^{-1}*y_{k_{\ell_{1}}}=x_{i_{1}}$. For the iteration step, assume
$g_{i_{1}},\ldots,g_{i_{s-1}}$ are determined so that the first $(s-1)$
objects and $(s-2)$ morphisms in
${\underline{g}}^{-1}*\rho_{IK}({\underline{y}})$ agree with those given by
${\underline{x}}=(x_{i_{1}},m_{i_{1}i_{2}},\ldots,m_{i_{|I|-1}i_{|I|}},x_{i_{|I|}})$.
Then we see that $g_{s}\in G_{i_{s}}$ is uniquely determined by matching
$m_{i_{s-1}i_{s}}\in{\rm Mor}_{\mathcal{X}}(x_{i_{s-1}},x_{i_{s}})$ with the
corresponding morphism in ${\underline{g}}^{-1}*\rho_{IK}({\underline{y}})$,

$\displaystyle m_{i_{s-1}i_{s}}\;=\;g_{i_{s-1}}\circ n^{K}_{i_{s-1}i_{s}}\circ
g_{i_{s}}^{-1}$ $\displaystyle\quad\Longleftrightarrow\quad$
$\displaystyle{\rm id}_{x_{i_{s}}}\circ
g_{i_{s}}\;=\;(m_{{i_{s-1}i_{s}}})^{-1}\circ g_{i_{s-1}}\circ
n^{K}_{i_{s-1}i_{s}}\in{\rm Mor}_{\mathcal{X}}(x_{i_{s}},y_{k_{\ell_{s}}}).$

This indeed determines $g_{i_{s}}\in G_{i_{s}}$ since
$x_{i_{s}},y_{k_{\ell_{s}}}\in U_{i_{s}}$ and the local uniformizer
$(U_{i_{s}},G_{i_{s}})$ comes with a bijection
${\Gamma}_{i_{s}}:G_{i_{s}}\times U_{i_{s}}\to{\rm
Mor}_{\mathcal{X}}(U_{i_{s}},U_{i_{s}}),(g,x)\mapsto{\rm id}_{x}\circ g$.
Finally, the local uniformizer property

$(s\times t)\circ{\Gamma}_{i_{s}}:G_{i_{s}}\times U_{i_{s}}\to U_{i_{s}}\times
U_{i_{s}},\quad(g,x)\mapsto(x,g*x)$

implies matching for the $s$-th object $y_{k_{\ell_{s}}}=t({\rm
id}_{x_{i_{s}}}\circ g_{i_{s}})=g_{i_{s}}*x_{i_{s}}$. ∎

######  Remark 5.2.8.

Let $C\subset|{\mathcal{X}}_{\mathcal{V}}|=|{\mathcal{X}}|$ be any set such
that its topology as a subspace of $|{\mathcal{X}}|$ is second countable. We
claim that there is a constant $K$ such that each sufficiently small open
subset in $C$ pulls back to a collection of at most $K$ disjoint open sets in
$\pi_{{\mathcal{X}}_{\mathcal{V}}}^{-1}(C)$. This holds because
$|X_{\mathcal{V}}|$ is covered by the footprints of the finite number of
uniformizers $(V_{J},G_{J}),J\subset A,$, and each $|V_{J}|$ has this property
with $K:=|G_{J}|$. Hence any countable collection of open subsets of $C$ that
form a basis for the subspace topology of $C$ pulls back to a countable
collection of open subsets of $\pi_{{\mathcal{X}}_{\mathcal{V}}}^{-1}(C)$ that
form a basis for the topology of $\pi_{{\mathcal{X}}_{\mathcal{V}}}^{-1}(C)$.
Thus the inverse image $\pi_{{\mathcal{X}}_{\mathcal{V}}}^{-1}(C)$ of any
second countable subset
$C\subset|{\mathcal{X}}_{\mathcal{V}}|=|{\mathcal{X}}|$ is second countable in
the subspace topology. Note that this might not hold for the map
$X\to|{\mathcal{X}}|$. ${\Diamond}$

######  Remark 5.2.9.

There is an analogous étale groupoid ${\mathcal{X}}_{\mathcal{U}}$ with
objects $\bigsqcup_{I}U_{I}$, where $U_{I}$ is as in Lemma 5.2.2, and whose
morphisms are generated by the projections $(\rho_{IJ}:U_{J}\to
U_{I})_{I\subset J}$ and group actions $G_{J}\times U_{J}\to U_{J}$. However,
the full space of morphisms in ${\mathcal{X}}_{\mathcal{U}}$ is rather
complicated to describe explicitly because there are morphisms from $U_{I}$ to
$U_{J}$ whenever $|U_{I\cup J}|=F_{I\cup J}\neq\emptyset.$ (See [MW3, Mc1] for
a discussion of such categories in the finite dimensional case.) Further, the
functor $\psi$ extends to an étale equivalence
$\psi:{\mathcal{X}}_{\mathcal{U}}\to{\mathcal{X}}$. The category
${\mathcal{X}}_{\mathcal{U}}$ per se plays no role in the current paper.
${\Diamond}$

###  5.3. The strong bundle
${\mathcal{W}}_{\mathcal{V}}\to{\mathcal{X}}_{\mathcal{V}}$

We now explain the structure of the pullback bundle
${\mathcal{W}}_{\mathcal{V}}:=\psi^{*}{\mathcal{W}}$ in the top row of diagram
(2.1.5) defined in Lemma 3.4.4. Since we need to establish the properties of
strong bundle and sc-Fredholm section functor, this subsection works
explicitly with sc-groupoids and polyfolds. However, many results extend to
general étale proper categories.

Throughout we consider a sc-Fredholm section functor
$f:{\mathcal{X}}\to{\mathcal{W}}$ of a strong bundle
$P:{\mathcal{W}}\to{\mathcal{X}}$ over an ep-groupoid ${\mathcal{X}}$ with
compact zero set $|f^{-1}(0)|$ as in Definition 2.1.1. The bundle is given by
structure maps $(P:W\to X,\mu:{\bf X}{\vphantom{\times}}_{s}{\times}_{P}W\to
W)$, where $P:W\to X$ is a strong bundle over the M-polyfold of objects $X$,
and $\mu$ defines the action of the morphisms of ${\mathcal{X}}$ on the object
space $W$ of ${\mathcal{W}}$. The following result describes the structure of
${\mathcal{W}}_{\mathcal{V}}$ and the perturbation theory of $f_{\mathcal{V}}$
in detail.

######  Theorem 5.3.1.

Let $(U_{i},G_{i})_{i=1,\ldots,N}$ be a finite collection of local
uniformizers whose footprints $F_{i}:=|U_{i}|$ cover
$S:=|f^{-1}(0)|\subset\bigcup_{i=1}^{N}F_{i}$, and let
$\bigl{(}F^{\prime}_{I}\subset|{\mathcal{X}}|\bigr{)}_{I\subset\\{1,\ldots,N\\}}$
be a cover reduction as in Lemma 5.2.4, with corresponding étale data
$(V_{J},G_{J},\rho_{IJ})$ as in Lemma 5.2.5. Then pullback with the étale
functor $\psi:{\mathcal{X}}_{\mathcal{V}}\to{\mathcal{X}}$ from Theorem 5.2.6
induces a sc-Fredholm section functor
$f_{\mathcal{V}}:{\mathcal{X}}_{\mathcal{V}}\to{\mathcal{W}}_{\mathcal{V}}$ of
a strong bundle
$(P_{\mathcal{V}}:{\mathcal{W}}_{\mathcal{V}}\to{\mathcal{X}}_{\mathcal{V}},\mu_{\mathcal{V}})$
over the reduced ep-groupoid ${\mathcal{X}}_{\mathcal{V}}$, whose perturbation
theory is equivalent to that of $f$. In detail:

  * (i)

The strong bundle
$P_{\mathcal{V}}:{\mathcal{W}}_{\mathcal{V}}\to{\mathcal{X}}_{\mathcal{V}}$ is
the pullback of $P:{\mathcal{W}}\to{\mathcal{X}}$ under $\psi$ as in Lemma
3.4.4. Using the specific structure from Theorem 5.2.6, this pullback is given
by

$\displaystyle P_{\mathcal{V}}:$
$\displaystyle\;\psi^{*}W:=X_{\mathcal{V}}\,{\vphantom{\times}}_{\psi}{\times}_{P}W\to
X_{\mathcal{V}},\quad
P_{\mathcal{V}}(I,{\underline{x}},w)=(I,{\underline{x}}),$
$\displaystyle\mu_{\mathcal{V}}:$ $\displaystyle\;{\bf
X}_{\mathcal{V}}\,{\vphantom{\times}}_{s}{\times}_{P_{\mathcal{V}}}\psi^{*}W\to\psi^{*}W,\quad\mu_{\mathcal{V}}(I,J,{\underline{y}},{\underline{g}},w)=\bigl{(}t(I,J,{\underline{y}},{\underline{g}}),\mu(\psi(I,J,{\underline{y}},{\underline{g}}),w)\bigr{)},$

where $(I,J,{\underline{y}},{\underline{g}},w)$ is an abbreviation for
$\bigl{(}(I,J,{\underline{y}},{\underline{g}}),s(I,J,{\underline{y}},{\underline{g}}),w)\bigr{)}\in{\bf
X}_{\mathcal{V}}\,{\vphantom{\times}}_{s}{\times}_{P_{\mathcal{V}}}\psi^{*}W$.
The resulting groupoid ${\mathcal{W}}_{\mathcal{V}}=\psi^{*}{\mathcal{W}}$ is
given by

$\displaystyle W_{\mathcal{V}}=\psi^{*}W$
$\displaystyle=\textstyle\bigsqcup_{I\subset
A}W_{I}=\textstyle\bigsqcup_{I\subset
A}\bigl{\\{}\bigl{(}I,{\underline{x}}=(x_{i_{1}},\ldots),w\bigr{)}\,\big{|}\,{\underline{x}}\in
V_{I},w\in W_{x_{i_{1}}}\bigr{\\}},$ $\displaystyle{\bf
W}_{\mathcal{V}}(W_{I},W_{J})$
$\displaystyle=\bigl{\\{}\bigl{(}I,J,{\underline{x}},{\underline{g}},w\bigr{)}\,\big{|}\,(I,J,{\underline{x}},{\underline{g}})\in{\rm
Mor}_{{\mathcal{X}}_{\mathcal{V}}}(V_{I},V_{J}),\;w\in
W_{\psi(s(I,J,{\underline{x}},{\underline{g}}))}\bigr{\\}}$

with the structure maps

(5.3.1) $\displaystyle s(I,J,{\underline{x}},{\underline{g}},w)$
$\displaystyle=\bigl{(}s(I,J,{\underline{x}},{\underline{g}}),w\bigr{)},\qquad
t(I,J,{\underline{x}},{\underline{g}},w)=\bigl{(}t(I,J,{\underline{x}},{\underline{g}}),\widehat{\rho}_{IJ}({\underline{x}},{\underline{g}})w\bigr{)},$
$\displaystyle(I,J,{\underline{x}},{\underline{g}},w)$