# Polyfold fundamental classes and globally structured multivalued
perturbations

Dusa McDuff  Department of Mathematics, Barnard College, Columbia University
<EMAIL_ADDRESS>and  Katrin Wehrheim  Department of Mathematics,
University of California, Berkeley<EMAIL_ADDRESS>
(Date: June 21, 2024)

###### Abstract.

Work of Hofer–Wysocki–Zehnder has shown that many spaces of pseudoholomorphic
curves that arise when studying symplectic manifolds may be described as the
zero set of a polyfold Fredholm section. This framework has many analytic
advantages. However the methods they develop to extract useful topological
information from it are rather cumbersome. This paper develops a general
construction of a finite dimensional space of multivalued perturbations of a
polyfold Fredholm section such that almost all elements are regularizing.
These perturbation are globally structured and explicitly described, and, in
cases where the moduli space has no formal boundary, permit a transparent
definition of its (rational Čech) fundamental class.

###### Key words and phrases:

polyfold, sc-Fredholm section, virtual fundamental class, Fredholm
stabilization

###### 1991 Mathematics Subject Classification:

46,53,58

This material is based upon work supported by the National Science Foundation
under Grants No.DMS-1708916, and DMS-1440140 while the first author was in
residence at the Mathematical Sciences Research Institute in Berkeley,
California, during the Spring 2018 semester.

###### Contents

  1. 1 Introduction
    1. 1.1 Overview of constructions
  2. 2 Terminology and main results
    1. 2.1 Statement of main theorems
    2. 2.2 Scale Calculus
    3. 2.3 Orientation Conventions
  3. 3 Étale categories and groupoid completions
    1. 3.1 Étale categories
    2. 3.2 Preuniformizers, properness, and groupoid completions
    3. 3.3 Group actions on étale categories
    4. 3.4 Bundles, sections and structured multisections
    5. 3.5 Compactness control
  4. 4 General construction principles
    1. 4.1 The structure of the categories ${\mathcal{X}}_{\mathcal{V}}$ and ${\mathcal{X}}_{\mathcal{V}}^{{\smallsetminus}G}$
    2. 4.2 The category ${{\mathcal{Q}}}$ and its relation to ${\mathcal{X}}_{\mathcal{V}}$
    3. 4.3 Bundles over ${\mathcal{X}}_{\mathcal{V}}^{{\smallsetminus}G}$ and ${\mathcal{X}}_{\mathcal{V}}$
  5. 5 Proofs of Main Theorems
    1. 5.1 Fundamental classes induced by Fredholm sections over polyfolds
    2. 5.2 The construction of ${\mathcal{X}}_{\mathcal{V}}$ from ${\mathcal{X}}$
    3. 5.3 The strong bundle ${\mathcal{W}}_{\mathcal{V}}\to{\mathcal{X}}_{\mathcal{V}}$
    4. 5.4 Fredholm stabilization over reduced ep-groupoids
  6. A A note on rational Čech homology

##  1\. Introduction

Polyfold theory, initiated by Hofer-Wysocki-Zehnder in the early 2000s,
generalizes the notion of a smooth section of an orbibundle to obtain a
nonlinear Fredholm theory for functors between groupoids that promises to make
the use of pseudoholomorphic curves in symplectic geometry more widely
accessible by combining the following features:

  * (a)

The theory applies to compactified moduli spaces ${\overline{{\mathcal{M}}}}$
of pseudoholomorphic curves in the sense that
${\overline{{\mathcal{M}}}}\simeq\sigma^{-1}(0)$ is homeomorphic to the
solution set of a so-called scale-smooth (or sc-) Fredholm section
$\sigma:{\mathcal{B}}\to{\mathcal{E}}$ of a strong polyfold bundle
${\mathcal{E}}\to{\mathcal{B}}$. 111While expected for all known types of
compact moduli spaces in symplectic geometry, only Gromov-Witten moduli spaces
currently have a published description [HWZ2] of this form. There is work in
progress [FH, L] on the moduli spaces of SFT, Floer theories, and Fukaya
categories.

  * (b)

The theory provides a transparent approach to regularizing such moduli spaces
by (multivalued) perturbations of the section $\sigma$ that achieve
transversality while preserving compactness and most geometric characteristics
of the perturbed solution set.222This theory is fully developed and documented
in [HWZ] and earlier publications cited therein.

Two key points about this theory contrast sharply with the usual treatments of
Fredholm operators on Banach spaces.333For an introduction to scale calculus –
the notion of differentiability on Banach spaces used by [HWZ] – see e.g.
[FFGW].

$\bullet$ It is based on a new notion of sc-smoothness, which is designed
firstly so that there is an appropriate analog of Fredholm theory, and
secondly so that reparametrization groups (such as ${\rm PSL}(2,{\mathbb{C}})$
act sc-smoothly on spaces $E$ of sc-smooth maps $u:{\mathbb{C}}P^{1}\to M$.
This is in sharp contrast to the action of such groups on Banach spaces. For
example the rotation group $S^{1}$ does not act even continuously on the space
of $L^{2}$-maps $S^{1}\to{\mathbb{R}}$.

$\bullet$ The fact that $M$-polyfolds are modelled on the image $r(U)$ of an
open subset $U\subset E$ under a scale smooth retraction (rather than simply
on the open set $U$ itself) allows one to describe a neighbourhood of a nodal
curve that includes stable maps with topologically different domains as an
open subset of a single $M$-polyfold.

Thus polyfold theory provides good local models for the moduli space (i.e. the
solution set of the given operator), and the challenge is to understand the
global structure. There are other theories with analogous features for
different notions of “Fredholm section”, in particular various versions of
Kuranishi theory [FOOO, MW3, Pa] which combines local finite dimensional
models into a global section functor $\sigma$ between categories (that are
generally neither étale nor groupoids). Although the current ideas may be used
to construct a (tame) smooth Kuranishi atlas that models the zero set of
${\sigma}$, in this paper, we work only in the infinite dimensional setting,
constructing a global stabilization of the operator ${\sigma}$, namely a
finite dimensional family of multi-sections such that a generic element is a
transverse perturbation.

Beyond Gromov-Witten theory, most moduli spaces in symplectic geometry have
formal boundary, and in fact come in ‘coherent families’ whose boundary strata
are Cartesian or fiber products of other moduli spaces in the family.
Extracting algebraic invariants – such as Floer homology, or a well defined
SFT algebra – from such families of moduli spaces requires ‘coherent
perturbations’ that preserve the identifications of boundary strata. The
corresponding challenge of extending multisections prescribed on boundaries is
present in both polyfold and Kuranishi theory, and is resolved in [HWZ] by a
class of so-called ‘structurable multisections’. However, this involves rather
elaborate definitions and constructions, and at present does not yield a
transparent general existence result for coherent perturbations.

This problem of constructing coherent multivalued perturbations is simplified
by our construction of a global Fredholm stabilization in Theorem 2.1.2. This
provides a finite dimensional family of perturbations that is sufficiently
large to achieve transversality, yet is multivalued in the simplest possible
way: each perturbation is the orbit of a single-valued section under a global
finite group action. This type of multisection has natural extensions from
boundaries and corners, so that a general existence proof of coherent
perturbations reduces to the project of extending the global stabilization
construction to coherent setups.

###  1.1. Overview of constructions

As defined in [HWZ, §16], a polyfold ${\mathcal{B}}$ is a topological space
that is homeomorphic to the realization of a topological groupoid
${\mathcal{X}}$ that is equipped with a special kind of smooth structure
called a sc-smooth structure. A local model (without formal boundary) for a
global Fredholm section $\sigma:{\mathcal{B}}\to{\mathcal{E}}$ with zero set
$\sigma^{-1}(0)={\overline{{\mathcal{M}}}}$ is given by the quotient by a
finite group $G$ of a $G$-equivariant scale-smooth section

$U\to{\mathcal{R}}=\bigcup_{x\in
U}\\{x\\}\times\Pi_{x}({\mathbb{F}}),x\mapsto(x,f(x)).$

Its base $U=r(\widetilde{U})$ is the image of a scale-smooth retraction
$r:\widetilde{U}\to\widetilde{U}$, $r\circ r=r$, on an open subset
$\widetilde{U}\subset{\mathbb{E}}$ of a scale-Banach space. The total space
${\mathcal{R}}$ is the image of a scale-smooth retraction
$\widetilde{U}\times{\mathbb{F}}\to\widetilde{U}\times{\mathbb{F}},(x,y)\mapsto(r(x),\Pi_{x}(y))$
given by linear projections $\Pi_{x}:{\mathbb{F}}\to{\mathbb{F}}$ on another
scale-Banach space. The polyfold-Fredholm property of this section requires
the existence of a scale-Fredholm444The Fredholm property in scale calculus
implies classical Fredholmness of the linearizations, but due to the lack of
classical differentiability requires a ‘contraction germ normal form’. In
practice [HWZ2, W] it follows from ‘classical differentiability in all but
finitely many directions’.  extension
$\widetilde{f}:\widetilde{U}\to{\mathbb{F}}$ with

$\widetilde{f}^{-1}(0)=f^{-1}(0),\quad\ker{\rm d}_{x}\widetilde{f}=\ker{\rm
d}_{x}f,\quad\text{\raise
3.44444pt\hbox{$\scriptstyle\Pi_{x}({\mathbb{F}})\\!$}/\lower
3.44444pt\hbox{$\\!\scriptstyle{\rm im\,}{\rm d}_{x}f$}}\simeq\text{\raise
3.44444pt\hbox{$\scriptstyle{\mathbb{F}}\\!$}/\lower
3.44444pt\hbox{$\\!\scriptstyle{\rm im\,}{\rm d}_{x}\widetilde{f}$}}\,,$

and the resulting local chart for the moduli space is an embedding
$f^{-1}(0)/G\subset\sigma^{-1}(0)={\overline{{\mathcal{M}}}}$.

The crucial construction that yields both transverse perturbations of the
polyfold-Fredholm section $\sigma$ and its finite dimensional reductions
(Kuranishi charts) is that of a local Fredholm stabilization near
$x\in\sigma^{-1}(0)$. Here we construct a finite dimensional vector space $E$
as the product of a representative of the cokernel
$\scriptstyle\Pi_{x}({\mathbb{F}})\\!$/$\\!\scriptstyle{\rm im\,}{\rm d}_{x}f$
with the group $G$ that acts on $U$ as above, so that $G$ acts diagonally on
$U\times E$. We then define a $G$-equivariant bundle map $\tau:U\times
E\to{\mathcal{R}}$ covering the identity, so that $\sigma-\tau$ is transverse
to the zero section of ${\mathcal{R}}$ near $(x,0)\in U\times E$. (For details
see Lemma 5.4.1.) Each $e\in E$ then yields a local multivalued perturbation
$U\ni x\mapsto\bigl{(}\tau(x,ge)\bigr{)}_{g\in G}\subset{\mathcal{R}}$, which
yields $G$-equivariant transversality of
$\sigma|_{U}-\bigl{(}\tau(\cdot,ge)\bigr{)}_{g\in G}$ for regular values of
the map $(\sigma|_{U}-\tau)^{-1}(0)\to E$. Alternatively, the same
construction can be viewed as giving a local finite dimensional reduction

$\displaystyle f_{\tau}\,:$
$\displaystyle\;U_{\tau}:=(\sigma|_{U}-\tau)^{-1}(0)\to E,\quad(e,x)\mapsto
e,$ $\displaystyle\quad\text{with}\quad
f_{\tau}^{-1}(0)/G\simeq(\sigma^{-1}(0)\cap
U)/G\subset{\overline{{\mathcal{M}}}}.$

Once this local structure is understood, the main difficulty with both the
Kuranishi atlas and polyfold approaches is to patch such local perturbations
into a globally transverse perturbation, while preserving compactness of the
perturbed solution space.

The key idea of the current paper is to model the base polyfold
${\mathcal{B}}$ by a groupoid ${\mathcal{X}}_{\mathcal{V}}$ whose structure is
so transparent that the pullback of the Fredholm section
$\sigma:{\mathcal{B}}\to{\mathcal{E}}$ has a natural family of globally
transverse perturbations. Here are the main constructions:

  * (i)

Given finitely many local polyfold charts
$U_{i}/G_{i}\hookrightarrow{\mathcal{B}}$ covering a compact subset
${\overline{{\mathcal{M}}}}\subset\bigcup_{i}U_{i}/G_{i}$, Theorem 5.2.6
constructs an étale groupoid ${\mathcal{X}}_{\mathcal{V}}$ with explicitly
described object and morphism spaces and whose realization
$|{\mathcal{X}}_{\mathcal{V}}|$ is equivalent as polyfold to an open
neighbourhood $|V|\subset{\mathcal{B}}$ of ${\overline{{\mathcal{M}}}}$.

Moreover, Theorem 5.3.1 constructs a Fredholm section functor
$f_{\mathcal{V}}:{\mathcal{X}}_{\mathcal{V}}\to{\mathcal{W}}_{\mathcal{V}}$
such that the perturbation theory for
$|f_{\mathcal{V}}^{-1}(0)|\simeq{\overline{{\mathcal{M}}}}$ is equivalent to
that for $\sigma$ in the sense that appropriate perturbations for each section
have naturally identified solution sets.

  * (ii)

Given data as in (i) and local Fredholm stabilizations $\tau_{i}:U_{i}\times
E_{i}\to{\mathcal{E}}|_{U_{i}}$, Theorem 2.1.2 constructs a global Fredholm
stabilization as follows: There is an étale (but generally not proper)
subgroupoid ${\mathcal{X}}_{\mathcal{V}}^{{\smallsetminus}G}$ of
${\mathcal{X}}_{\mathcal{V}}$ obtained by removing all morphisms that give
rise to isotropy. It carries an action of $G:=\prod G_{i}$ such that
$|{\mathcal{X}}_{\mathcal{V}}^{{\smallsetminus}G}|\,/G\simeq|{\mathcal{X}}_{\mathcal{V}}|$.
We then construct a finite dimensional vector space $E:=\prod_{i}E_{i}$ and a
$G$-equivariant functor
$\tau:{\mathcal{X}}_{\mathcal{V}}^{{\smallsetminus}G}\times
E\to{\mathcal{W}}_{\mathcal{V}}$ so that $f_{\mathcal{V}}-\tau$ is transverse
to the zero section.

  * (iii)

Given data as in (ii), Corollary 2.1.3 shows that $\tau$ applied to generic,
sufficiently small $e\in E$ induces canonically structured sc+-multisection
functors
${\Lambda}_{{\mathcal{V}},e}:{\mathcal{W}}_{\mathcal{V}}\to{\mathbb{Q}}^{\geq
0}$ and ${\Lambda}_{{\underline{e}}}:{\mathcal{W}}|_{V}\to{\mathbb{Q}}^{\geq
0}$ that satisfy the requirements of the polyfold perturbation scheme and have
easily described solution sets.

  * (iv)

When ${\overline{{\mathcal{M}}}}$ has no formal boundary, Theorem 5.1.4
defines a polyfold fundamental class in the rational Çech homology of
${\overline{{\mathcal{M}}}}$ as the inverse limit of the fundamental classes
of the perturbed solution sets. In this setting, (iii) identifies the polyfold
fundamental classes
$[{\overline{{\mathcal{M}}}}]_{f_{\mathcal{V}}}=[{\overline{{\mathcal{M}}}}]_{\sigma}$,
where the latter depends on the initial polyfold Fredholm section
${\sigma}:{\mathcal{B}}\to{\mathcal{E}}$.555 The book [HWZ] does not attempt
to define such a fundamental class, instead showing in this situation that
there is a well defined notion of integration of sc-smooth forms over
${\overline{{\mathcal{M}}}}$.

As outlined in [Mc2, Rmk.1.3.8], the current results easily adapt to construct
a Kuranishi atlas 666Another abstract construction of this type, without
identification of perturbations, can be found in [Y]. ${\mathcal{K}}_{\tau}$
for ${\overline{{\mathcal{M}}}}$ whose perturbation theory is equivalent to
the polyfold perturbation theory for $f_{\mathcal{V}}$ and thus for $\sigma$.

Note to the reader: This paper contains many technical details that readers
should not attempt to master until the outlines of the construction are clear.
The main theorems are clearly stated in §2.1. The language here is very
formal; readers can consult §2.2 and §3.1 for background definitions. We
recommend that the reader goes next to §4.1, where the main constructions are
defined in an abstract setting. This section uses the language developed in
§3.1–3.3 concerning étale categories and groupoid completions. In §5, we first
show in Theorem 5.1.4 that the perturbed zero sets of polyfold Fredholm
sections carry a fundamental class in rational Čech homology, and then
complete the proofs of the main theorems by applying the abstract
constructions developed in §4 to the polyfold setting.

Acknowledgement: This paper is the joint work of Dusa McDuff and Katrin
Wehrheim. However this final version was prepared by the first named author,
who takes responsibility for all inaccuracies and omissions. She warmly thanks
Zhengyi Zhou for his many careful comments on an earlier version.

##  2\. Terminology and main results

###  2.1. Statement of main theorems

A sc-Fredholm section $\sigma:{\mathcal{B}}\to{\mathcal{E}}$ of a strong
polyfold bundle is a continuous map between topological spaces together with
an equivalence class of functors $f:{\mathcal{X}}\to{\mathcal{W}}$ between
topological groupoids, whose realization
$|f|:|{\mathcal{X}}|\to|{\mathcal{W}}|$ together with homeomorphisms
$|{\mathcal{X}}|\simeq{\mathcal{B}}$, $|{\mathcal{W}}|\simeq{\mathcal{E}}$
induces $\sigma$. Throughout this paper, we will work with a fixed
representative $f:{\mathcal{X}}\to{\mathcal{W}}$ that is a sc-Fredholm section
functor of a strong bundle ${\mathcal{P}}:{\mathcal{W}}\to{\mathcal{X}}$ over
an ep-groupoid ${\mathcal{X}}$ modeled on M-polyfolds.777Questions of
equivalence between such representatives are addressed in [HWZ, §10-11]. To
summarize these notions we use conventions of [HWZ] in denoting object and
morphism spaces as ${\rm Obj}_{\mathcal{X}}=X$ and ${\rm
Mor}_{\mathcal{X}}={\bf X}$; but we will differentiate between object space
$X$ and groupoid ${\mathcal{X}}=(X,{\bf X})$. In particular, the realization
$|{\mathcal{X}}|=X/\\!\\!\sim\,$ is the quotient of $X$ by the equivalence
relation generated by the morphisms ${\bf X}$, equipped with the quotient
topology. The following generalizes the notions of orbifolds, orbibundles, and
smooth sections to infinite dimensions and scale calculus. The underlying
analytic notions of M-polyfold, sc-Fredholm, etc. are reviewed in §2.2 along
with the finite dimensional cases.

######  Definition 2.1.1.

  * (i)

An ep-groupoid as in [HWZ, Def.7.1.3] is a groupoid ${\mathcal{X}}=(X,{\bf
X})$ equipped with M-polyfold structures888 As in other regularization
constructions of polyfold theory, we assume that $X$ admits sc-smooth
partitions of unity. This can be guaranteed by working with M-polyfold charts
in Hilbert spaces; see [HWZ, §5.5]. In classical PDE applications that means
working with Sobolev spaces of class $L^{p}$ with $p=2$. Moreover, the
boundaries of $X$ and ${\bf X}$ are assumed to be tame in the sense of [HWZ,
§2.5]. In the known applications, this is guaranteed by retracts of splicing
form; see [HWZ, Lemma 2.5.5]. on the object and morphism sets so that

    * •

source and target maps $s,t:{\bf X}\to X$ are local sc-diffeomorphisms and the
unit $X\to{\bf X},q\mapsto{\rm id}_{q}$, inverse ${\bf X}\to{\bf X},m\mapsto
m^{-1}$, and composition ${\bf X}{\vphantom{\times}}_{t}{\times}_{s}{\bf
X}\to{\bf X},(m,m^{\prime})\mapsto m\circ m^{\prime}$ are sc-smooth maps (i.e.
the groupoid is étale);

    * •

every $x\in X$ has a neighbourhood $N(x)$ so that $t:s^{-1}\bigl{(}{\rm
cl}_{X}(N(x))\bigr{)}\to X$ is proper.

In addition, we will throughout assume that the realization $|{\mathcal{X}}|$
is paracompact so that $|{\mathcal{X}}|$ is metrizable by [HWZ, Thm.7.3.1]. We
call an ep-groupoid ${\mathcal{X}}=(X,{\bf X})$ nonsingular if the isotropy
groups $G_{x}:=s^{-1}(x)\cap t^{-1}(x)\subset{\bf X}$ are trivial for all
$x\in X$. Finally, we will say that ${\mathcal{X}}$ has no formal boundary if
the boundary $\partial X$ of its object set as defined in [HWZ, §2.4] is
empty.

  * (ii)

A strong bundle as in [HWZ, Def.8.3.1]999This summarizes the discussion
following [HWZ, Def.8.3.1]; for more details see Definition 3.4.1. over the
ep-groupoid ${\mathcal{X}}$ is a pair $(P,\mu)$ of a strong bundle $P:W\to X$
and a strong bundle map $\mu:{\bf X}{\vphantom{\times}}_{s}{\times}_{P}W\to W$
so that $P$ lifts to a functor ${\mathcal{P}}:{\mathcal{W}}\to{\mathcal{X}}$
given by ${\mathcal{P}}(w)=P(w)$, ${\mathcal{P}}(m,w)=m$. Here ${\mathcal{W}}$
is the ep-groupoid given by ${\rm Obj}_{\mathcal{W}}=W$, ${\rm
Mor}_{\mathcal{W}}={\bf X}{\vphantom{\times}}_{s}{\times}_{P}W$, $s(m,w)=w$,
$t(m,w)=\mu(m,w)$, ${\rm id}_{w}=({\rm id}_{P(w)},w)$, and
$(m,w)\circ(m^{\prime},w^{\prime})=\bigl{(}m\circ m^{\prime},w\bigr{)}$ if
$w^{\prime}=\mu(m,w)$.

We will thus often denote strong bundles by
$({\mathcal{P}}:{\mathcal{W}}\to{\mathcal{X}},\mu)$ or simply
${\mathcal{P}}:{\mathcal{W}}\to{\mathcal{X}}$.

  * (iii)

A sc-Fredholm section functor of the strong bundle
${\mathcal{P}}:{\mathcal{W}}\to{\mathcal{X}}$ as in [HWZ, Def.8.3.4] is a
functor $f:{\mathcal{X}}\to{\mathcal{W}}$ that is sc-smooth on object and
morphism spaces, satisfies ${\mathcal{P}}\circ f={\rm id}_{\mathcal{X}}$, and
is such that $f:X\to W$ is sc-Fredholm on the M-polyfold $X$.

  * (iv)

An orientation of a sc-Fredholm section functor $f$ as in [HWZ, Def.12.5.4] is
a continuous section ${\mathfrak{o}}:X_{\infty}\to{\mathcal{O}}_{f}$ of the
orientation bundle over the dense subset of smooth objects $X_{\infty}\subset
X$, which is compatible with morphisms as specified in §2.3.

Given a sc-Fredholm section $f:{\mathcal{X}}\to{\mathcal{W}}$ that describes a
compact moduli space ${\overline{{\mathcal{M}}}}\simeq|f^{-1}(0)|$, Lemma
5.4.1 constructs local Fredholm stabilizations near any $x\in f^{-1}(0)$, as
explained in §1.1. Now a finite covering of $|f^{-1}(0)|$ by such local
stabilizations can be used in two ways: On the one hand, the various fattened
solution sets resulting from each local stabilization can be related by
appropriate transition data to form a Kuranishi atlas for
${\overline{{\mathcal{M}}}}$.On the other hand, after an appropriate
refinement of the polyfold model, we can combine a covering of
${\overline{{\mathcal{M}}}}\simeq|f^{-1}(0)|$ by local Fredholm stabilizations
into a global finite dimensional stabilization. This is our first main result
(proven in §5.2–5.4):

######  Theorem 2.1.2 (Global stabilization).

Let $f:{\mathcal{X}}\to{\mathcal{W}}$ be a sc-Fredholm section functor with
compact solution set, Then choices of local uniformizers and Fredholm
stabilizations $(U_{i},G_{i},\tau_{i}:X\times
E_{i}\to{\mathcal{W}})_{i\in\\{1,\ldots,N\\}}$ covering $|f^{-1}(0)|$, a
partition of unity, compatible cover reductions as specified in Theorem 5.4.2,
and suitable compactness controlling data $(N,{\mathcal{U}})$ as in Definition
3.5.1 induce

  * -

an open subset $V\subset X$ such that $|V|\subset|{\mathcal{X}}|$ is an open
neighbourhood of $|f^{-1}(0)|$ that contains $|{\rm cl}_{X}({\mathcal{U}})|$,

  * -

a finite group $G:=\prod_{i=1}^{N}G_{i}$ and a finite dimensional vector space
$E:=\prod_{i=1}^{N}E_{i}$ with $G$-action, and and

  * -

a commutative diagram of étale groupoids and functors

(2.1.5)

with the following properties:

  * (i)

${\mathcal{X}}|_{V}$ and ${\mathcal{W}}|_{V}$ are the full subcategories of
${\mathcal{X}}$ resp. ${\mathcal{W}}$ with objects $V$ resp.
$W|_{V}=P^{-1}(V)$ as in Lemma 3.4.4. In particular their realizations are
naturally identified with
$|{\mathcal{X}}|_{V}|\simeq|V|\subset|{\mathcal{X}}|$ and
$|{\mathcal{W}}|_{V}|\simeq|{\mathcal{P}}|^{-1}(|V|)\subset|{\mathcal{W}}|$.

  * (ii)

${\mathcal{X}}_{\mathcal{V}}$ is an ep-groupoid constructed in Theorem 5.2.6,
covered by finitely many local uniformizers
$(V_{I},G_{I})_{I\subset\\{1,\ldots,N\\}}$ so that ${\rm
Obj}_{{\mathcal{X}}_{\mathcal{V}}}=\bigsqcup_{I}V_{I}$ and ${\rm
Mor}_{{\mathcal{X}}_{\mathcal{V}}}(V_{I},V_{I})=G_{I}\times V_{I}$ with
$G_{I}=\prod_{i\in I}G_{i}$. It is equipped with an inner action of $G$ in the
sense of Definition 3.3.4, in particular lifting the trivial action on
$|{\mathcal{X}}_{\mathcal{V}}|$. Further, the action on each $V_{I}\subset{\rm
Obj}_{{\mathcal{X}}_{\mathcal{V}}}$ is given by the natural projection from
$G\times V_{I}$ to the local uniformizer morphisms $G_{I}\times V_{I}={\rm
Mor}_{{\mathcal{X}}_{\mathcal{V}}}(V_{I},V_{I})$ induced by the projection
$G\to G_{I}$.

The functor $\psi:{\mathcal{X}}_{\mathcal{V}}\to{\mathcal{X}}|_{V}$, given by
natural maps $V_{I=\\{i_{1}<\cdot\cdot\\}}\to U_{i_{1}}$, is finite-to-one,
surjective on objects, and an equivalence of ep-groupoids in the sense of
[HWZ, Def.10.1.1], in particular inducing a homeomorphism
${|\psi|:|{\mathcal{X}}_{\mathcal{V}}|\stackrel{{\scriptstyle\simeq}}{{\to}}|{\mathcal{X}}|_{V}|=|V|}$.

  * (iii)

The bundle
${\mathcal{P}}_{\mathcal{V}}:{\mathcal{W}}_{\mathcal{V}}:=\psi^{*}{\mathcal{W}}\to{\mathcal{X}}_{\mathcal{V}}$
and its section $f_{\mathcal{V}}=\psi^{*}f$ are the pullbacks in the sense of
Lemma 3.4.4 of ${\mathcal{P}}$ and $f$, as spelled out in Theorem 5.3.1. The
bundle ${\mathcal{W}}_{\mathcal{V}}$ is equipped with the natural lift of the
inner $G$ action on ${\mathcal{X}}_{\mathcal{V}}$ from Lemma 3.3.6, with
respect to which both ${\mathcal{P}}_{\mathcal{V}}$ and $f_{\mathcal{V}}$ are
equivariant. The natural lift of $\psi$ to the pullback bundle functor
$\Psi:{\mathcal{W}}_{\mathcal{V}}\to{\mathcal{W}}|_{V}$ of Lemma 3.4.4 (iii)
is a strong bundle equivalence in the sense of [HWZ, Def.10.4.1], and $|\psi|$
restricts to a homeomorphism
$|f_{\mathcal{V}}^{-1}(0)|\stackrel{{\scriptstyle\simeq}}{{\to}}|f^{-1}(0)|$.
The section $f_{\mathcal{V}}$ is sc-Fredholm, and the strong bundle
equivalence $\Psi$ pulls back any choice of orientation of $f$ to an
orientation of $f_{\mathcal{V}}$.

  * (iv)

The étale groupoid ${\mathcal{X}}_{\mathcal{V}}^{{\smallsetminus}G}$
constructed in Proposition 4.1.2 is a (nonproper) nonsingular subgroupoid of
${\mathcal{X}}_{\mathcal{V}}$ with the same objects ${\rm
Obj}_{{\mathcal{X}}_{\mathcal{V}}^{{\smallsetminus}G}}=\bigsqcup_{I}V_{I}$ but
where enough morphisms have been removed so that
${\mathcal{X}}_{\mathcal{V}}^{{\smallsetminus}G}$ has at most one morphism
between any two objects.101010 Although we do not attempt to formulate a sense
in which ${\mathcal{X}}_{\mathcal{V}}^{{\smallsetminus}G}$ is unique, it has a
very simple structure: It is the groupoid completion of an étale poset
${{{\mathcal{Q}}}}$, that is, the object space $\bigsqcup_{I}V_{I}$ has a
natural partial order $\preccurlyeq$ such that ${\rm
Mor}_{{{{\mathcal{Q}}}}}(x,y)$ is nonempty exactly if $x\preccurlyeq y$; see
Lemma 4.2.1.

The removed morphisms are parts of the group action. Indeed,
${\mathcal{X}}_{\mathcal{V}}^{{\smallsetminus}G}$ is equipped with an action
of $G$ so that the inclusion functor
${\iota}:{\mathcal{X}}_{\mathcal{V}}^{{\smallsetminus}G}\to{\mathcal{X}}_{\mathcal{V}}$
is $G$-equivariant in the sense of Definition 3.3.1 and induces a
homeomorphism
$|{\iota}|:|{\mathcal{X}}_{\mathcal{V}}^{{\smallsetminus}G}|\,/G\stackrel{{\scriptstyle\simeq}}{{\to}}|{\mathcal{X}}_{\mathcal{V}}|$.

  * (v)

The category ${\mathcal{X}}_{\mathcal{V}}^{{\smallsetminus}G}\times E$ is the
product of ${\mathcal{X}}_{\mathcal{V}}^{{\smallsetminus}G}$ with the category
with object space $E$ and only identity morphisms. Then the functor
$\tau:{\mathcal{X}}_{\mathcal{V}}^{{\smallsetminus}G}\times
E\to{\mathcal{W}}_{\mathcal{V}}$ constructed in Theorem 5.4.2 is
$G$-equivariant, a strong bundle map in the sense of [HWZ, Def.2.6.1], and a
compactness-controlled Fredholm stabilization in the following sense:

The data
$\bigl{(}N_{\mathcal{V}}:=N\circ\Psi,{\mathcal{U}}_{\mathcal{V}}:=\psi^{-1}({\mathcal{U}})\bigr{)}$
controls compactness of $f_{\mathcal{V}}$ and there exists a neighbourhood
$O_{\tau}\subset E$ of $0$ such that the family of maps
$\bigl{(}\tau_{\underline{e}}:{\rm Obj}_{{\mathcal{X}}_{\mathcal{V}}}\to{\rm
Obj}_{{\mathcal{W}}_{\mathcal{V}}},{\underline{o}}\mapsto\tau({\underline{o}},{\underline{e}})\bigr{)}_{{\underline{e}}\in
E}$ – while generally not functorial – is
$(N_{\mathcal{V}},{\mathcal{U}}_{\mathcal{V}})$-regular as follows:

    * (1)

$N_{\mathcal{V}}(\tau_{\underline{e}}({\underline{o}}))<1$ for all
${\underline{e}}\in O_{\tau}$, ${\underline{o}}\in{\rm
Obj}_{{\mathcal{X}}_{\mathcal{V}}}={\rm
Obj}_{{\mathcal{X}}_{\mathcal{V}}^{{\smallsetminus}G}}$;

    * (2)

${\rm supp\,}\tau_{\underline{e}}={\rm cl}\\{{\underline{o}}\in{\rm
Obj}_{{\mathcal{X}}_{\mathcal{V}}}\,|\,\tau_{\underline{e}}({\underline{o}})\neq
0_{\underline{o}}\\}\subset{\mathcal{U}}_{\mathcal{V}}$ for all
${\underline{e}}\in O_{\tau}$;

    * (3)

the map ${\rm Obj}_{{\mathcal{X}}_{\mathcal{V}}}\times O_{\tau}\to{\rm
Obj}_{{\mathcal{W}}_{\mathcal{V}}},({\underline{o}},{\underline{e}})\mapsto
f_{\mathcal{V}}({\underline{o}})-\tau({\underline{o}},{\underline{e}})$ is
transverse to the zero section of ${\mathcal{W}}_{\mathcal{V}}$ and in general
position over the boundary.

Full details and proofs are given in Theorems 5.2.6, 5.3.1, and 5.4.2. The
construction of such a global finite dimensional stabilization is the common
technical work behind two major points of view on how to construct the
“virtual” fundamental class of a compact moduli space
${\overline{{\mathcal{M}}}}\simeq|f^{-1}(0)|$ without formal boundary:
Moreover, as sketched in [Mc2, Rmk.1.3.8], the global stabilization leads
directly to the construction of a Kuranishi atlas as well as other finite
dimensional models. On the other hand the following Corollary shows how the
stabilization also yields a collection of multivalued perturbations whose
solution sets are compact weighted branched orbifolds and can be used for an
inverse limit construction of $[{\overline{{\mathcal{M}}}}]$ in Theorem 5.1.4.

This construction of transverse perturbations from a global stabilization is
the same line of argument as in both general polyfold theory and more
classical regularization approaches such as [MS].111111 For example, the
stabilization in the trivial isotropy case is the map ${\mathbb{R}}^{m}\times
X\to W$ in the proof of [HWZ, Thm.5.3.5], and is the section
${\mathcal{B}}\times{\mathcal{J}}\to{\mathcal{E}}$ in the proof of [MS,
Prop.3.2.1]. In polyfold theory (and any other abstract construction of
perturbations) the main technical complications arise from nontrivial
isotropy. This is resolved locally by working with multivalued perturbations,
but their patching and extension from the boundary is surprisingly nontrivial;
see [HWZ, §14]. We avoid the technicalities of patching or extending
multivalued perturbations by following the strategy of [MW3, §3.2]: Our
technical work is in the construction of an appropriate atlas, given by the
étale groupoid ${\mathcal{X}}_{\mathcal{V}}^{{\smallsetminus}G}$ that is both
reduced (to organize overlaps of charts) and pruned (to remove isotropy), and
the global action of a group $G$ that captures all isotropy. Then the global
stabilization $\tau:{\mathcal{X}}_{\mathcal{V}}^{{\smallsetminus}G}\times
E\to{\mathcal{W}}_{\mathcal{V}}$ gives rise to a family of single-valued
sections

$f_{\mathcal{V}}-\tau(\cdot,{\underline{e}}):X_{\mathcal{V}}:={\rm
Obj}_{{\mathcal{X}}_{\mathcal{V}}}\to{\rm Obj}_{{\mathcal{W}}_{\mathcal{V}}}$

of the topological bundle formed by the object spaces of the bundle
${\mathcal{W}}_{\mathcal{V}}\to{\mathcal{X}}_{\mathcal{V}}$, which fail
functoriality only in their lack of $G$-equivariance.121212 Note here that,
while we construct the stabilization functor $\tau$ to be $G$-equivariant, the
single-valued perturbation $f_{\mathcal{V}}-\tau(\cdot,{\underline{e}})$ is
typically not itself $G$-equivariant because $G$ acts nontrivially on $E$.
Taking the $G$-orbit of such a section yields a transverse multivalued
perturbation of $f_{\mathcal{V}}$ which is more globally symmetric and
structured than the complicated hierarchy of [HWZ, §13]. Our notion of a
globally structured multisection is given in Definition 3.4.5. This notion
should allow considerable simplification of the intricate extension
constructions in [HWZ, §14].

The following is a detailed description of these perturbations and perturbed
zero sets. Here we denote by ${\mathbb{Q}}^{\geq 0}$ the category with objects
${\mathbb{Q}}\cap[0,\infty)$ and only identity morphisms.131313 This category
is denoted ${\mathbb{Q}}^{+}$ in [HWZ].

######  Corollary 2.1.3.

Let $\tau:{\mathcal{X}}_{\mathcal{V}}^{{\smallsetminus}G}\times
E\to{\mathcal{W}}_{{\mathcal{V}}}$ be a stabilization functor as in Theorem
2.1.2.

  * (i)

Each ${\underline{e}}\in E$ induces sc+-multisection functors
${\Lambda}_{{\mathcal{V}},{\underline{e}}}:{\mathcal{W}}_{\mathcal{V}}\to{\mathbb{Q}}^{\geq
0}$ given by ${\Lambda}_{{\mathcal{V}},{\underline{e}}}(w):=\tfrac{1}{\\#G}\
\\#\\{{\underline{h}}\in
G\,|\,\tau(P_{\mathcal{V}}(w),{\underline{h}}*{\underline{e}})=w\\}$ and
${\Lambda}_{{\underline{e}}}:=\Psi_{*}{\Lambda}_{{\mathcal{V}},{\underline{e}}}:{\mathcal{W}}|_{V}\to{\mathbb{Q}}^{\geq
0}$ given by pushforward [HWZ, Lemma 11.5.2]. Their support (i.e. “graph”) is

$\displaystyle{\rm supp\,}{\Lambda}_{{\mathcal{V}},{\underline{e}}}$
$\displaystyle=G*\tau(X_{\mathcal{V}}\times\\{{\underline{e}}\\})={\textstyle\bigcup_{{\underline{g}}\in
G}}\;\tau(X_{\mathcal{V}}\times\\{{\underline{g}}*{\underline{e}}\\}),$
$\displaystyle{\rm supp\,}{\Lambda}_{\underline{e}}$ $\displaystyle=\Psi({\rm
supp\,}{\Lambda}_{{\mathcal{V}},{\underline{e}}}).$

These multisections are structurable in the sense of [HWZ, §13.3], and
${\Lambda}_{{\mathcal{V}},{\underline{e}}}$ is globally structured in the
sense of Definition 3.4.5.

  * (ii)

There is a comeagre (and thus dense) subset $O^{\pitchfork}_{\tau}\subset
O_{\tau}$ such that for all ${\underline{e}}\in O^{\pitchfork}_{\tau}$ the
functors
$\Theta_{{\mathcal{V}},{\underline{e}}}:={\Lambda}_{{\mathcal{V}},{\underline{e}}}\circ
f_{\mathcal{V}}:{\mathcal{X}}_{\mathcal{V}}\to{\mathbb{Q}}^{\geq 0}$ and
$\Theta_{\underline{e}}:={\Lambda}_{{\underline{e}}}\circ
f:{\mathcal{X}}|_{V}\to{\mathbb{Q}}^{\geq 0}$ are transversal and in general
position over the boundary in the sense of [HWZ, §15.2]. This gives both the
structure of compact, tame branched ep+-subgroupoids in the sense of [HWZ,
§9.1].

  * (iii)

If $f$ is oriented, then for every ${\underline{e}}\in O^{\pitchfork}_{\tau}$
both $\Theta_{{\mathcal{V}},{\underline{e}}}$ and $\Theta_{\underline{e}}$
inherit natural orientations.

  * (iv)

Suppose, in addition to (iii), that $f$ has constant Fredholm index
$d\in{\mathbb{Z}}$, and $\psi_{f}:|f^{-1}(0)|\to{\overline{{\mathcal{M}}}}$ is
a homeomorphism to a compact Hausdorff space ${\overline{{\mathcal{M}}}}$.
Then the perturbed solution sets

$\displaystyle|{\rm
supp\,}\Theta_{{\mathcal{V}},{\underline{e}}}|=|\\{{\underline{x}}\in
X_{\mathcal{V}}\,|\,\Theta_{{\mathcal{V}},{\underline{e}}}({\underline{x}})>0\\}|\subset|{\mathcal{X}}_{\mathcal{V}}|,$
$\displaystyle|{\rm supp\,}\Theta_{\underline{e}}|=|\\{x\in
X|_{V}\,|\,\Theta_{\underline{e}}(x)>0\\}|\subset|V|$

carry natural fundamental classes denoted
$[\Theta_{{\mathcal{V}},{\underline{e}}}],[\Theta_{\underline{e}}]$ that
belong to $\check{H}_{d}({\overline{{\mathcal{M}}}};{\mathbb{Q}})$ if
${\overline{{\mathcal{M}}}}$ has no formal boundary, and to the relative group
$\check{H}^{\infty}_{d}({\overline{{\mathcal{M}}}}{\smallsetminus}{\partial}{\overline{{\mathcal{M}}}};{\mathbb{Q}})$
if ${\overline{{\mathcal{M}}}}$ has formal boundary
${\partial}{\overline{{\mathcal{M}}}}$.141414See (A.3) for notation. Further,
if there is no formal boundary $\psi_{f_{\mathcal{V}}}:=\psi_{f}\circ|\psi|$
is a homeomorphism $|f_{\mathcal{V}}^{-1}(0)|\to{\overline{{\mathcal{M}}}}$,
and the perturbations in (ii) induce the same fundamental classes as defined
in Theorem 5.1.4,

$[{\overline{{\mathcal{M}}}}]_{f}\;=\;(\psi_{f})_{*}\;\underset{\leftarrow}{\lim}\,(\psi_{k})_{*}\bigl{[}\Theta_{{\underline{e}}_{k}}\bigr{]}\;=\;(\psi_{f_{\mathcal{V}}})_{*}\;\underset{\leftarrow}{\lim}\,(\psi_{{\mathcal{V}},k})_{*}\bigl{[}\Theta_{{\mathcal{V}},{\underline{e}}_{k}}\bigr{]}\;=\;[{\overline{{\mathcal{M}}}}]_{f_{\mathcal{V}}}\;\in\;\check{H}_{d}({\overline{{\mathcal{M}}}};{\mathbb{Q}}).$

  * (v)

In particular, in the situation of (iv) there is a well defined notion of
integration over the perturbed solution sets, which satisfies Stokes’ Theorem
in the sense of [HWZ, Thm.15.4.3], and the results can be identified via the
homeomorphism $|\psi|\,|_{|{\rm
supp\,}\Theta_{{\mathcal{V}},{\underline{e}}}|}:|{\rm
supp\,}\Theta_{{\mathcal{V}},{\underline{e}}}|\to|{\rm
supp\,}\Theta_{\underline{e}}|$.

In this Corollary, the identity for the support of the multisection in (i)
follows from $G$-equivariance of the functor $\tau$; the rest of the proof is
given at the end of §5.4.

###  2.2. Scale Calculus

The purpose of this subsection is to give a very brief summary of the key
notions of scale calculus (M-polyfold, strong bundle, and sc-Fredholm) that
were developed in [HWZ] to replace the notions of Banach manifolds, Banach
bundles, and Fredholm sections – which turned out not to be good fits for
global descriptions of moduli spaces of pseudoholomorphic curves. For a more
detailed exposition see [FFGW]; for an application-centered introduction see
[BDSVW].

An M-polyfold without boundary, as defined in [HWZ, Def.2.3.4], is analogous
to the notion of a Banach manifold: While the latter are locally homeomorphic
to open subsets of a Banach space, an M-polyfold is locally homeomorphic to
the image of a retract $r:U\to U$ of an open subset $U\subset E$ of a Banach
space $E$.151515To ensure sc-smooth partitions of unity, we assume $E$ to be a
Hilbert space as in [HWZ, Cor.5.5.17].  While $r$ is generally not classically
differentiable, it is required to be scale-smooth (or ‘sc-smooth’) with
respect to a scale structure (or ‘sc-structure’) on $E$ – see [FFGW, §2.2] for
an introduction.

Unless otherwise specified, we will allow M-polyfolds to have “tame” boundary,
arising from local models in $[0,\infty)^{k}\times E$ satisfying the tameness
condition [HWZ, Def.2.5.2]. Each such M-polyfold161616Any (M-)polyfold in this
paper is a “tame (M-)polyfold” in the language of [HWZ]. See [FFGW, §5.3] for
an explanation of the additional condition – previously called “neatness”.
carries a well defined degeneracy index $d_{X}:X\to{\mathbb{N}}_{0}$ given in
local models by counting the number of coordinates in $[0,\infty)^{k}$ that
equal to zero. With that, the boundary of an M-polyfold is defined as
$\partial X:=\\{x\in X\,|\,d_{X}(x)\geq 1\\}$, and each connected component
$Y\subset\\{x\in X\,|\,d_{X}(x)=1\\}$ inherits the structure of an M-polyfold
with degeneracy $d_{Y}=d_{X}-1$ by [HWZ, Thm.2.5.10]. We will call such
$Y\subset\partial X$ a boundary stratum, while its closure is called a face in
[HWZ]. For $l\geq 2$, the connected components of $\\{x\in
X\,|\,d_{X}(x)=l\\}$ – called corner strata – then inherit sc-smooth
structures by local descriptions as intersections of local faces as in [HWZ,
Prop.2.5.16].

As with boundary and corners, most other notions of scale calculus arise by
generalizing classical differential geometry with the two notions of sc-
calculus and retracts. For example, a sc-smooth map $f:X\to Y$ between
M-polyfolds $X,Y$ is called local sc-diffeomorphism if for every $x\in X$ and
sufficiently small open neighbourhood $O\subset X$ of $x$ (we can assume a
retract chart $O\simeq r(U)$) the image $f(O)\subset Y$ is open and there is a
sc-smooth map $f_{O}^{-1}:f(O)\to O$ that is inverse to $f$, i.e.
$f_{O}^{-1}\circ f={\rm id}_{O}$ and $f\circ f_{O}^{-1}={\rm id}_{f(O)}$.

Given the notion of ep-groupoid in Definition 2.1.1, we can also view
M-polyfolds as the realizations $|{\mathcal{X}}|$ of nonsingular ep-groupoids
${\mathcal{X}}$. On the other hand, the following remark specializes
Definition 2.1.1 to scale calculus on finite dimensional Banach spaces.

######  Remark 2.2.1.

The classical notion of manifolds (with boundary and corners) is the class of
M-polyfolds whose local models use finite dimensional Banach spaces $E$. In
that case scale calculus coincides with classical calculus and retracts are
necessarily trivial by [HWZ, Prop.2.1.2].

Thus the definition of an orbifold as the realization $|{\mathcal{X}}|$ of an
étale proper groupoid ${\mathcal{X}}$ is the special case of Definition 2.1.1
(i) in which the object space $X$ and morphism space ${\bf X}$ are finite
dimensional manifolds. Indeed, sc-smoothness on finite dimensional manifolds
is classical smoothness by [HWZ, Prop.1.2.1]), the étale property implies that
unit, inverse, and composition are sc-diffeomorphisms by Lemma 3.1.5, and the
properness condition on locally compact object spaces is equivalent to the
classical condition of $s\times t:{\bf X}\to X\times X$ being proper by Lemma
3.2.5.

Similarly, the definition of an orbibundle as the realization
$|{\mathcal{P}}|:|{\mathcal{W}}|\to|{\mathcal{X}}|$ of a smooth functor
between étale proper groupoids is the special case of Definition 2.1.1 (ii)
where $W\to X$ is a finite rank vector bundle over a manifold. Here finite
rank bundles are the special case of the strong bundle notion below for which
the fiber ambient space $F$ is finite dimensional, and hence a smooth family
of projections on $F$ has images of constant dimension.

In this special case of an orbibundle
$|{\mathcal{P}}|:|{\mathcal{W}}|\to|{\mathcal{X}}|$, the sc-Fredholm sections
in Definition 2.1.1 (iii), as specified below, are precisely the classically
smooth sections. ${\Diamond}$

Back to trivial isotropy but general M-polyfolds, a strong bundle, as defined
in [HWZ, Def.2.6.5], is a sc-smooth surjection $P:W\to X$ between M-polyfolds
with linear structures on each fiber $W_{x}=P^{-1}(x)$ for $x\in X$, and an
equivalence class of compatible strong bundle charts. Here the notion of
‘strong’ encodes a subbundle $W\supset W[1]\to X$ equipped with another
topology and sc-smooth structure such that the fiber inclusions
$W[1]_{x}\hookrightarrow W_{x}$ are dense and compact in the following sense:
Local trivializations of $W$ are of the form $\bigcup_{u\in O}{\rm
im\,}\Gamma(u)\to O$, where $O=r(U)\subset E$ is a retract as above and
$\Gamma(u):F\to F$ is a family of projections on another Banach space $F$.
Scale-smoothness of $(u,h)\mapsto\Gamma(u)h$ is required with respect to the
product sc-structure on $E\times F$, but also with respect to a shifted sc-
structure on $E\times F_{1}$, where $F_{1}\subset F$ is a compact dense
embedding of another Banach space (part of the sc-structure on $F$). In this
local trivialization, $W[1]\subset W$ is the subset $\bigcup_{u\in
O}\Gamma(u)F_{1}$, equipped with the topology induced by its inclusion in
$E\times F_{1}$.

Local trivializations are compatible if their transition maps are strong
bundle maps. Here a sc-smooth map $\Phi:W\to W^{\prime}$ between two strong
bundles (for example local models as above) is called a strong bundle
map171717This notion generalizes [HWZ, Def.2.6.1] as described on [HWZ, p.67].
if it covers a sc-smooth map on the base, is linear on fibers, and restricts
to a sc-smooth map $W[1]\to W^{\prime}[1]$.

Finally, a sc-Fredholm section $f:X\to W$ of a strong bundle over an
M-polyfold as in [HWZ, Def.3.1.16] is a sc-smooth map satisfying $P\circ
f={\rm id}_{X}$ and sc-Fredholm properties which encode elliptic regularity
and a nonlinear contraction property [HWZ, Def.3.1.6, 3.1.7, 3.1.11]. The
latter is a stronger condition than the classical notion of the linearized
operator being Fredholm, and is crucial to ensure an implicit function theorem
as shown in [FWZ] – since the differentials of a sc-smooth section with
Fredholm linearizations may not vary continuously with the basepoint.

Now an M-polyfold model for a compact moduli space
${\overline{{\mathcal{M}}}}$ (which we generalize to allow for nontrivial
isotropy in Definition 5.1.2) would consist of a sc-Fredholm section $f:X\to
W$ and a homeomorphism $\psi:f^{-1}(0)\to{\overline{{\mathcal{M}}}}$. The
regularization of such (relatively simple) moduli spaces is achieved in [HWZ,
§5.3] by showing that different transverse perturbations of $f$ have cobordant
solution spaces $(f-s)^{-1}(0)\sim(f-s^{\prime})^{-1}(0)$ when perturbations
are restricted to compactness-controlled (see §3.5) sc+-sections
$s,s^{\prime}:X\to W$. This serves as introduction to the last crucial scale
calculus notion: An sc+-section of a strong bundle $W\to X$ is a section that
is sc-smooth $X\to W[1]$ relative to the shifted sc-structure on the dense
subbundle $W[1]\subset W$. Since the fibers of $W[1]$ embed compactly into
$W$, these sections can be viewed as compact perturbations of the sc-Fredholm
section $f$. In particular, adding them does not affect the Fredholm index by
[HWZ, Thm.3.1.18].

###  2.3. Orientation Conventions

The goal of this subsection (which can be omitted on first reading) is to
spell out the notion of an orientation in [HWZ, Def.12.5.4] for a sc-Fredholm
section functor $f:{\mathcal{X}}\to{\mathcal{W}}$ over an ep-groupoid in
sufficient detail to make sense of the constructions in this paper. This will
mainly be a matter of constructing the orientation bundle over the M-polyfold
of objects $X$, which will only be possible over a dense subset of so-called
“smooth points”:

M-polyfolds inherit from their sc-smooth structure a filtration
$X=X_{0}\supset X_{1}\supset\ldots\supset
X_{\infty}=\bigcap_{i\in{\mathbb{N}}_{0}}X_{i}$ of subsets with their own
(metrizable, paracompact) topology [HWZ, Thm.2.3.10]. Viewed as subsets, each
of them are dense in $X$ [HWZ, p.289].181818This density holds by definition
[HWZ, Def.1.1.1] for $X={\mathbb{E}}=(E_{m})_{m\in{\mathbb{N}}_{0}}$ a sc-
Banach space. In a local chart $X\supset O\simeq r(U)$ it holds since every
$x\in O$ is an image $x=r(y)$ of a map $r:U\to U$ on an open subset $U\subset
E_{0}$ of a sc-Banach space, where $r$ is sc-continuous in the sense that
$r:U\cap E_{m}\to U\cap E_{m}$ is continuous for all $m\in{\mathbb{N}}_{0}$.
Now an approximating sequence $E_{m}\ni y_{i}\to y$ lies in $U$ for
sufficiently large $i$ and yields an approximation $E_{m}\cap r(U)\ni
r(y_{i})\to r(y)=x$. The same is true for any polyfold – the realization
$|{\mathcal{X}}|$ of an ep-groupoid – since the equivalence relation by sc-
diffeomorphisms preserves the filtration on the object set, resulting in a
filtration
$|{\mathcal{X}}|=|X_{0}|\supset|X_{1}|\supset\ldots\supset|X_{\infty}|=\bigcap_{i\in{\mathbb{N}}_{0}}|X_{i}|$
of dense subsets.

Now the orientation bundle of a sc-Fredholm section $f:X\to W$ on object level
– where $W$ is a strong bundle over an M-polyfold $X$ – is constructed in
[HWZ, §6, §12.5] as a local homeomorphism ${\mathcal{O}}_{f}\to X_{\infty}$
between Hausdorff spaces, where the fiber of ${\mathcal{O}}_{f}$ over $x\in
X_{\infty}$ consists of the two possible orientations of the line bundle ${\rm
DET}(f,x)\to{\rm LIN}(f,x)$ over a convex set of admissible linearizations of
$f$ at $x$ [HWZ, Def.6.2.1].

Here the fiber of ${\rm DET}(f,x)$ over each admissible linearization $S:{\rm
T}_{x}X\to W_{x}$ is the classical determinant line $\det(S)=\Lambda^{\rm
max}\,\ker S\otimes\bigl{(}\Lambda^{\rm max}\,\bigl{(}\text{\raise
3.44444pt\hbox{$\scriptstyle W_{x}\\!$}/\lower
3.44444pt\hbox{$\\!\scriptstyle{\rm im\,}S$}}\bigr{)}\bigr{)}^{*}$. When
$f(x)=0$ then the canonical linearization $S={\rm D}f(x):{\rm T}_{x}X\to
W_{x}$ is admissible, and all other admissible linearizations differ by a
compact operator (see the discussion of sc+-operators after [HWZ, Def.1.1.5]).
When $f(x)\neq 0$ then all admissible linearizations can be expressed as
canonical linearizations $S={\rm D}(f-s)(x):{\rm T}_{x}X\to W_{x}$ for some
nonlinear sc+-section $s:X\to W$ with $s(x)=f(x)$ (see [FFGW, Def.6.1.8] for
the notion of sc+-section).

Next, each ${\rm DET}(f,x)$ is given the structure of a line bundle by viewing
it as the restriction of the classical determinant line bundle to a subset of
the Fredholm operators ${\rm T}_{x}X\to W_{x}$. From the abundance of more or
less natural conventions for topologizing this determinant bundle (as
discussed in [Z]) we use the conventions in [HWZ, Prop.6.5.8]: Given a
Fredholm operator $T:E\to F$ between Banach spaces $E,F$, we can topologize
the determinant line bundle $\bigcup_{\|S-T\|<{\varepsilon}}\det(S)$ in a
neighbourhood of $T$ as follows: The choice of any good left-projection
$P:F\to F$ – satisfying in particular ${\rm im\,}PT={\rm im\,}P$, see [HWZ,
Def.6.4.3] – induces an exact sequence in [HWZ, (6.9)],

$E_{(T,P)}\;:\quad 0\to\ker(T)\to\ker(PT)\to\text{\raise
3.44444pt\hbox{$\scriptstyle F\\!$}/\lower 3.44444pt\hbox{$\\!\scriptstyle{\rm
im\,}PT$}}\to\text{\raise 3.44444pt\hbox{$\scriptstyle F\\!$}/\lower
3.44444pt\hbox{$\\!\scriptstyle{\rm im\,}T$}}\to 0.$

Then the isomorphism $\Phi_{E}$ of [HWZ, Lemma 6.3.3] applied to $E=E_{(T,P)}$
yields an isomorphism ${\gamma}_{T}^{P}:\det(T)\to\det(PT)$ as in [HWZ,
(6.12)]. This extends to isomorphisms ${\gamma}_{S}^{P}:\det(S)\to\det(PS)$
for $\|S-T\|\leq{\varepsilon}$ guaranteed by [HWZ, Prop.6.5.1 (1)], and these
are used to pull back the local determinant bundle structure of
$\bigcup_{\|S-T\|<{\varepsilon}}\\{S\\}\times\det(PS)$ constructed in [HWZ,
Prop.6.5.4]. Independence of choices and continuity of transition maps is
proven in [HWZ, §6.4-5].191919 Here we must take into account the fact that
the sign in [HWZ, Prop.6.4.15] needs to be corrected to
$\det(c_{1},...,c_{m},-r_{1},...,-r_{l})$. This correction makes this
proposition coherent with the rest of [HWZ, §6.4] as well as [MW2, (8.1.9)].

This defines the orientation bundle ${\mathcal{O}}_{f}$ as a set with a 2-to-1
map to $X_{\infty}$. Topologizing it requires a notion of continuous extension
of orientations when the basepoint $x\in X_{\infty}$ varies, which is given in
[HWZ, Thm.6.6.11].

Now an orientation of a sc-Fredholm section functor
$f:{\mathcal{X}}\to{\mathcal{W}}$ over an ep-groupoid ${\mathcal{X}}$ is a
continuous section
${\mathfrak{o}}:X_{\infty}\to{\mathcal{O}}_{f},x\mapsto{\mathfrak{o}}_{x}$ of
the orientation bundle over the smooth objects $X_{\infty}$, that is
compatible with the morphisms of ${\mathcal{X}}$ in the sense that
$\phi_{*}{\mathfrak{o}}_{s(\phi)}={\mathfrak{o}}_{t(\phi)}$ for all
$\phi\in{\bf X}$. Here $\phi_{*}$ is the second of the following conventions
worth spelling out:

General isomorphism convention: Any commutative diagram

of Fredholm operators $S:E\to F,S^{\prime}:E^{\prime}\to F^{\prime}$ and
isomorphisms $I:E\to E^{\prime}$, $J:F\to F^{\prime}$ induces a commutative
diagram for each choice of good left-projection $P:F\to F$ for $S$

Here $P^{\prime}:=JPJ^{-1}$ is the corresponding good left-projection for
$S^{\prime}$, which combines the exact sequences $E_{(S,P)}$ and
$E_{(S^{\prime},P^{\prime})}$ of type [HWZ, (6.9)] into a diagram of type
[HWZ, (6.5)]. Now [HWZ, Lemma 6.3.4] induces a natural commutative diagram
intertwining the isomorphisms $\gamma^{P}_{S}:\det(S)\to\det(PS)$ and
$\gamma^{P^{\prime}}_{S^{\prime}}:\det(S^{\prime})\to\det(P^{\prime}S^{\prime})$
used in the construction of the determinant bundles near $S$ and $S^{\prime}$.
We denote the natural isomorphism $\det(S)\to\det(S^{\prime})$ in this diagram
by

$\displaystyle(I,J)_{*}:\;\det(S)$ $\displaystyle\to\det(S^{\prime}),\;\;$
$\displaystyle(e_{1}\wedge\ldots e_{k})\otimes(f_{1}\wedge\ldots f_{c})^{*}$
$\displaystyle\mapsto(I(e_{1})\wedge\ldots
I(e_{k}))\otimes(J(f_{1})\wedge\ldots J(f_{c}))^{*}.$

Note that this map is well defined since we assumed $JS=S^{\prime}I$, and
hence $I$ resp. $J$ restrict to isomorphisms between kernels resp. cokernels
of the two Fredholm operators.

Morphism convention: Since source and target maps ${\bf X}\to X$ are local sc-
diffeomorphisms, every morphism $\phi\in{\bf X}$ extends to a local sc-
diffeomorphism $\phi:U_{x}\to U_{y}$ between neighbourhoods
$U_{x},U_{y}\subset X$ of $x=s(\phi)$ and $y=t(\phi)$ as in [HWZ, §7.1]. Its
differential ${\rm d}\phi(x):{\rm T}_{x}X\to{\rm T}_{y}X$ is a sc-isomorphism
for smooth points $x\in X_{\infty}$. At smooth points, $\phi$ also lifts to a
strong bundle isomorphism $\mu(\phi,\cdot):W|_{U_{x}}\to W|_{U_{y}}$, and
functoriality of $f$ implies $f\circ\phi=\mu(\phi,f)$ on $U_{x}$. Now we can
combine these maps to obtain a bijection between admissible linearizations

$\phi_{*}:{\rm LIN}(f,x)\to{\rm LIN}(f,y),S\mapsto\mu(\phi,\cdot)\circ
S\circ({\rm d}\phi(x))^{-1}.$

Since these linearizations satisfy the commuting diagram relation
$\mu(\phi,\cdot)\circ S=\phi_{*}S\circ{\rm d}\phi(x)$ with isomorphisms
$I={\rm d}\phi(x)$ between their domains and $J=\mu(\phi,\cdot)$ between their
target spaces, we can use the general isomorphism convention to lift
$\phi_{*}$ to a bundle isomorphism $\phi_{*}:{\rm DET}(f,x)\to{\rm DET}(f,y)$
given by $(I,J)_{*}:\det(S)\to\det(\phi_{*}S)$ on each fiber. This in turn
induces a bijection of orientations
$\phi_{*}:({\mathcal{O}}_{f})_{x}\to({\mathcal{O}}_{f})_{y}$ given by
restricting the isomorphisms $I={\rm d}\phi(x)$ and $J:=\mu(\phi,\cdot)$ to
the kernels and cokernels of the linearizations.

Equivalence convention: If $\psi:{\mathcal{Y}}\to{\mathcal{X}}|_{V}$ is an
equivalence of ep-groupoids [HWZ, Def.10.1.1] and
$f:{\mathcal{X}}\to{\mathcal{W}}$ is sc-Fredholm, then any orientation of $f$
has a natural pullback to an orientation of the pullback section
$\widetilde{f}:=\psi^{*}f:{\mathcal{Y}}\to\psi^{*}{\mathcal{W}}$ as follows:

The section $\widetilde{f}$ satisfies $\Psi\circ\widetilde{f}=f\circ\psi$ by
Lemma 3.4.4 and is sc-Fredholm by Lemma 3.5.3. Here the pullback bundle
functor $\Psi:\psi^{*}{\mathcal{W}}\to{\mathcal{W}}$ is a strong bundle
equivalence, in particular restricts to isomorphisms between the fibers
$\Psi_{y}:(\psi^{*}{\mathcal{W}})_{y}\to{\mathcal{W}}_{\psi(y)}$, whereas
$\psi$ is a local sc-diffeomorphism, hence induces isomorphisms ${\rm
d}\psi(y):{\rm T}_{y}Y\to{\rm T}_{\psi(y)}X$ between the domains of the
linearizations of $\widetilde{f}$ and $f$. For each $y\in Y$ these induce a
bijection between the sets of admissible linearizations

$\psi^{*}:{\rm LIN}(f,\psi(y))\to{\rm
LIN}(\widetilde{f},y),S\mapsto\Psi_{y}^{-1}\circ S\circ{\rm d}\psi(y).$

Indeed, given a sc+-section $s$ with $s(\psi(y))=f(\psi(y))$, its pullback
$\widetilde{s}:=\psi^{*}s$ is a sc+-section with
$\widetilde{s}(y)=\widetilde{f}(y)$, and the corresponding linearizations
satisfy

$\Psi_{y}\circ{\rm D}_{y}(\widetilde{f}-\widetilde{s})={\rm
D}_{\psi(y)}(f-s)\circ{\rm d}\psi(y).$

Since this pullback of linearizations satisfies the commuting diagram relation
$\Psi_{y}\circ\psi^{*}S=S\circ{\rm d}\psi(y)$ with isomorphisms $I={\rm
d}\psi(y)$ between their domains and $J=\Psi_{y}$ between their target spaces,
the general isomorphism convention lifts $\psi^{*}$ to a bundle isomorphism
$\psi^{*}:{\rm DET}(f,\psi(y))\to{\rm DET}(\widetilde{f},y)$ given by
${(I,J)_{*}}^{-1}:\det(S)\to\det(\psi^{*}S)$ on each fiber. This then induces
a bijection of orientations
$\psi^{*}:({\mathcal{O}}_{f})_{\psi(y)}\to({\mathcal{O}}_{\widetilde{f}})_{y}$
given by restricting the isomorphisms $I={\rm d}\psi(y)$ and $J=\Psi_{y}$ to
the kernels and cokernels of the linearizations.

Now any orientation ${\mathfrak{o}}:X_{\infty}\to{\mathcal{O}}_{f}$ of $f$
induces a section

$\psi^{*}{\mathfrak{o}}:=\psi^{*}\circ{\mathfrak{o}}\circ\psi:Y_{\infty}\to{\mathcal{O}}_{\widetilde{f}},$

which is an orientation of $\widetilde{f}$ by [HWZ, Thm.12.5.7 (2)] – called
the pullback orientation.

Solution set convention: Given an oriented sc-Fredholm section functor
$f:{\mathcal{X}}\to{\mathcal{W}}$ and a suitably transverse – possibly multi-
valued – perturbation of $f$, the perturbed solution set inherits a natural
orientation. In the ep-groupoid context this is formulated precisely and
proven in [HWZ, §15.4], with local orientations constructed from the following
single-valued, trivial isotropy case in [HWZ, Thm.6.6.18]202020Here we
generalize the conventions of [HWZ, §6.6] as indicated in the beginning of
[HWZ, §15.4].:

Suppose $f:X\to W$ is an oriented sc-Fredholm section over an M-polyfold,
$s:X\to W$ is a sc+-section, and $x\in(f-s)^{-1}(0)\subset X_{\infty}$ is a
perturbed solution.212121The inclusion of perturbed solution sets in
$X_{\infty}$ is guaranteed by the regularizing property of sc-Fredholm
sections, which is preserved by sc+-perturbations. If $f-s$ is transverse at
$x$, that is if the linearization ${\rm D}_{x}(f-s):{\rm T}_{x}X\to W_{x}$ is
surjective, then $\ker{\rm D}_{x}(f-s)$ inherits a natural orientation since
$S:={\rm D}_{x}(f-s)\in{\rm LIN}(f,x)$ is an admissible linearization and so
the given orientation of

$\det(S)=\Lambda^{\rm max}\,\ker S\otimes(\Lambda^{\rm max}\,\text{\raise
3.44444pt\hbox{$\scriptstyle W_{x}\\!$}/\lower
3.44444pt\hbox{$\\!\scriptstyle{\rm im\,}S$}})^{*}\cong\Lambda^{\rm
max}\,\ker{\rm D}_{x}(f-s)$

transfers to the kernel by canonical identification. If $(f-s)^{-1}(0)$ also
inherits a manifold structure (which requires transversality at all perturbed
solutions and good or general position for solutions at the boundary), then
the induced pointwise orientations of its tangent spaces ${\rm
T}_{x}(f-s)^{-1}(0)=\ker{\rm D}_{x}(f-s)$ form a continuous orientation of the
tangent bundle since the orientation propagation that topologizes
${\mathcal{O}}_{f}\to X_{\infty}$ in [HWZ, Thm.6.6.11] agrees with classical
parallel transport on the tangent bundle over $(f-s)^{-1}(0)$.

More precisely, let $\phi:[0,1]\to(f-s)^{-1}(0)$ be a smooth path along which
$f-s$ is transverse (for boundary considerations see [HWZ]). Then the polyfold
construction of orientation propagation
$\phi_{*}:({\mathcal{O}}_{f})_{\phi(0)}\to({\mathcal{O}}_{f})_{\phi(1)}$ in
[HWZ, Lemma 6.6.4-6] simplifies:

  * -

We can choose the sc+-section $[0,1]\times X\to W,(t,x)\mapsto s(x)$ since
$s(\phi(t))=f(\phi(t))$.

  * -

The section $F:[0,1]\times X\to W,(t,x)\mapsto f(x)-s(x)$ is transverse along
the graph of $\phi$.

  * -

On the line bundle $\Lambda^{\rm
max}\,L=\bigcup_{t\in[0,1]}\\{t\\}\times\Lambda^{\rm max}\,L_{t}$ with
$L_{t}=\ker{\rm D}_{\phi(t)}(f-s)$ the parallel transport is the classical
orientation propagation along the path $\phi:[0,1]\to(f-s)^{-1}(0)$.

  * -

The identification of $\det(\ker{\rm D}_{\phi(t)}(f-s))\to\Lambda^{\rm
max}\,L_{t}\otimes(\Lambda^{\rm max}\,{\mathbb{R}}^{k})^{*}=\Lambda^{\rm
max}\,L_{t}$ in [HWZ, (6.28)] is the identity map since $k=0$.

So the polyfold orientation propagation $\phi_{*}$ in this setting agrees with
classical orientation propagation on $(f-s)^{-1}(0)$, as claimed. Thus an
orientation of $f$ induces an orientation of any appropriately transverse
perturbed solution set $(f-s)^{-1}(0)$.

######  Remark 2.3.1.

The orientation conventions for equivalences and solution sets interact as
follows: Consider a sc-Fredholm section functor
$f:{\mathcal{X}}\to{\mathcal{W}}$ with orientation
${\mathfrak{o}}:X_{\infty}\to{\mathcal{O}}_{f}$. Then any local perturbed
solution set $(f-s)^{-1}(0)\cap U$ inherits the structure of an oriented
manifold when $s:X\to W$ is a (not necessarily functorial) sc+-section such
that $(f-s)|_{U}$ is transverse over an open subset $U\subset X$.

Now consider an equivalence of ep-groupoids
$\psi:{\mathcal{Y}}\to{\mathcal{X}}|_{V}$, and the pullback section
$\widetilde{f}:=\psi^{*}f:{\mathcal{Y}}\to\psi^{*}{\mathcal{W}}$ with pullback
orientation
$\psi^{*}{\mathfrak{o}}=\psi^{*}\circ{\mathfrak{o}}\circ\psi:Y_{\infty}\to{\mathcal{O}}_{\widetilde{f}}$.
Suppose that the above perturbation is given on a set
$U=\psi(\widetilde{U})\subset V$ that is the image of a sufficiently small
open subset $\widetilde{U}\subset Y$ on which $\psi$ restricts to a sc-
diffeomorphism. Then
$\widetilde{s}:=\psi^{*}s:{\mathcal{Y}}\to\psi^{*}{\mathcal{W}}$ is a
sc+-section with $(\widetilde{f}-\widetilde{s})|_{\widetilde{U}}$ transverse,
and $\psi$ restricts to an diffeomorphism between the manifolds
$(\widetilde{f}-\widetilde{s})^{-1}(0)\cap\widetilde{U}\to(f-s)^{-1}(0)\cap U$
that preserves orientations.

Indeed, the pullback construction guarantees that $\psi$ maps solutions

$y\in(\widetilde{f}-\widetilde{s})^{-1}(0)\cap\widetilde{U}\subset Y_{\infty}$

to solutions

$x:=\psi(y)\in(f-s)^{-1}(0)\cap U\subset X_{\infty},$

sc-diffeomorphisms restrict to diffeomorphisms in finite dimensions, and its
linearization ${\rm d}\psi(y):{\rm
T}_{y}(\widetilde{f}-\widetilde{s})^{-1}(0)\to{\rm T}_{x}(f-s)^{-1}(0)$ is
compatible with the orientations induced by $\psi^{*}{\mathfrak{o}}$ and
${\mathfrak{o}}$ as follows:

In the absence of cokernels, ${\rm T}_{x}(f-s)^{-1}(0)=\ker S$ is oriented by
the natural identification $\Lambda^{\rm max}\,\ker S=\det(S)$ with the
determinant of the linearization $S:={\rm D}_{x}(f-s)\in{\rm
LIN}(f,{\psi(y)})$, which is oriented by ${\mathfrak{o}}(x)$. Similarly, ${\rm
T}_{y}(\widetilde{f}-\widetilde{s})^{-1}(0)=\ker\psi^{*}S$ is oriented by the
natural identification $\Lambda^{\rm max}\,\ker\psi^{*}S=\det(\psi^{*}S)$ with
the determinant of the linearization $\psi^{*}S=\Psi_{y}^{-1}\circ S\circ{\rm
d}\psi(y)\in{\rm LIN}(\widetilde{f},y)$, which is oriented by
$(\psi^{*}{\mathfrak{o}})(y)=\psi^{*}({\mathfrak{o}}(\psi(y)))$. Here, due to
the trivial cokernel,
$\psi^{*}:({\mathcal{O}}_{f})_{\psi(y)}\to({\mathcal{O}}_{\widetilde{f}})_{y}$
is given by restricting the isomorphism ${\rm d}\psi(y)$ to the kernels. So
the polyfold-theoretic identification $\psi^{*}$ between the orientation
${\mathfrak{o}}(x)$ of ${\rm T}_{x}(f-s)^{-1}(0)$ and the orientation
$(\psi^{*}{\mathfrak{o}})(y)=\psi^{*}({\mathfrak{o}}(x))$ of ${\rm
T}_{y}(\widetilde{f}-\widetilde{s})^{-1}(0)$ coincides with the differential-
geometric identification induced by ${\rm d}\psi(y)$. ${\Diamond}$

Product convention: Given $f:{\mathcal{X}}\to{\mathcal{W}}$ sc-Fredholm with
orientation ${\mathfrak{o}}:X_{\infty}\to{\mathcal{O}}_{f}$, its constant
extension
$\widehat{f}:[0,1]\times{\mathcal{X}}\to[0,1]\times{\mathcal{W}},(t,x)\mapsto(t,f(x))$
inherits a natural orientation $\widehat{\mathfrak{o}}:[0,1]\times
X_{\infty}\to{\mathcal{O}}_{\widehat{f}}$ specified in [HWZ, Rmk.6.6.19] as
follows:

The map on linearizations ${\rm LIN}(f,x)\to{\rm
LIN}(\widehat{f},(t,x)),S\mapsto S\Pi$ given by composition with the
projection $\Pi:{\rm T}_{(t,x)}[0,1]\times X={\mathbb{R}}\times{\rm
T}_{x}X\to{\rm T}_{x}X$ lifts to a bundle map ${\rm DET}(f,x)\to{\rm
DET}(\widehat{f},(t,x))$ given on each fiber by

$\det(S)\to\det(S\Pi),(a_{1}\wedge\ldots\wedge
a_{k})\otimes(b_{1}\wedge\ldots\wedge b_{c})^{*}\mapsto(1\wedge
a_{1}\wedge\ldots\wedge a_{k})\otimes(b_{1}\wedge\ldots\wedge b_{c})^{*},$

where $1\in{\mathbb{R}}={\rm T}_{t}[0,1]$ represents the standard orientation
of the interval $[0,1]$.

Boundary convention: Given an oriented sc-Fredholm section functor
$f:{\mathcal{X}}\to{\mathcal{W}}$ and a boundary stratum $Y\subset\partial X$,
the restriction $f|_{Y}$ inherits a natural orientation as follows:

Each $y\in Y_{\infty}$, considered as boundary point of $X$ has a local model
$(0,0)\in r(U)$, where $r:U\to U$ is a sc-smooth retraction on an open subset
$U\subset[0,\infty)\times E$. Here tameness guarantees
$U_{\partial}:=U\cap\\{0\\}\times E=\\{u\in U\,|\,d_{U}(u)=1\\}$ to be
$r$-invariant, inducing a local model $(0,0)\in r(U_{\partial})$ for $y\in Y$
with ${\rm T}_{y}Y\simeq{\rm D}_{(0,0)}r(\\{0\\}\times E)\subset{\rm im\,}{\rm
D}_{(0,0)}r\simeq{\rm T}_{y}X$. To find an outward vector $\nu\in{\rm T}_{y}X$
that splits ${\rm T}_{y}X={\mathbb{R}}\nu\oplus{\rm T}_{y}Y$ we need the
second tameness condition: This guarantees a subspace $A\subset\\{0\\}\times
E$ such that ${\mathbb{R}}\times E={\rm im\,}{\rm D}_{(0,0)}r\oplus A$, and
hence we can define $0\neq\nu\in{\rm T}_{y}X\simeq{\rm im\,}{\rm D}_{(0,0)}r$
to be the unique vector in the decomposition $(-1,0)=\nu+a$ with $a\in A$.
Here a different choice of complement $A^{\prime}\neq A$ induces another
vector $\nu^{\prime}\in{\rm im\,}{\rm D}_{(0,0)}r$ with
$\nu^{\prime}-\nu\in{\rm im\,}{\rm D}_{(0,0)}r\cap\\{0\\}\times E\simeq{\rm
T}_{y}Y$.

To restrict the orientation of $f$ to an orientation of $f|_{Y}$ at the point
$y\in Y$, choose a linearization $S={\rm D}_{y}(f-s)\in{\rm LIN}(f,y)$ with
${\rm D}_{y}(f-s)(-\nu)=0$ using Lemma 2.3.3 below and the fact that the
direct sum ${\mathbb{R}}\times E={\rm im\,}{\rm D}_{(0,0)}r\oplus A$ is in
fact a sc-splitting, so that $(-1,0)\in{\mathbb{R}}\times E_{\infty}$ yields
$-\nu=(1,a)\in{\mathbb{R}}\times E_{\infty}$. This linearization induces a
restricted linearization $S_{\partial}:={\rm D}_{y}(f-s)|_{{\rm
T}_{y}Y}\in{\rm LIN}(f|_{Y},y)$ with ${\mathbb{R}}\nu\oplus\ker
S_{\partial}=\ker S$ and ${\rm im\,}S_{\partial}={\rm im\,}S$, so that we can
define an isomorphism $\det S_{\partial}\to\det S$ by mapping
$(a_{1}\wedge\ldots\wedge a_{k})\otimes(b_{1}\wedge\ldots\wedge b_{c})^{*}$ to
$(\nu\wedge a_{1}\wedge\ldots\wedge a_{k})\otimes(b_{1}\wedge\ldots\wedge
b_{c})^{*}$.

A different choice of outward vector $\nu^{\prime}\in\nu+{\rm T}_{y}Y$ gives
rise to a continuous family $\nu_{\tau}:=\nu+\tau(\nu^{\prime}-\nu)\in({\rm
T}_{y}X)_{\infty}$ and the construction of Lemma 2.3.3 yields a continuous
family of sc+-sections $s_{\tau}$ with $s_{\tau}|_{Y}$ independent of
$\tau\in[0,1]$ and ${\rm D}_{y}(f-s_{\tau})(\nu_{\tau})=0$. So we obtain a
continuous family of isomorphism $\det S_{\partial}\to\det S_{\tau}$ mapping

$(a_{1}\wedge\ldots\wedge a_{k})\otimes(b_{1}\wedge\ldots\wedge b_{c})^{*}$

to

$(\nu_{\tau}\wedge a_{1}\wedge\ldots\wedge
a_{k})\otimes(b_{1}\wedge\ldots\wedge b_{c})^{*},$

and can conclude that the induced (inverse) map from orientations of $f$ to
orientations of $f|_{Y}$ at $y$ is independent of choices – except for the
crucial ordering and outward convention that we fixed here. This convention is
moreover compatible with the boundary orientation convention in [HWZ,
Def.9.3.8] for solution sets (resulting from multi-valued perturbations).

An analogous extension construction for the stabilizations in [HWZ, Lemma
6.6.4-6] shows that this pointwise convention is compatible with orientation
propagation, thus induces a well defined orientation of $f|_{Y}$.

######  Remark 2.3.2.

The orientation conventions for products and boundary restrictions interact as
follows: Given a sc-Fredholm section functor $f:{\mathcal{X}}\to{\mathcal{W}}$
with orientation ${\mathfrak{o}}:X_{\infty}\to{\mathcal{O}}_{f}$, the
restrictions of its constant extension
$\widehat{f}:[0,1]\times{\mathcal{X}}\to{\mathcal{W}}$ to the boundary strata
${\mathcal{X}}\to\\{i\\}\cong{\mathcal{X}}$ for $i=0,1$ are
$\widehat{f}|_{\\{0\\}\times{\mathcal{X}}}\cong f$ with orientation
$-{\mathfrak{o}}$ and $\widehat{f}|_{\\{1\\}\times{\mathcal{X}}}\cong f$ with
orientation ${\mathfrak{o}}$. This identification results from the
isomorphisms $\det(S)\to\det(S\Pi=\widehat{S})$ and
$\det(\widehat{S})\to\det(\widehat{S}_{\partial}=S)$ mapping
$(a_{1}\wedge\ldots\wedge a_{k})\otimes(b_{1}\wedge\ldots\wedge
b_{c})^{*}\mapsto(1\wedge a_{1}\wedge\ldots\wedge
a_{k})\otimes(b_{1}\wedge\ldots\wedge b_{c})^{*}$ and $(\nu\wedge
a_{1}\wedge\ldots\wedge a_{k})\otimes(b_{1}\wedge\ldots\wedge
b_{c})^{*}\mapsto(a_{1}\wedge\ldots\wedge
a_{k})\otimes(b_{1}\wedge\ldots\wedge b_{c})^{*}$ with outward vector $\nu=-1$
for $i=0\in\partial[0,1]$ and outward vector $\nu=1$ for
$i=1\in\partial[0,1]$. ${\Diamond}$

To state and prove existence of the sc+-sections used in the boundary
convention we assume familiarity with [HWZ, ch.1-2].

######  Lemma 2.3.3.

Consider a strong bundle retraction

$R:U\times F\to U\times F,(u,f)\mapsto(r(u),{\Gamma}(u)f)$

covering a sc-smooth retraction $r:U\to U$ on an open neighbourhood
$U\subset[0,1)\times E$ of $(0,0)=r(0,0)$, and a sc-smooth section of this
local bundle given by $f:r(U)\to F$ satisfying $f(u)\in{\Gamma}(u)F$. Given
any $\nu=(1,a)\in{\rm im\,}{\rm D}_{(0,0)}r\cap{\mathbb{R}}\times E_{\infty}$
there exists a sc+-section $s$ of this local bundle such that $s(0,0)=f(0,0)$
and ${\rm D}_{(0,0)}(f-s)(\nu)=0$.

###### Proof.

Note that we have $\nu=\dot{\gamma}(0)$ for the sc-smooth path
$\gamma:[0,{\varepsilon})\to r(U),t\mapsto r(t\nu)$ as in [HWZ, Lemma 2.4.9]
for some ${\varepsilon}>0$. Since $[0,{\varepsilon})$ consists of smooth
points and sc-continuous maps preserve sc-levels, this path takes values in
the smooth points, $\gamma(t)=r(t\nu)\in[0,1)\times E_{\infty}$. Moreover, sc-
smoothness with this finite dimensional domain is equivalent to smoothness as
map $[0,{\varepsilon})\to[0,1)\times E_{\ell}$ for any
$\ell\in{\mathbb{N}}_{0}$. So composing this path with the sc-smooth map $f$
yields a path $[0,{\varepsilon})\to F_{\infty},t\mapsto f(\gamma(t))$ that is
sc-smooth w.r.t. any shift of the sc-structure on $F$.

We now construct a map $s:r(U)\to F$ by $s(t,e):={\Gamma}(t,e)f(\gamma(t))$
for $(t,e)\in r(U)$. This is a section since $s(u)\in{\Gamma}(u)F$. It is sc-
smooth and sc+ since its extension $\widetilde{s}:U\to F,u\mapsto s(r(u))$ can
be viewed for each $\ell\in{\mathbb{N}}_{0}$ as the composition of sc-smooth
maps $r:U\to U$, the linear map $U\to
U\times{\mathbb{R}},(t,e)\mapsto((t,e),t)$, the map $U\times{\mathbb{R}}\to
U\times F_{l},(u,t)\mapsto(u,f(\gamma(t)))$, and $U\times F_{l}\to
F_{l},(u,f)\mapsto\Gamma(u)f$. It satisfies the base point condition

$s(0,0)={\Gamma}(0,0)f(\gamma(0))={\Gamma}(0,0)f(0,0)=f(0,0),$

and we can compute with $g(t):={{\rm pr}}_{\mathbb{R}}(\gamma(t))$

$\displaystyle{\rm D}_{(0,0)}(f-s)(\nu)$ $\displaystyle=\tfrac{{\rm d}}{{\rm
d}t}\big{|}_{t=0}(f-s)(\gamma(t))$ $\displaystyle=\lim_{t\to
0}\tfrac{f(\gamma(t))-s(\gamma(t))}{t}$ $\displaystyle=\lim_{t\to
0}{\Gamma}(\gamma(t))\tfrac{f(\gamma(t))-f(\gamma(g(t)))}{t}=0.$

To prove the last step, we first claim that
$\gamma(t)=r(t\nu)=t\nu+O(t^{2})\in{\mathbb{R}}\times E_{0}$ and hence
$g(t)=t+O(t^{2})$. To check this note that $r:U\cap{\mathbb{R}}\times
E_{1}\to{\mathbb{R}}\times E_{0}$ is continuously differentiable so that
$r(t\nu)=r(0,0)+t{\rm D}_{(0,0)}r(\nu)+O(t^{2})$. Here $r(0,0)=(0,0)$ and
${\rm D}_{(0,0)}r(\nu)=\nu$ follow from $r\in r(U)$ resp. $\nu\in{\rm
im\,}{\rm D}_{(0,0)}r$ and the retraction property $r\circ r=r$. Next, sc-
smoothness of $f\circ{\gamma}$ includes continuous differentiability as map
$[0,\frac{1}{2}{\varepsilon}]\to F_{0}$ with uniformly bounded derivative.
Thus we can estimate for $t\to 0$

$t^{-1}\|f(\gamma(t))-f(\gamma(g(t)))\|_{F_{0}}\;\leq\;t^{-1}C|t-g(t)|\;=\;Ct^{-1}O(t^{2})=O(t)\;\to\;0.$

Finally, continuity of $\Gamma:U\times F_{0}\to F_{0}$ confirms the last step
in ${\rm D}_{(0,0)}(f-s)(\nu)=0$. ∎

##  3\. Étale categories and groupoid completions

This section develops the general theory of étale categories in order to have
a suitable language in which to discuss the structure and properties of the
groupoids ${\mathcal{X}}_{\mathcal{V}}$ and
${\mathcal{X}}_{\mathcal{V}}^{{\smallsetminus}G}$ that appear in Theorem
2.1.2. After a general introduction to étale categories in §3.1, §3.2 defines
the notion of preuniformizer, generalizing the local uniformizers considered
in [HWZ]. Lemma 3.2.5 gives conditions that imply the properness of a groupoid
${\mathcal{X}}$ and hence, by Corollary 3.2.6, the Hausdorffness of
$|{\mathcal{X}}|$. This subsection also develops the notion of the groupoid
completion of an étale category, which is used to construct
${\mathcal{X}}_{\mathcal{V}}^{{\smallsetminus}G}$. Then §3.3 discusses the
properties of group actions on étale categories such as the action of $G$ on
${\mathcal{X}}_{\mathcal{V}}^{{\smallsetminus}G}$ in Theorem 2.1.2. The next
subsection explains the basic properties of bundles and their sections in more
detail, with special mention of the properties of strong bundles in the
polyfold setting, that is in étale class (d). Globally structured
multisections are defined in Definition 3.4.5, and Proposition 3.4.11 shows
that such sections are structurable in the sense of [HWZ, Def.13.3.6].
Finally, §3.5 concentrates on the polyfold case, summarizing the notions
required for ensuring the compactness of perturbed zero sets.

###  3.1. Étale categories

Throughout this paper we will work with topological categories. Thus we
consider small categories ${\mathcal{B}}$ whose sets $B:={\rm
Obj}_{\mathcal{B}}$ and ${\bf B}:={\rm Mor}_{{\mathcal{B}}}$ of objects and
morphisms are equipped with topologies such that the source and target maps
$s,t:{\bf B}\to B$ as well as unit $u:B\to{\bf B}$ and composition

$c:\\{(m,m^{\prime})\in{\bf B}\times{\bf B}\,|\,t(m)=s(m^{\prime})\\}\to{\bf
B},\quad(m,m^{\prime})\mapsto m\circ m^{\prime},$

are continuous. Further, any functor between topological categories is assumed
continuous on object and morphism spaces. When working with a groupoid
${\mathcal{X}}=(X,{\bf X})$ we also assume that the inverse operation $i:{\bf
X}\to{\bf X}$ is continuous, i.e. ${\mathcal{X}}$ is a topological groupoid.
For notational clarity we reserve the letter ${\mathcal{X}}$ to denote étale
groupoids throughout this paper. In that case the relation

$x_{1}\sim_{\mathcal{X}}x_{2}\quad:\Longleftrightarrow\quad{\rm
Mor}_{\mathcal{X}}(x_{1},x_{2})\neq\emptyset$

is an equivalence relation, and this defines the realization
$|{\mathcal{X}}|:=\text{\raise 3.44444pt\hbox{$\scriptstyle X\\!$}/\lower
3.44444pt\hbox{$\\!\scriptstyle\sim_{\mathcal{X}}$}}$ as topological space,
equipped with the quotient topology. Here we denote the set of morphisms
between subsets $U_{1},U_{2}\subset X$, for example $U_{i}=\\{x_{i}\\}$, by

${\rm Mor}_{\mathcal{X}}(U_{1},U_{2})\,:=\;s^{-1}(U_{1})\cap
t^{-1}(U_{2})\;\subset\;{\bf X}.$

For more general categories ${\mathcal{B}}$ which may not have inverses, the
realization $|{\mathcal{B}}|:=\text{\raise 3.44444pt\hbox{$\scriptstyle
B\\!$}/\lower 3.44444pt\hbox{$\\!\scriptstyle\sim_{\mathcal{B}}$}}$ is defined
by the equivalence relation $\sim_{\mathcal{B}}$ on $B$ that is generated by
the morphisms. Again, we equip $|{\mathcal{B}}|$ with the quotient topology
such that the quotient map is continuous,

(3.1.1)
$\displaystyle\pi_{\mathcal{B}}\,:\;B\;\to\;|{\mathcal{B}}|=\text{\raise
3.44444pt\hbox{$\scriptstyle B\\!$}/\lower
3.44444pt\hbox{$\\!\scriptstyle\sim_{\mathcal{B}}$}},\qquad
x\;\mapsto\;|x|:=[x].$

For any subset $V\subset B$ we denote its image in the realization by
$|V|:=\pi_{\mathcal{B}}(V)\subset|{\mathcal{B}}|$, and note that it carries
two natural topologies which may not coincide222222 For example, consider
${\mathcal{B}}$ with objects $B=[0,1]\times\\{\pm 1\\}$ and non-identity
morphisms $(x,-1)\mapsto(x,+1)$ and
$V=([0,1/2]\times\\{-1\\})\cup((1/2,1]\times\\{1\\})\subset B$. Then
$\pi_{\mathcal{B}}(V)=|{\mathcal{B}}|$ is homeomorphic to the interval
$[0,1]$, while $\scriptstyle V\\!$/$\\!\scriptstyle\sim_{\mathcal{B}}$ is
disconnected in the quotient topology.: the relative topology from the
inclusion into $|{\mathcal{B}}|$ and the quotient topology from the
identification $|V|\simeq\text{\raise 3.44444pt\hbox{$\scriptstyle
V\\!$}/\lower 3.44444pt\hbox{$\\!\scriptstyle\sim_{\mathcal{B}}$}}$. However,
when $V\subset B$ is open or saturated (that is
$\pi_{\mathcal{B}}^{-1}(\pi_{\mathcal{B}}(V))=V$) then the topologies agree by
Lemma 3.1.7.

In the following, we will always work with categories that are étale in the
sense of the following Definition 3.1.2. For that purpose we fix a class of
metrizable topological spaces and a corresponding class of local
homeomorphisms that is closed under compositions and taking local inverses. In
particular, all our results will hold for any of the following classes of
étale spaces and étale maps:

  * (a)

metrizable topological spaces and local homeomorphisms;

  * (b)

topological manifolds (assumed to be finite dimensional and second countable –
possibly with specified boundary and corner structure, see below) and local
homeomorphisms;

  * (c)

smooth manifolds (assumed to be finite dimensional and second countable –
possibly with boundary and corners, see below) and local diffeomorphisms;

  * (d)

M-polyfolds and local sc-diffeomorphisms (see §2.2).

Unless otherwise specified, we allow manifolds and M-polyfolds to have
boundary and corners, arising from local models in $[0,\infty)^{k}\times E$ as
in e.g. [FFGW, §5.3], giving rise to a degeneracy index
$d_{X}:X\to{\mathbb{N}}_{0}$. The corresponding étale maps are required to
restrict to local homeomorphisms resp. diffeomorphisms on each boundary and
corner stratum – that is connected components of $\\{x\in
X\,|\,d_{X}(x)=\ell\\}$ for fixed $\ell\geq 1$.

######  Remark 3.1.1.

  * (i)

A map $\phi:X\to Y$ is called a local homeomorphism (or diffeomorphism) if for
each $x\in X$ there exist open neighbourhoods232323 We always assume that
neighbourhoods are open sets; sometimes, for clarity, we mention this
explicitly. $N_{x}\subset X$ of $x$ and $N_{y}\subset Y$ of $y:=\phi(x)$ so
that the restriction $\phi|_{N_{x}}:N_{x}\to N_{y}$ is a homeomorphism (or
diffeomorphism) with respect to the induced topology (or smooth structure) on
the open subsets $N_{x},N_{y}$.

  * (ii)

This convention implies that every étale map is open. Indeed, given an open
subset $U\subset X$, we check that its image $\phi(U)\subset Y$ is open by
finding for each image $y=\phi(x)$ of some $x\in U$ an open neighbourhood
$N^{\prime}_{y}\subset Y$ contained in $\phi(U)$. We obtain
$N^{\prime}_{y}:=\phi(N_{x}\cap U)$ from a choice of open neighbourhoods
$N_{x}\subset X$ of $x$ and $N_{y}\subset Y$ of $y$ so that
$\phi|_{N_{x}}:N_{x}\to N_{y}$ is a homeomorphism. Here
$N^{\prime}_{y}=\phi(N_{x}\cap U)$ is open in $N_{y}$ because it is the image
under a homeomorphism of the open subset $N_{x}\cap U\subset N_{x}$. Finally,
openness of $N^{\prime}_{y}\subset N_{y}$ implies openness of
$N^{\prime}_{y}\subset Y$ since $N_{y}$ is open.

  * (iii)

The topological categories associated to Kuranishi atlases or structures are
almost never étale, as the local homeomorphism property of source and target
maps rules out overlaps between Kuranishi charts whose domains have different
dimension. ${\Diamond}$

Once such a class is chosen, we will refer to étale spaces and étale maps as
the spaces and maps in this class, and define the notion of étale category for
any chosen class as follows.

######  Definition 3.1.2.

A topological category ${\mathcal{B}}$ is said to be étale w.r.t. a chosen
class as above, if its object and morphism spaces $B,{\bf B}$ are étale spaces
and the structure maps $s,t,c,u$ are étale maps. If ${\mathcal{B}}$ is a
groupoid, then we require in addition that the inverse $i$ is étale.

######  Remark 3.1.3.

The requirement that composition be étale makes sense since each étale class
is closed under fiber products in the following sense: Three étale spaces
$U,V,W$ and two étale maps $t:U\to W,s:V\to W$ yield a fiber product
$U{\vphantom{\times}}_{t}{\times}_{s}V=\big{\\{}(x,y)\in U\times V\ \big{|}\
t(x)=s(y)\big{\\}}$ that also is an étale space in this class. Further, the
projections $U{\vphantom{\times}}_{t}{\times}_{s}V\to U$ and
$U{\vphantom{\times}}_{t}{\times}_{s}V\to V$ are étale maps in the same class.

Indeed, a neighbourhood of $(x_{0},y_{0})\in
U{\vphantom{\times}}_{t}{\times}_{s}V$ consists of points $(x,y)$ where
$x\approx x_{0}$ and $y=s\circ t^{-1}(x)$ for a suitable local inverse
$t^{-1}$ to $t$. A more detailed statement and proof including boundary and
corner structures is [HWZ, Prop.5.2.20] for the polyfold case. ${\Diamond}$

We will moreover use the following standard notions.

######  Definition 3.1.4.

  * (i)

A functor $f:{\mathcal{B}}\to{\mathcal{B}}^{\prime}$ between étale categories
is called étale if the induced maps on object and morphism spaces are étale.

  * (ii)

A category ${\mathcal{B}}$ is said to be nonsingular if for any two objects
$x,y\in B$ there is at most one morphism in ${\rm Mor}_{\mathcal{B}}(x,y)$.

  * (iii)

A nonsingular category ${\mathcal{B}}$ is called a poset if the object space
$B$ is equipped with a partial order $\preccurlyeq$ such that $x\preccurlyeq
y$ iff ${\rm Mor}_{{\mathcal{B}}}(x,y)\neq\emptyset$.

  * (iv)

If for some $x\in B$ the semigroup242424 A semigroup is a set $S$ with an
associative operation $S\times S\to S$ that has an identity element but is not
required to have inverses. ${\rm Mor}_{\mathcal{B}}(x,x)$ is a group, then we
call $G_{x}:={\rm Mor}_{\mathcal{B}}(x,x)$ the isotropy group at $x$.

  * (v)

A topological category ${\mathcal{B}}$ is said to be locally injective if
$\pi_{\mathcal{B}}:B\to|{\mathcal{B}}|$ is locally injective, i.e. every $x\in
B$ has a neighbourhood $N_{x}$ such that
$\pi_{\mathcal{B}}:N_{x}\to|{\mathcal{B}}|$ is injective.

  * (vi)

A groupoid ${\mathcal{X}}$ is called proper if it satisfies the properness
condition of Definition 2.1.1(i), i.e. every $x\in X$ has a neighbourhood
$N(x)$ such that the restricted target map

$t:s^{-1}\bigl{(}{\rm cl}_{X}(N(x))\bigr{)}\to X$

is proper.252525 A map $f:X\to Y$ is said to be proper if $f^{-1}(K)$ is
compact for every compact subset $K\subset Y$.

  * (vii)

An étale functor $f:{\mathcal{Y}}\to{\mathcal{X}}$ between groupoids is said
to be an equivalence if it induces isomorphisms ${\rm
Mor}_{{\mathcal{Y}}}(y,y^{\prime})\simeq{\rm
Mor}_{{\mathcal{X}}}(f(y),f(y^{\prime}))$ between morphism sets and a
homeomorphism $|f|:|{\mathcal{Y}}|\to|{\mathcal{X}}|$ of the realizations.

The various properness notions in the literature are compared in Lemma 3.2.5,
and Corollary 3.2.6 shows that the realization of a proper étale groupoid
${\mathcal{X}}$ is Hausdorff.

If we consider an étale proper groupoid using the class of M-polyfolds and
local sc-diffeomorphisms, then the above definition is not quite the same as
the notion of ep-groupoid in Definition 2.1.1, which requires only sc-
smoothness of the maps $c,u,i$, rather than local sc-diffeomorphisms. However,
the next lemma shows that étale proper groupoids in this class are the same as
ep-groupoids in the sense of polyfold theory [HWZ, Ch.7].262626 For clarity,
we will use the word ep-groupoid only if we are working in the class of
M-polyfolds.

######  Lemma 3.1.5.

Let ${\mathcal{B}}$ be an étale category modelled on M-polyfolds in the sense
of Definition 2.1.1. Then the unit and composition maps are local sc-
diffeomorphisms. If in addition ${\mathcal{B}}$ is a groupoid then the inverse
map is also a sc-diffeomorphism. Hence ${\mathcal{B}}$ is étale in the sense
of Definition 3.1.2.

###### Proof.

It suffices to exhibit local inverses of the structure maps of
${\mathcal{B}}$. For the unit $u:B\to{\bf B},x\mapsto{\rm id}_{x}$ we know
that $s\circ u={\rm id}_{B}$, where $s:{\bf B}\to B$, is a local sc-
diffeomorphism; hence $s$ provides the local inverse for $u$. For the
composition $c:{\bf B}{\vphantom{\times}}_{t}{\times}_{s}{\bf B}\to{\bf
B},(m,m^{\prime})\mapsto m\circ m^{\prime}$ we can use the fact that $s\circ
c=s\circ{\rm pr}_{1}$. Here ${\rm pr}_{1}:{\bf
B}{\vphantom{\times}}_{t}{\times}_{s}{\bf B}\to{\bf B}$ is projection to the
first factor, which is a local sc-diffeomorphism by construction of the fiber
product ${\bf B}{\vphantom{\times}}_{t}{\times}_{s}{\bf B}$. Since the source
map $s$ also is a local sc-diffeomorphism, this shows that the composition $c$
is a local sc-diffeomorphism. Finally, in the case of a groupoid the inverse
$i:{\bf X}\to{\bf X},m\mapsto m^{-1}$ is its own inverse map since inverses
are unique in any groupoid. ∎

######  Remark 3.1.6.

Let ${\mathcal{B}}=(B,{\bf B})$ be a category. For any subset $V\subset B$ we
obtain a restricted category ${\mathcal{B}}|_{V}$ by taking the full
subcategory of ${\mathcal{B}}$ with objects $V$. If ${\mathcal{B}}$ is étale
and $V$ is open, then Lemma 3.1.7 below identifies the realization
$|{\mathcal{B}}|_{V}|=\text{\raise 3.44444pt\hbox{$\scriptstyle V\\!$}/\lower
3.44444pt\hbox{$\\!\scriptstyle\sim_{\mathcal{B}}$}}$ (with the quotient
topology) with the subset $|V|=\pi_{\mathcal{B}}(V)\subset|{\mathcal{B}}|$
(with the relative topology) as topological spaces. Moreover, the restriction
${\mathcal{B}}|_{V}$ to an open subset $V$ is also étale, since ${\bf
B}|_{V}:=s^{-1}(V)\cap t^{-1}(V)\subset{\bf B}$ is an open subset, hence
inherits étale structure, and all structure maps in ${\mathcal{B}}|_{V}$ are
restrictions of the étale structure maps in ${\mathcal{B}}$. Furthermore,
Lemma 3.2.5 considers an étale proper groupoid ${\mathcal{X}}$ and shows that
the restriction ${\mathcal{X}}|_{V}$ to an open subset $V\subset X$ is also an
étale proper groupoid.

Although the above results apply when $V\subset B$ is any open set, in
applications such as Theorem 2.1.2, $V$ is saturated, i.e. of the form
$\pi_{\mathcal{B}}^{-1}(\pi_{\mathcal{B}}(V))$. ${\Diamond}$

We will often consider subsets $Z\subset B$ that are not open but saturated.
For groupoids as in [HWZ, p.341], this is equivalent to requiring that the set
is closed under morphisms. (For example, the zero set $Z=f^{-1}(0)\subset X$
of any sc-Fredholm section $f:{\mathcal{X}}\to{\mathcal{W}}$ is saturated.) In
general categories, we call $Z\subset B$ saturated if
$Z=\pi_{\mathcal{B}}^{-1}(\pi_{\mathcal{B}}(Z))$. In that case, as noted
above, $|Z|$ carries both the relative topology given by the embedding
$|Z|:=\pi_{\mathcal{B}}(Z)\subset|{\mathcal{B}}|$ and the quotient topology
given by the quotient map $Z\to\text{\raise 3.44444pt\hbox{$\scriptstyle
Z\\!$}/\lower
3.44444pt\hbox{$\\!\scriptstyle\sim_{\mathcal{B}}$}}=|{\mathcal{Z}}|$ to the
realization of the full subcategory ${\mathcal{Z}}$ of ${\mathcal{B}}$ with
objects $Z$. Fortunately, the two topologies agree in étale settings (unlike
the Kuranishi situation in [MW1, Def.3.1.14]).

######  Lemma 3.1.7.

Suppose $Z\subset B$ is an open or saturated subset of the objects in an étale
category ${\mathcal{B}}=(B,{\bf B})$. Then the quotient topology on
$|Z|=\text{\raise 3.44444pt\hbox{$\scriptstyle Z\\!$}/\lower
3.44444pt\hbox{$\\!\scriptstyle\sim_{\mathcal{B}}$}}$ agrees with the relative
topology on $|Z|\subset|{\mathcal{B}}|$.

###### Proof.

In both cases, we will identify $\text{\raise 3.44444pt\hbox{$\scriptstyle
Z\\!$}/\lower
3.44444pt\hbox{$\\!\scriptstyle\sim_{\mathcal{B}}$}}\cong\pi_{\mathcal{B}}(Z)$
so that the quotient map ${\rm pr}_{Z}:Z\to\text{\raise
3.44444pt\hbox{$\scriptstyle Z\\!$}/\lower
3.44444pt\hbox{$\\!\scriptstyle\sim_{\mathcal{B}}$}}$ coincides with the
restriction of $\pi_{\mathcal{B}}$ to $Z$. If $Z$ is open, then by Lemma 3.1.8
below, every open subset of $Z$ projects to an open subset of
$|{\mathcal{B}}|$. In particular, $\pi_{\mathcal{B}}(Z)\subset|{\mathcal{B}}|$
is open. Thus the openness of $O\subset\pi_{\mathcal{B}}(Z)$ in the relative
topology is equivalent to the openness of $O\subset|{\mathcal{B}}|$, and
(using the lemma again) to the openness of $\pi_{\mathcal{B}}^{-1}(O)\cap
Z=\pi_{\mathcal{Z}}^{-1}(O)$, and hence to the openness of $O$ in the quotient
topology on $\scriptstyle Z\\!$/$\\!\scriptstyle\sim_{\mathcal{B}}$.

In the case of a saturated subset $Z\subset B$, openness of
$O\subset\pi_{\mathcal{B}}(Z)$ in the relative topology means
$O=\pi_{\mathcal{B}}(Z)\cap V$ for some open set $V\subset|{\mathcal{B}}|$. By
definition of the quotient topology on $|{\mathcal{B}}|$ that means
$\pi_{\mathcal{B}}^{-1}(V)\subset B$ is open, and thus
$Z\cap\pi_{\mathcal{B}}^{-1}(V)\subset B$ is an open subset in the relative
topology of $Z\subset B$. Moreover, we have ${\rm
pr}_{Z}^{-1}(O)=\pi_{\mathcal{B}}^{-1}(\pi_{\mathcal{B}}(Z)\cap
V)=Z\cap\pi_{\mathcal{B}}^{-1}(V)$ since $Z$ is saturated. This shows that
${\rm pr}_{Z}^{-1}(O)\subset Z$ is open, and thus $O\subset\text{\raise
3.44444pt\hbox{$\scriptstyle Z\\!$}/\lower
3.44444pt\hbox{$\\!\scriptstyle\sim_{\mathcal{B}}$}}$ is open in the quotient
topology.

Conversely, openness of $O\subset\text{\raise 3.44444pt\hbox{$\scriptstyle
Z\\!$}/\lower 3.44444pt\hbox{$\\!\scriptstyle\sim_{\mathcal{B}}$}}$ in the
quotient topology means ${\rm pr}_{Z}^{-1}(O)\subset Z$ is open. Here
$Z\subset B$ is equipped with the relative topology, so ${\rm
pr}_{Z}^{-1}(O)=Z\cap U$ for some open subset $U\subset B$. Now
$\pi_{\mathcal{B}}(U)\subset|{\mathcal{B}}|$ is open by Lemma 3.1.8 below.
Moreover, we have $\pi_{\mathcal{B}}(Z\cap
U)=\pi_{\mathcal{B}}(Z)\cap\pi_{\mathcal{B}}(U)$ since $Z$ is saturated. This
identifies $O=\pi_{\mathcal{B}}({\rm
pr}_{Z}^{-1}(O))\subset\pi_{\mathcal{B}}(Z)$ as the intersection
$O=\pi_{\mathcal{B}}(Z)\cap\pi_{\mathcal{B}}(U)$ with an open set, thus as
open set in the relative topology of
$\pi_{\mathcal{B}}(Z)\subset|{\mathcal{B}}|$. ∎

Here and throughout, topological considerations in étale categories and étale
proper groupoids are simplified by the following.

######  Lemma 3.1.8.

  * (i)

The projection $\pi_{\mathcal{B}}:B\to|{\mathcal{B}}|$ is an open map in any
étale category.

  * (ii)

For any étale proper groupoid ${\mathcal{X}}$ we have ${\rm
cl}_{|{\mathcal{X}}|}(|U|)=|{\rm cl}_{X}(U)|$ when $U\subset X$ is saturated
or $U\subset X$ is precompact.

###### Proof.

(i) is a generalization of [HWZ, Prop.7.1.8]. To prove it, let $U$ be an open
subset of $B$. Then $\pi_{\mathcal{B}}(U)$ is open in $|{\mathcal{B}}|$ iff
$\pi_{\mathcal{B}}^{-1}(\pi_{\mathcal{B}}(U))$ is open in $B$, i.e. iff every
$y\in\pi_{\mathcal{B}}^{-1}(\pi_{\mathcal{B}}(U)$ has an open neighbourhood
$U_{y}\subset\pi_{\mathcal{B}}^{-1}(\pi_{\mathcal{B}}(U))$. For all such $y$
there is a finite chain of morphisms $m_{1},\cdots,m_{k}$ in ${\mathcal{B}}$
with $s(m_{1})=y,\ t(m_{k})=y_{k}\in U$ as follows

$y=:y_{0}\;\stackrel{{\scriptstyle
m_{1}}}{{\rightarrow}}\;y_{1}\;\stackrel{{\scriptstyle
m_{2}}}{{\leftarrow}}\;y_{3}\;\cdots\stackrel{{\scriptstyle
m_{k}}}{{\rightarrow}}\;y_{k}\in U.$

(To achieve such a zigzag chain from an arbitrary chain, one can insert
appropriate identity morphisms, and/or compose pairs of morphisms that go in
the same direction.) Since each $m_{i}$ extends to an injective étale map from
a neighbourhood of its source onto a neighbourhood of its target, the end
point $y_{k}=y$ has an open neighbourhood $N_{y}$ in $B$ consisting of points
that are equivalent to some point in $x\in U$. Hence $\pi_{\mathcal{B}}(U)$ is
open, because every point $y\in\pi_{\mathcal{B}}^{-1}(\pi_{\mathcal{B}}(U))$
has a neighbourhood
$N_{y}\subset\pi_{\mathcal{B}}^{-1}(\pi_{\mathcal{B}}(U))$.

(ii) is proven by the same arguments as in [HWZ, Lemma 12.4.4]. ∎

We end this section by explaining the notation used in [HWZ] for bundles. This
is sufficient preparation for the discussion of group actions on bundles in
§3.3. A much fuller treatment of the basic definitions is given in §3.4.

######  Definition 3.1.9.

A bundle ${\mathcal{P}}:{\mathcal{W}}\to{\mathcal{X}}$ over an étale groupoid
${\mathcal{X}}$ is given by an étale space $W$ with continuous surjections
$P:W\to X$ and $\mu:{\bf X}{\vphantom{\times}}_{s}{\times}_{P}W\to W$ such
that

  * (i)

$P$ is a locally trivial vector bundle in the appropriate category.

  * (ii)

$\mu$ is étale and is a lift of the target map $t:{\bf X}\to X$ in the sense
that $P\circ\mu(\cdot,w)=t$ for all $w\in W$. Further, it is compatible with
the relevant linear and groupoidal structures.

As in [HWZ, §8.3], these conditions ensure that $P$ lifts to a functor
${\mathcal{P}}:{\mathcal{W}}\to{\mathcal{X}}$ given by
${\mathcal{P}}(w)=P(w),\;{\mathcal{P}}(m,w)=m$, where ${\mathcal{W}}=(W,{\bf
W})$ is the étale groupoid with objects $W$ and

(3.1.2) $\displaystyle{\bf W}={\bf
X}{\vphantom{\times}}_{s}{\times}_{P}W,\quad(s\times t)(m,w)=(w,\mu(m,w)),$
$\displaystyle{\rm id}_{w}=({\rm
id}_{P(w)},w),\quad(m,w)\circ(m^{\prime},w^{\prime})=\bigl{(}m^{\prime}\circ
m,w\bigr{)}\quad\mbox{ if }w^{\prime}=\mu(m,w).$

A section of $(P,\mu)$ is an étale map $f:X\to W$ that satisfies $P\circ
f={\rm id}_{X}$, and is compatible with morphisms in the sense that
$f(t(m))=\mu(m,f(s(m)))$ for all $m\in{\bf X}$.

One can check as in Lemma 3.4.3 that each section $f:X\to W$ extends to a
unique étale functor $f:{\mathcal{X}}\to{\mathcal{W}}$.

###  3.2. Preuniformizers, properness, and groupoid completions

An important property of ep-groupoids is the existence of local uniformizers
[HWZ, Def.7.1.20] centered at any $x\in X$ – roughly speaking, a neighbourhood
$\simeq\text{\raise 3.44444pt\hbox{$\scriptstyle U\\!$}/\lower
3.44444pt\hbox{$\\!\scriptstyle G_{x}$}}$ of $|x|\in|{\mathcal{X}}|$ given as
the quotient of a natural action of the isotropy group $G_{x}:={\rm
Mor}_{\mathcal{X}}(x,x)$ on a neighbourhood $U\subset X$ of $x$. We will
generalize this notion to étale categories – i.e. allowing any étale class
from §3.1, and not requiring groupoid structure. Throughout, we consider left
actions of a group $G$ on a space $U$, that is $(hg)*x=h*(g*x)$ for $h,g\in
G,x\in U$.

######  Definition 3.2.1.

A preuniformizer for an étale category ${\mathcal{B}}=(B,{\bf B})$ consists of

  * (i)

an open subset $U\subset B$;

  * (ii)

an étale action $G\times U\to U,(g,y)\mapsto g*y$ of a finite group $G$;

  * (iii)

an étale injection ${\Gamma}:G\times U\to{\rm
Mor}_{\mathcal{B}}(U,U)\subset{\bf B}$ satisfying the following:

    * (a)

${\Gamma}$ identifies the morphisms in its image with the group action in the
sense that272727The inversion of order in the last identity compared with
[HWZ] results from our use of categorical notation for the composition $\circ$
of morphisms, while group actions are left actions both here and in [HWZ].

(3.2.1) $\displaystyle s\bigl{(}{\Gamma}(g,y)\bigr{)}=y,\qquad
t\bigl{(}{\Gamma}(g,y)\bigr{)}=g*y,\qquad{\Gamma}(hg,y)={\Gamma}(g,y)\circ{\Gamma}(h,g*y);$

    * (b)

${\Gamma}(G\times U)$ contains all connected components of ${\rm
Mor}_{\mathcal{B}}(U,U)$ that have a morphism $m$ with $s(m)=t(m)$;

    * (c)

the induced map $|{\Gamma}|:\text{\raise 3.44444pt\hbox{$\scriptstyle
U\\!$}/\lower 3.44444pt\hbox{$\\!\scriptstyle G$}}\to|{\mathcal{B}}|$ is
locally injective and has bounded fibers in the sense that
$\sup_{p\in|U|}\\#\\{[x]\in\text{\raise 3.44444pt\hbox{$\scriptstyle
U\\!$}/\lower 3.44444pt\hbox{$\\!\scriptstyle G$}}\,|\,|x|=p\\}<\infty$.

We will abbreviate this data by $(U,G,{\Gamma})$ or even $(U,G)$, and also
identify it with the induced faithful functor of the translation
groupoid282828 A category of the form $(U,G\times U)$ is called a translation
groupoid in [HWZ, p.294 in §7.1].  $(U,G\times U)$ into the subcategory
$(U,{\rm Mor}_{\mathcal{B}}(U,U))$ of ${\mathcal{B}}$. The induced map
$|{\Gamma}|:\text{\raise 3.44444pt\hbox{$\scriptstyle U\\!$}/\lower
3.44444pt\hbox{$\\!\scriptstyle G$}}\to|{\mathcal{B}}|$ is called the
footprint map.

If ${\mathcal{X}}$ is an étale groupoid and ${\Gamma}:G\times U\to{\rm
Mor}_{\mathcal{X}}(U,U)$ is surjective, then, as in [HWZ, Def.7.1.20]
$(U,G,{\Gamma})$ is called a local uniformizer. A local uniformizer around
$x\in X$ is a local uniformizer $(U,G,{\Gamma})$ such that $x\in U$ and we
have $t\circ{\Gamma}(\cdot,x)\equiv x$. In particular, ${\Gamma}(\cdot,x):G\to
G_{x}={\rm Mor}_{\mathcal{X}}(x,x)$ identifies $G$ with the isotropy group of
$x$.

######  Remark 3.2.2.

  * (i)

If ${\mathcal{X}}$ is an ep-groupoid, then local uniformizers for
${\mathcal{X}}$ exist near any object $x\in X$ by [HWZ, Prop.7.1.19], where
the isotropy group $G\simeq{\rm Mor}_{\mathcal{X}}(x,x)$ is finite by [HWZ,
Prop.7.1.12]. Moreover, by lifting a neighbourhood basis of
$|x|\in|{\mathcal{X}}|$, we may choose the domain $U$ to be arbitrarily small.
This proof, although given in the polyfold case, also applies to the étale
proper groupoids in the other étale categories considered here.

  * (ii)

Our notion of preuniformizer is significantly more general than that of a
local uniformizer; in particular the footprint map $|{\Gamma}|:\text{\raise
3.44444pt\hbox{$\scriptstyle U\\!$}/\lower 3.44444pt\hbox{$\\!\scriptstyle
G$}}\to|{\mathcal{B}}|$ is not in general injective – though it is assumed
locally injective, and is locally surjective by Lemma 3.2.3 (ii).

However, our notion of local uniformizer is the same as in [HWZ, Def.7.1.20];
in particular, in this case $|{\Gamma}|$ is a homeomorphism to its image by
Lemma 3.2.3 (iii). ${\Diamond}$

Here are some general properties of preuniformizers and thus local
uniformizers.

######  Lemma 3.2.3.

The following holds for any preuniformizer $(U,G,{\Gamma})$ in an étale
category ${\mathcal{B}}$.

  * (i)

For all $x\in U$, the semigroup292929See the footnote in Definition 3.1.4
(iv). ${\rm Mor}_{\mathcal{B}}(x,x)$ is isomorphic via ${\Gamma}(\cdot,x)$ to
the subgroup $G_{x}=\\{g\in G\,\big{|}\,t({\Gamma}(g,x))=x\\}\subset G$. In
particular, each ${\rm Mor}_{\mathcal{B}}(x,x)$ is a group, and these groups
are isomorphic as $x$ varies in an orbit of $G$.

  * (ii)

The subset $|U|\subset|{\mathcal{B}}|$ is open and the induced map
$|{\Gamma}|:\text{\raise 3.44444pt\hbox{$\scriptstyle U\\!$}/\lower
3.44444pt\hbox{$\\!\scriptstyle G$}}\to|U|$ is a local homeomorphism with
respect to the quotient topology on $\scriptstyle U\\!$/$\\!\scriptstyle G$
and the subspace topology on $|U|\subset|{\mathcal{B}}|$.

  * (iii)

If ${\mathcal{B}}$ is a groupoid, and $(U,G,{\Gamma})$ is a local uniformizer,
then $|{\Gamma}|:\text{\raise 3.44444pt\hbox{$\scriptstyle U\\!$}/\lower
3.44444pt\hbox{$\\!\scriptstyle G$}}\to|U|$ is injective and hence a
homeomorphism.

###### Proof.

Since ${\Gamma}$ intertwines products in $G$ with composition of morphisms, it
induces an injective homomorphism ${\Gamma}_{x}:G_{x}\to{\rm
Mor}_{\mathcal{B}}(x,x)\subset G_{x}$. Condition (iii)(b) in Definition 3.2.1
implies that ${\Gamma}_{x}:G_{x}\to{\rm Mor}_{\mathcal{B}}(x,x)$ is surjective
and hence an isomorphism, which implies that $G_{x}\simeq{\rm
Mor}_{\mathcal{B}}(x,x)$ for all $x\in U$. The second claim in (i) then holds
because it holds for the translation groupoid $(U,G\times U)$.

To prove (ii), notice first that $|U|$ is open in $|{\mathcal{B}}|$ because
the subset $U\subset B$ is open by condition (i) in Definition 3.2.1, and the
étale property of ${\mathcal{B}}$ implies that $\pi_{\mathcal{B}}$ is an open
map by Lemma 3.1.8. Further, the quotient map ${\sigma}:U\to\text{\raise
3.44444pt\hbox{$\scriptstyle U\\!$}/\lower 3.44444pt\hbox{$\\!\scriptstyle
G$}}$ is open. Since the morphisms in the translation groupoid $(U,G\times U)$
inject into those of ${\mathcal{B}}$, each $G$-orbit maps to a single
equivalence class of $\sim_{\mathcal{B}}$. Hence there is an induced
surjective map $|{\Gamma}|:\text{\raise 3.44444pt\hbox{$\scriptstyle
U\\!$}/\lower 3.44444pt\hbox{$\\!\scriptstyle G$}}\to|U|$. To see that
$|{\Gamma}|$ is continuous, consider $O^{\prime}:=|{\Gamma}|^{-1}(O)$ where
$O\subset|U|$ is open. Then
${\sigma}^{-1}(O^{\prime})=\pi_{\mathcal{B}}^{-1}(O)\cap U$ is an open set,
since ${\sigma}:U\to\text{\raise 3.44444pt\hbox{$\scriptstyle U\\!$}/\lower
3.44444pt\hbox{$\\!\scriptstyle G$}}$ is an open map, so that
$O^{\prime}={\sigma}({\sigma}^{-1}(O^{\prime}))$ is also an open set.

It remains to show that each $y\in U$ has an open neighbourhood $N$ such that
$|{\Gamma}||_{{\sigma}(N)}$ is a homeomorphism to its image. To prove this, we
adapt the arguments in [HWZ, Prop.7.1.15,7.1.19]. Given $y\in U$ choose an
open neighbourhood $N\subset U$ such that $|{\Gamma}|:{\sigma}(N)\to|U|$ is
injective. By shrinking $N$ we may suppose that it is $G_{y}$-invariant and
disjoint from its images under all $g\in G{\smallsetminus}G_{y}$. Then
${\sigma}:N\to{\sigma}(N)\subset\text{\raise 3.44444pt\hbox{$\scriptstyle
U\\!$}/\lower 3.44444pt\hbox{$\\!\scriptstyle G$}}$ factors through a
homeomorphism $\text{\raise 3.44444pt\hbox{$\scriptstyle N\\!$}/\lower
3.44444pt\hbox{$\\!\scriptstyle
G_{y}$}}\stackrel{{\scriptstyle\simeq}}{{\to}}{\sigma}(N)$. We claim that the
continuous map $|{\Gamma}||_{{\sigma}(N)}:{\sigma}(N)\to|N|$ is a
homeomorphism onto $|N|\subset|U|$ (which is an open set since
$\pi_{\mathcal{B}}$ is an open map). Since $|{\Gamma}||_{{\sigma}(N)}$ is
injective by construction we need only check that its inverse is continuous.
Thus for each open set $O\subset{\sigma}(N)\subset\text{\raise
3.44444pt\hbox{$\scriptstyle U\\!$}/\lower 3.44444pt\hbox{$\\!\scriptstyle
G$}}$ we must check that $|{\Gamma}|(O)$ is open. But the open set
${\sigma}^{-1}(O)$ maps onto $O$, so that
$|{\Gamma}|(O)=\pi_{\mathcal{B}}({\sigma}^{-1}(O))$ and hence is open because
$\pi_{\mathcal{B}}$ is an open map. This completes the proof of (ii).

The hypothesis that ${\mathcal{B}}$ is a groupoid implies that two points
$x,y$ in $U$ are equivalent under $\sim_{\mathcal{B}}$ if and only if there is
a morphism $m\in{\bf B}$ with $s(m)=x,t(m)=y$. Since this morphism lies in the
full subcategory ${\mathcal{B}}|_{U}$ which is isomorphic to the translation
groupoid $(U,G\times U)$, the map $\text{\raise 3.44444pt\hbox{$\scriptstyle
U\\!$}/\lower 3.44444pt\hbox{$\\!\scriptstyle G$}}\to|U|$ is injective. Hence
(iii) follows from (ii). ∎

We now give an example of an étale category ${\mathcal{B}}$ that is the union
of two preuniformizers, both with trivial groups $G$. We show in (ii) that
although $|{\mathcal{B}}|$ is not usually Hausdorff, the category
${\mathcal{B}}$ is proper in the sense of Definition 3.1.4, which illustrates
the fact that properness is not a useful notion in the context of étale
categories that are not groupoids. The setup is slightly more elaborate than
is needed here to make our point, but it is a special case of the
constructions discussed in §4.2.

######  Example 3.2.4.

  * (i)

Let ${\widetilde{V}}\subset U\subset{\mathbb{R}}^{k}$ be nested open subsets,
and suppose that $G$ is a finite group that acts freely on ${\widetilde{V}}$
in such a way that the action extends freely to an open neighbourhood
$U^{\prime}\subset U$ of the closure ${\rm cl}({\widetilde{V}})$ of
${\widetilde{V}}$ in $U$. Let $\rho:{\widetilde{V}}\to V:=\text{\raise
3.44444pt\hbox{$\scriptstyle{\widetilde{V}}\\!$}/\lower
3.44444pt\hbox{$\\!\scriptstyle G$}}$ be the quotient map, and consider the
étale category ${\mathcal{B}}$ with objects $V\sqcup U$ and morphisms
consisting of the identity morphisms together with a component303030 To be
consistent with the notation in [MW1, MW2, MW3] we will interpret these
morphisms as going from $V$ to $U$, though in some ways it might be more
natural to think of them as going in the other direction. ${\rm
Mor}_{\mathcal{B}}(V,U)\simeq{\widetilde{V}}$ of morphisms from $V$ to $U$
that “quotients out by the action of $G$ on ${\widetilde{V}}$.” More
precisely, if we denote the elements of ${\widetilde{V}}$ by $y$ and suppress
the inclusion map ${\widetilde{V}}\to U$ we have

${\rm Mor}_{\mathcal{B}}(V,U)\simeq{\widetilde{V}},\quad s\times
t\,:{\widetilde{V}}\to V\times U,\;\;y\mapsto\bigl{(}\rho(y),y\bigr{)}.$

While the projection map $\pi_{\mathcal{B}}|_{V}$ is injective (where
$\pi_{\mathcal{B}}$ is as in (3.1.1)), the map $\pi_{\mathcal{B}}|_{U}$ is
not, because the points in $|{\widetilde{V}}|=|V|\subset|{\mathcal{B}}|$ have
$\\#G$ preimages in $U$. Nevertheless, the map $\pi_{\mathcal{B}}|_{U}$ is
locally injective because $G$ acts freely on $U^{\prime}$. Thus
${\mathcal{B}}$ is covered by two preuniformizers with domains $V,U$
respectively and trivial groups $G=\\{{\rm id}\\}$.

  * (ii)

We claim that if ${\mathcal{B}}$ is as in (i) its realization
$|{\mathcal{B}}|$ is not Hausdorff unless $G=\\{{\rm id}\\}$ or the frontier
${\rm Fr}_{U}({\widetilde{V}}):={\rm
cl}({\widetilde{V}}){\smallsetminus}{\widetilde{V}}$ is empty. To see this,
suppose given $x\in{\rm Fr}_{U}({\widetilde{V}})\subset U$ and ${\rm id}\neq
g\in G$. Then $x,g(x)$ are two distinct points in $U{\smallsetminus}V$, and
have distinct images $|x|,|g(x)|$ in $|{\mathcal{B}}|$. Now choose any
neighbourhoods $N_{|x|},N_{|g(x)|}$ of $|x|,|g(x)|$ in $|{\mathcal{B}}|$ and
consider their inverse images $N_{x}$, $N_{g(x)}$ in $U$. Since these sets are
open and contain $x,g(x)$ respectively, both $g(N_{x})$ and $N_{g(x)}$ are
neighbourhoods of $g(x)$ in $U$. Therefore $g(N_{x})\cap N_{g(x)}$ is a
neighbourhood of $g(x)$, and hence must have nonempty intersection with
${\widetilde{V}}$. But if $y\in g(N_{x})\cap N_{g(x)}\cap{\widetilde{V}}$,
then $|y|\in N_{|x|}\cap N_{|g(x)|}$ because $|N_{x}|=N_{|x|}$ and
$|N_{g(x}|=N_{|g(x)|}$ by construction. Therefore $N_{|x|},N_{|g(x)|}$ are
never disjoint.

On the other hand, the map $s\times t\,:{\widetilde{V}}\to V\times U$ is
proper because the map $\rho:{\widetilde{V}}\to V$ is surjective and proper.
Further, every point $x\in B=V\sqcup U$ does have a neighbourhood $N(x)$ such
that the map $t:s^{-1}({\rm cl}_{B}N(x))\to B$ is proper. Indeed, if $x\in U$
and we choose $N(x)\subset U$, the only morphisms in $s^{-1}({\rm
cl}_{B}N(x))$ are identity maps, and the existence of $N(x)$ is clear.
Further, since $\rho({\widetilde{V}})=V$ then for any $x\in V$ we may take
$N(x)\subset V$ to be compact, so that $s^{-1}({\rm cl}_{B}N(x))=({\rm
cl}_{V}N(x))\sqcup\rho^{-1}({\rm cl}_{V}N(x))$; thus $t:s^{-1}({\rm
cl}_{B}N(x))\to V\sqcup U$ is proper. ${\Diamond}$

We next discuss various conditions on a topological groupoid
${\mathcal{X}}:=(X,{\mathcal{X}})$ that guarantee that its realization is
Hausdorff. We will assume that the object space $X$ is first countable, but we
do not assume that ${\mathcal{X}}$ is étale.

######  Lemma 3.2.5.

Let ${\mathcal{X}}$ be a topological groupoid such that $X$ is first
countable.

  * (i)

If the space $X$ is locally compact and $s\times t:{\bf X}\to X\times X$ is
proper, then ${\mathcal{X}}$ is proper in the sense of Definition 3.1.4.

  * (ii)

If ${\mathcal{X}}$ is proper in the sense of Definition 3.1.4, then the map
$s\times t:{\bf X}\to X\times X$ is proper and has closed graph.

  * (iii)

If ${\mathcal{X}}$ is étale and has local uniformizer $(U,G)$, then for every
closed subset $C$ of $U$ such that $|C|$ is closed in $|{\mathcal{X}}|$ the
map $t:s^{-1}(C)\to X$ is proper.

  * (iv)

If the map $\pi:X\to|{\mathcal{X}}|$ is open313131For étale groupoids this is
guaranteed by Lemma 3.1.8 (i). and $s\times t:{\bf X}\to X\times X$ has closed
graph, then $|{\mathcal{X}}|$ is Hausdorff.

###### Proof.

To prove (i) suppose that $s\times t:{\bf X}\to X\times X$ is proper. Then,
the local compactness assumption on $X$ provides for any $x\in X$ a
neighbourhood $N(x)\subset X$ whose closure ${\rm cl}_{X}(N(x))$ is compact.
Now, given a compact subset $K\subset X$, its preimage under
$t:s^{-1}\bigl{(}{\rm cl}_{X}(N(x))\bigr{)}\to X$ is

$s^{-1}\bigl{(}{\rm cl}_{X}(N(x))\bigr{)}\cap t^{-1}(K)=(s\times
t)^{-1}\bigl{(}{\rm cl}_{X}(N(x))\times K\bigr{)}\subset{\bf X},$

which is also compact by properness of $s\times t$ and compactness of the
Cartesian product ${\rm cl}_{X}(N(x))\times K\subset X\times X$.

To prove the first claim in (ii), suppose that every $x\in X$ has a
neighbourhood $N(x)\subset X$ such that $t:s^{-1}\bigl{(}{\rm
cl}_{X}(N(x))\bigr{)}\to X$ is proper, and denote by $({\rm pr}_{i})_{i=1,2}$
the projection of $X\times X$ onto its $i$th factor. Then the preimage in
${\bf X}$ of a compact subset $K\subset X\times X$ under $s\times t:{\bf X}\to
X\times X$ is compact because $s^{-1}\bigl{(}{\rm cl}_{X}(N(x))\bigr{)}\cap
t^{-1}\bigl{(}{\rm pr}_{2}(K)\bigr{)}$ is compact for every $x$ by choice of
$N(x)$, and the compactness of ${\rm pr}_{1}(K)$ implies that it has a finite
covering by sets of the form $N(x)$.

To prove the second claim in (ii), it suffices to observe that if $s\times t$
does not have closed graph, there is a convergent sequence
$(x_{k},y_{k})\to(x_{\infty},y_{\infty})$ in $X\times X$, with
$(x_{k},y_{k})\in{\rm im\,}(s\times t)$, but
$(x_{\infty},y_{\infty})\notin{\rm im\,}(s\times t)$. Then the compact set
$K=\\{(x_{k},y_{k})|k\in{\mathbb{N}}\\}\cup\\{(x_{\infty},y_{\infty})\\}$ does
not have compact preimage.

The statement in (iii) generalizes [HWZ, Prop.7.5.10], which characterizes the
“proper” subsets of an ep-groupoid ${\mathcal{X}}$ (subsets $C$ for which the
map $t:s^{-1}(C)\to X$ is proper). Here we do not assume ${\mathcal{X}}$ to be
proper, and so need to give a direct proof. Let $K\subset X$ be compact. Since
$U\cap\pi^{-1}(|C|)=\bigcup_{g\in G}gC$, it suffices to prove the claim when
$C=U\cap\pi^{-1}(|C|)$. Further, because $X$ is first countable, it suffices
to prove that every sequence $\phi_{k}\in s^{-1}(C)\cap t^{-1}(K)$ has a
convergent subsequence. Since $K$ is compact, we can assume that
$y_{k}:=t(\phi_{k})\to y_{\infty}\in K$. Consider the sequence
$x_{k}:=s(\phi_{k})\in C\cap\pi^{-1}(\pi(K))$. Since $|C|\cap|K|$ is the
intersection of a compact set with a closed set and hence compact, and since
$\pi:U\to|U|$ quotients out by the action of the finite group $G$, the set
$C\cap\pi^{-1}(\pi(K))=C\cap\pi^{-1}(|C|\cap|K|)$ is also compact. Hence,
without loss of generality we may assume that $x_{k}$ also converges, with
limit $x_{\infty}$. Since $\pi:C\to|C|$ is continuous and $|x_{k}|=|y_{k}|$ by
construction, we have $|x_{\infty}|=\lim|x_{k}|=\lim|y_{k}|=|y_{\infty}|$,
where here we make essential use of the fact that $|U|$ is Hausdorff, which
holds because the map $\psi:\text{\raise 3.44444pt\hbox{$\scriptstyle
U\\!$}/\lower 3.44444pt\hbox{$\\!\scriptstyle G$}}\to|U|$ is a homeomorphism
by Lemma 3.2.3 (iii). Therefore there is an element $m_{\infty}\in{\rm
Mor}_{\mathcal{X}}(x_{\infty},y_{\infty})$, and it is now easy to find a
convergent subsequence of $(\phi_{k})$. Indeed, because the maps $s,t$ are
étale, there is a convergent sequence $\phi_{k}^{\prime}\in{\bf X}$ with limit
$m_{\infty}$ and with $t(\phi^{\prime}_{k})=y_{k}$ and
$s(\phi_{k}^{\prime})\in U$ for all $k$. Since
$|s(\phi_{k}^{\prime})|=|y_{k}|=|x_{k}|$ for all $k$, there are elements
$g_{k}\in G$ such that

$x_{k}=g_{k}*s(\phi_{k}^{\prime})=s\bigl{(}{\Gamma}(g_{k},x)\circ\phi_{k}^{\prime}\bigr{)},$

which implies by the groupoid property of ${\mathcal{X}}$ that there are
elements $h_{k}\in G_{x_{k}}$ such that
$\phi_{k}={\Gamma}(h_{k},x)\circ{\Gamma}(g_{k},x)\circ\phi^{\prime}_{k}$ for
all $k$. Since there are only finitely many possibilities for $h_{k},g_{k}$,
the given sequence $\phi_{k}$ must have a convergent subsequence.

To prove (iv) suppose that $x,y$ are two points in $X$ such that every
neighbourhood of $|x|$ intersects every neighbourhood of $|y|$. Since the map
$\pi:X\to|{\mathcal{X}}|$ is open we may take neighbourhoods of the form
$\pi(N_{x}),\pi(N_{y})$ where $N_{x},N_{y}$ are open neighbourhoods of $x,y$
respectively, and hence find sequences $x_{k}\to x$ and $y_{k}\to y$ such that
$x_{k}\sim y_{k}$. Because ${\mathcal{X}}$ is a groupoid, $x_{k}\sim y_{k}$
exactly if there is a morphism $x_{k}\to y_{k}$, i.e. exactly if
$(x_{k},y_{k})\in s\times t({\bf X})\subset X\times X$. Since
$(x_{k},y_{k})\to(x,y)$, the closed graph condition implies that $(x,y)\in
s\times t\,({\bf X})$. Hence $|x|=|y|$, i.e. $|{\mathcal{X}}|$ is Hausdorff. A
similar argument applies if $\pi$ is closed, since every point of $X$ has a
countable basis of closed neighbourhoods. ∎

######  Corollary 3.2.6.

  * (i)

The realization $|{\mathcal{X}}|$ of any étale proper groupoid ${\mathcal{X}}$
is Hausdorff.

  * (ii)

Suppose the étale groupoid ${\mathcal{X}}$ has Hausdorff realization
$|{\mathcal{X}}|$, and is covered by a finite number of local uniformizers
$(U_{I},G_{I})_{I\in{\mathcal{I}}}$ in the sense that
$|{\mathcal{X}}|=\bigcup_{I\in{\mathcal{I}}}|U_{I}|$. Let
$U:=\bigcup_{I\in{\mathcal{I}}}U_{I}\subset X$, and define
${\mathcal{X}}|_{U}$ (as in Remark 3.1.6) to be the full subcategory of
${\mathcal{X}}$ with objects $U$. Then the categories ${\mathcal{X}}$ and
${\mathcal{X}}|_{U}$ are both proper.

###### Proof.

If ${\mathcal{X}}$ is étale, the projection $\pi:X\to|{\mathcal{X}}|$ is open
by Lemma 3.1.8, and the map $s\times t:{\bf X}\to X\times X$ has closed graph
by Lemma 3.2.5 (ii). Therefore (i) follows from Lemma 3.2.5 (iv).

Towards (ii), we first show that the realization $|{\mathcal{X}}|$ is
metrizable when we assume it to be Hausdorff and covered
$|{\mathcal{X}}|=\bigcup_{I\in{\mathcal{I}}}|U_{I}|$ by a finite number of
local uniformizers $(U_{I},G_{I})_{I\in{\mathcal{I}}}$. Note first that each
$U_{I}$ is metrizable since it is a subset of $X$, which is metrizable by
definition of the étale spaces (a)-(d). By the Nagata–Smirnov metrization
theorem (see [Mu, Thm.40.3], this is equivalent to saying that $U_{I}$ is
regular and has a basis for its topology that is a countable union of locally
finite collections of open sets. Both these properties are inherited by the
quotient $|U_{I}|=U_{I}/G_{I}$; regularity of the quotient follows from
Exercise 7(b) on p 199 of [Mu], while the second condition holds because the
projection $U_{I}\to|U_{I}|$ is open and globally finite-to-one so that the
image in $|U_{I}|$ of a locally finite collection of open sets in $U_{I}$ has
the same properties. Hence the quotient $|U_{I}|$ is metrizable. Thus, because
$|U_{I}|$ is open in $|{\mathcal{X}}|$ by Lemma 3.2.3 (ii), the space
$|{\mathcal{X}}|$ is locally metrizable. Further, because metrizability is
equivalent to paracompactness, Hausdorffness and local metrizability by
Smirnov’s metrizability theorem (see [Mu, Thm.42.1]), each $|U_{I}|$ is
paracompact. It follows that $|{\mathcal{X}}|$, which is the finite union of
the open sets $|U_{I}|$, is also paracompact. Since $|{\mathcal{X}}|$ is
Hausdorff by hypothesis, Smirnov’s metrizability theorem implies that
$|{\mathcal{X}}|$ is metrizable.

To prove properness of ${\mathcal{X}}|_{U}$ in (ii), we must check that every
$x\in U$ has a neighbourhood $N(x)$ so that $t:s^{-1}\bigl{(}{\rm
cl}_{X}(N(x))\bigr{)}\to X$ is proper. By assumption, we have $x\in U_{I}$ for
some local uniformizer $(U_{I},G_{I})$, and applying Lemma 3.2.5 (iii) to this
uniformizer guarantees properness of $t:s^{-1}(C)\to X$ for any closed subset
$C$ subset of $U_{I}$ such that $|C|\subset|{\mathcal{X}}|$ is closed. Thus it
suffices to find a closed neighbourhood $C\subset U_{I}$ of $x$ such that
$|C|$ is closed in $|{\mathcal{X}}|$. To this end, consider the closed set
$|{\mathcal{X}}|{\smallsetminus}|U_{I}|$. Since
$|x|\notin|{\mathcal{X}}|{\smallsetminus}|U_{I}|$ and $|{\mathcal{X}}|$ is
metrizable and hence normal (by [Mu, Thm.32.3]), the point $|x|$ (which is a
closed set since $|{\mathcal{X}}|$ is Hausdorff) has an open neighbourhood
$W\subset|{\mathcal{X}}|$ whose closure ${\rm cl}_{|{\mathcal{X}}|}(W)$ is
disjoint from $|{\mathcal{X}}|{\smallsetminus}|U_{I}|$ and hence is contained
in $|U_{I}|$. Therefore $C:=\pi^{-1}\bigl{(}{\rm
cl}_{|{\mathcal{X}}|}(W)\bigr{)}$ is contained in $U_{I}$. Moreover, because
${\rm cl}_{|{\mathcal{X}}|}(W)$ is a closed neighbourhood of $|x|$ in the open
set $|{\mathcal{X}}|$ its pullback $C$ to $U_{I}$ is a closed neighbourhood of
$x$ in $U_{I}$.

It is now straightforward to check that the category ${\mathcal{X}}$ itself is
proper. Indeed, by assumption, for every $y\in X$ there is some $x\in U_{I}$
such that $|x|=|y|$. Hence, as in the proof of Lemma 3.1.8(i) there is a local
étale bijection
$\phi:N^{\prime}(y)\stackrel{{\scriptstyle\simeq}}{{\to}}N^{\prime}(x)$ from a
neighbourhood $N^{\prime}(y)$ of $y$ onto a neighbourhood
$N^{\prime}(x)\subset U$ of $x$. Now choose a neighbourhood $N(x)$ of $X$ such
that ${\rm cl}_{X}(N(x))\subset N^{\prime}(x)$ and $t:s^{-1}\bigl{(}{\rm
cl}_{X}(N(x))\bigr{)}\to X$ is proper. Then the étale map

$s^{-1}\bigl{(}{\rm cl}_{X}(N(x))\bigr{)}\to s^{-1}\bigl{(}{\rm
cl}_{X}(N(y))\bigr{)},\quad m\mapsto\phi\circ m$

has an étale inverse. It follows that $t:s^{-1}\bigl{(}{\rm
cl}_{X}(N(y))\bigr{)}\to X$ is proper.

This completes the proof of the lemma. ∎

Finally, we discuss groupoid completions of étale categories ${\mathcal{B}}$.
For that purpose recall from Definition 3.1.4 that the sets $B_{x}:={\rm
Mor}_{\mathcal{B}}(x,x)$ are generally semigroups as they may lack inverses.

######  Definition 3.2.7.

Let ${\mathcal{B}}=(B,{\bf B})$ be a category. Then a pair
$(\widehat{\mathcal{B}},{\iota})$ consisting of a groupoid
$\widehat{\mathcal{B}}$ together with an injective functor
${\iota}:{\mathcal{B}}\to\widehat{{\mathcal{B}}}$ is called a groupoid
completion of the category ${\mathcal{B}}$ if the following holds:

  * (a)

${\iota}:B\to\widehat{B}$ is bijective and induces a bijection
$|{\mathcal{B}}|\to|\widehat{{\mathcal{B}}}|$,

  * (b)

for all $x\in B$, the functor ${\iota}$ induces an isomorphism
$B_{x}\to\widehat{B}_{x}:={\rm Mor}_{\widehat{\mathcal{B}}}(x,x)$.

If ${\mathcal{B}}$ is étale, then a pair $(\widehat{\mathcal{B}},{\iota})$ as
above is an étale groupoid completion if in addition

  * (c)

the category $\widehat{\mathcal{B}}$ has an étale structure so that $\iota$ is
étale.

######  Remark 3.2.8.

If ${\mathcal{B}}$ is étale and has étale groupoid completion
$\widehat{\mathcal{B}}$, then by conditions (a) and (c) we can identify the
object spaces of both categories. Then by (a) and the fact that the
realization of a category has the quotient topology, the two realization maps
$\pi_{\mathcal{B}}:B\to|{\mathcal{B}}|$ and
$\pi_{\widehat{\mathcal{B}}}:\widehat{B}\to|\widehat{\mathcal{B}}|$ can also
be identified, and have the same topological properties. For example, if one
is locally injective, so is the other. ${\Diamond}$

The following lemma establishes some useful properties of groupoid
completions.

######  Lemma 3.2.9.

  * (i)

If ${\mathcal{B}}$ has a groupoid completion, then each $B_{x}$ is a group,
and these groups are isomorphic as $x$ ranges over an equivalence class with
respect to $\sim_{\mathcal{B}}$.

  * (ii)

If ${\mathcal{B}}$ is étale and there is an injective étale functor
$f:{\mathcal{B}}\to{\mathcal{X}}$ to an étale groupoid ${\mathcal{X}}$ that
induces an isomorphism
$f:B_{x}\stackrel{{\scriptstyle\simeq}}{{\to}}G_{f(x)}:={\rm
Mor}_{\mathcal{X}}(x,x)$ for all $x\in B$ and an injection
$|f|:|{\mathcal{B}}|\to|{\mathcal{X}}|$ that is a homeomorphism to its image,
then ${\mathcal{B}}$ has an étale groupoid completion, namely
$({\mathcal{X}}|_{f(B)},f)$, where ${\mathcal{X}}|_{f(B)}$ is the full
subcategory of ${\mathcal{X}}$ with objects $f(B)$.

  * (iii)

A nonsingular category ${\mathcal{B}}$ has a unique groupoid completion
$\widehat{\mathcal{B}}$ up to functorial isomorphism. It is necessarily
nonsingular, and is étale and/or locally injective if ${\mathcal{B}}$ is.

  * (iv)

If the nonsingular category ${\mathcal{B}}$ is a poset such that each
equivalence class $[x]:=\\{y\in B\ |\ |x|=|y|\\}$ in $B$ has a unique minimum
with morphisms to all other elements. Then every functor
$f:{\mathcal{B}}\to{\mathcal{X}}$ whose target is a groupoid extends to a
functor $\widehat{f}:\widehat{\mathcal{B}}\to{\mathcal{X}}$. Further if
${\mathcal{B}},{\mathcal{X}}$ and $f:{\mathcal{B}}\to{\mathcal{X}}$ are étale,
then so is the functor $\widehat{f}$.

###### Proof.

To prove (i), note first that the isomorphisms $B_{x}\to\widehat{B}_{x}$ in
(b) provide inverses for $B_{x}$. Next, in any groupoid ${\mathcal{X}}$ the
morphism $m\in{\rm Mor}_{\mathcal{X}}(x,y)$ induces an isomorphism

$\phi_{m}:G_{x}\to G_{y},\quad{\alpha}\mapsto{\beta}:=m^{-1}\circ{\alpha}\circ
m:y\to x\to x\to y.$

Since the above identity may be written $m\circ{\beta}={\alpha}\circ m$, it
follows that if ${\mathcal{B}}$ has a groupoid completion the equation
$m\circ{\beta}={\alpha}\circ m$ in ${\bf B}$ must have a unique solution
${\beta}\in B_{y}$ for each ${\alpha}\in B_{x},m\in{\rm
Mor}_{\mathcal{B}}(x,y)$, and moreover that the assignment

(3.2.2)
$\displaystyle\phi_{m}:\quad{\alpha}\mapsto{\beta}=:\phi_{m}({\alpha}),\;\;\mbox{
where }m\circ{\beta}={\alpha}\circ m$

is a group isomorphism $B_{x}\to B_{y}$. This proves (i).

To prove (ii), notice that because $f$ is étale, $f(B)$ is an open subset of
$X$. Further, if ${\mathcal{X}}$ is an étale groupoid and $U\subset X$ is open
then ${\mathcal{X}}|_{U}$ is also an étale groupoid. Hence
$f:{\mathcal{B}}\to{\mathcal{X}}|_{f(B)}$ satisfies the conditions in
Definition 3.2.7.

To prove (iii), notice that by the definition of nonsingularity (see
Definition 3.1.2) we have $|{\rm Mor}_{\mathcal{B}}(x,y)|\leq 1$ for all
$x,y\in B$. Therefore condition (b) implies that any groupoid completion
$\widehat{\mathcal{B}}$ is also nonsingular. By condition (a) we must
therefore have

${\rm Mor}_{\widehat{\mathcal{B}}}:=\bigl{\\{}(x,y)\in B\times B\ \big{|}\
x\sim_{\mathcal{B}}y\bigr{\\}},$

with structure maps $s,t$ given by projection onto the first and second
factors respectively. Then $\widehat{\mathcal{B}}$ with this set of morphisms
is a groupoid. Next note that ${\rm Mor}_{\widehat{\mathcal{B}}}$ can be given
the structure of an étale space in such a way that all structure maps are
étale. But this follows because, as in the proof of Lemma 3.1.8, each added
morphism $y_{0}\to y_{k}$ in $\widehat{\mathcal{B}}$ is formed from a zigzag
chain $y_{0}\;\stackrel{{\scriptstyle
m_{1}}}{{\rightarrow}}\;y_{1}\;\stackrel{{\scriptstyle
m_{2}}}{{\leftarrow}}\;y_{3}\;\cdots\stackrel{{\scriptstyle
m_{k}}}{{\rightarrow}}\;y_{k}$ of morphisms in ${\mathcal{B}}$. Finally, the
claim about local injectivity holds by Remark 3.2.8.

In the situation of (iv), denote by $\widehat{\mathcal{B}}$ the unique
groupoid completion of ${\mathcal{B}}$ provided by (iii). For any pair $x,y\in
B$ with $x\sim_{\mathcal{B}}y$, denote by $x_{\min}\in B$ the root of the
equivalence class $[x]$ so that there are unique morphisms $m_{x}:x_{\min}\to
x,\ m_{y}:x_{\min}\to y$ in ${\mathcal{B}}$. If
$m_{xy}\in\widehat{\mathcal{B}}$ is the unique morphism in ${\rm
Mor}_{\widehat{\mathcal{B}}}(x,y)$, define
$\widehat{f}(m_{xy})=f(m_{x})^{-1}\circ f(m_{y})$. Then $\widehat{f}$ takes
identity maps to identity maps and is compatible with inverses. To see it is
compatible with composition, notice that if
$x\sim_{\mathcal{B}}y\sim_{\mathcal{B}}z$, then

$m_{xz}=m_{xy}\circ m_{yz}=\bigl{(}(m_{x})^{-1}\circ
m_{y}\bigr{)}\circ\bigl{(}(m_{y})^{-1}\circ m_{z}\bigr{)}=(m_{x})^{-1}\circ
m_{z}.$

It follows that

$\widehat{f}(m_{xz})=f(m_{x})^{-1}\circ f(m_{z})=f(m_{x})^{-1}\circ
f(m_{y})\circ f(m_{y})^{-1}\circ
f(m_{z})=\widehat{f}(m_{xy})\circ\widehat{f}(m_{yz}).$

Finally, if ${\mathcal{B}},{\mathcal{X}},f$ are étale, then the induced map
$\widehat{f}:{\bf B}\to\widehat{\bf B}$ is a composite of étale maps and hence
itself étale. ∎

######  Remark 3.2.10.

This notion of groupoid completion is rather weak. For example, if
${\mathcal{B}}$ is a nonsingular category and
$\phi:{\mathcal{B}}\to{\mathcal{X}}$ is any functor whose target
${\mathcal{X}}$ is a groupoid, it is not necessarily true that $\phi$ extends
to the groupoid completion $\widehat{\mathcal{B}}$. Suppose for example that
$B=\\{x_{1},x_{2},x_{3},x_{4}\\}$ and that there are morphisms $m_{ij}\in{\rm
Mor}_{\mathcal{B}}(x_{i},x_{j})$ for $(i,j)=(1,3),(1,4),(2,3),(2,4)$. Then
$\widehat{\mathcal{B}}$ has a unique morphism between any pair of objects.
However, in order for $\phi$ to extend to $\widehat{\mathcal{B}}$ we would
need $\phi(m_{13})\circ\phi(m_{23})^{-1}=\phi(m_{14})\circ\phi(m_{24})^{-1}$,
a relation that may or may not hold. ${\Diamond}$

The following example shows that groupoid completion generally does not
preserve properness, and thus realizations of étale categories are often not
Hausdorff.

######  Example 3.2.11.

Consider the étale category ${\mathcal{B}}$ constructed in Example 3.2.4 with
objects $B=V\sqcup U$ and ${\rm
Mor}_{{\mathcal{B}}}(V,U)\simeq{\widetilde{V}}\subset U$. This category is
nonsingular and $\pi_{\mathcal{B}}:B\to|{\mathcal{B}}|$ is locally injective,
hence ${\mathcal{B}}$ has a groupoid completion $\widehat{\mathcal{B}}$ by
Lemma 3.2.9 (iii). Besides the morphisms in ${\mathcal{B}}$,
$\widehat{\mathcal{B}}$ has morphisms from ${\widetilde{V}}\subset U$ to $V$
given by inverting the elements of ${\rm Mor}_{{\mathcal{B}}}(V,U)$ and also
morphisms with both source and target in $U$ given by the action of $G$ on
${\widetilde{V}}\subset U$:

$\displaystyle{\rm Mor}_{\widehat{\mathcal{B}}}(U,U)$
$\displaystyle=U\sqcup\bigl{(}{\widetilde{V}}\times(G{\smallsetminus}\\{{\rm
id}\\})\bigr{)},$ $\displaystyle s\times t(x,{\gamma})$
$\displaystyle=({\gamma}^{-1}*x,x),\quad x\in{\widetilde{V}},{\gamma}\in
G{\smallsetminus}\\{{\rm id}\\}.$

Thus $\widehat{\mathcal{B}}$ is étale. However, it follows from Example 3.2.4
(ii) that $|\widehat{\mathcal{B}}|=|{\mathcal{B}}|$ is not Hausdorff if ${\rm
Fr}_{U}({\widetilde{V}})\neq\emptyset$ and $G\neq{{\rm id}}$. In this case,
$\widehat{\mathcal{B}}$ is not proper, since properness implies Hausdorffness
by Lemma 3.2.5 (ii) and (iv) and Lemma 3.1.8. ${\Diamond}$

###  3.3. Group actions on étale categories

We begin with a natural notion of the action of a finite group on a category.
Here we denote by ${\rm Aut}({\mathcal{B}},{\mathcal{B}})$ the group of
invertible functors ${\mathcal{B}}\to{\mathcal{B}}$ on a given category
${\mathcal{B}}$. If ${\mathcal{B}}$ is topological, then we require functors
in ${\rm Aut}({\mathcal{B}},{\mathcal{B}})$ to act continuously on object and
morphism spaces. If ${\mathcal{B}}$ is étale, then we require functors in
${\rm Aut}({\mathcal{B}},{\mathcal{B}})$ to be étale.

######  Definition 3.3.1.

An action of a finite group $G$ on a (possibly topological or étale) category
${\mathcal{B}}$ is a homomorphism $F:G\to{\rm
Aut}({\mathcal{B}},{\mathcal{B}}),g\mapsto F_{g}$, that is satisfying
$F_{hg}=F_{g}\circ F_{h}$.

If $G$ acts on two (possibly topological or étale) categories
${\mathcal{B}},{\mathcal{B}}^{\prime}$ via the homomorphisms $F,F^{\prime}$,
then a (continuous or étale) functor
$\psi:{\mathcal{B}}\to{\mathcal{B}}^{\prime}$ is said to be $G$-equivariant if
$F_{g}\circ\psi=\psi\circ F^{\prime}_{g}$ for all $g\in G$.

######  Remark 3.3.2.

Above we write functor composition in categorical notation so that, for
example, $(F_{g}\circ F_{h})(x)=F_{h}(F_{g}(x))$ for $x\in B$. On the other
hand, the action $x\mapsto g*x$ of a group $G$ on a space $X$ satisfies the
identity $(hg)*x=h*(g*x)$.

######  Lemma 3.3.3.

Let $G$ be a finite group.

  * (i)

Any action of $G$ on a topological category ${\mathcal{B}}$ induces a
continuous action of $G$ on the realization $|{\mathcal{B}}|$ of
${\mathcal{B}}$.

  * (ii)

Let ${\mathcal{B}}=(B,{\bf B})$ be a (possibly topological resp. étale)
category. Then $G$ acts on ${\mathcal{B}}$ if and only if $G$ acts on the
spaces $B,{\bf B}$ by (continuous resp. étale) maps $F_{g}:x\mapsto
g*x,F_{g}:m\mapsto g*m$, in such a way that for all $x\in B,\;g,h\in
G,\;m,m^{\prime}\in{\bf B}$ we have

(3.3.1) $\displaystyle g*{\rm id}_{x}={\rm id}_{g*x},\quad$ $\displaystyle
g*(m\circ m^{\prime})=(g*m)\circ(g*m^{\prime}),$ $\displaystyle(s\times
t)(g*m)=\bigl{(}g*s(m),g*t(m)\bigr{)}.$

  * (iii)

Suppose ${\mathcal{X}}=(X,{\bf X})$ is a groupoid and ${\alpha}:G\times
X\to{\bf X}$ is a map with

(3.3.2) $\displaystyle t({\alpha}(g,x))=x,\quad{\alpha}({\rm id},x)={\rm
id}_{x},\quad{\alpha}(hg,x)={\alpha}(g,s({\alpha}(h,x)))\circ{\alpha}(h,x)$

for all $g,h\in G,\;x\in X$. Then this induces an action of $G$ on
${\mathcal{X}}$ as in (ii) with

(3.3.3) $g*x:=s({\alpha}(g^{-1},x)),\qquad
g*m:={\alpha}(g^{-1},s(m))\,\circ\,m\,\circ\,{\alpha}(g,g*t(m))$

for $x\in X$ and $m\in{\bf X}$. If ${\mathcal{X}}$ is a topological resp.
étale groupoid and ${\alpha}$ is continuous resp. étale, then this
construction yields an action of $G$ on the topological resp. étale category
${\mathcal{X}}$. Further, the induced action on $|{\mathcal{X}}|$ is trivial.

  * (iii′)

Suppose ${\mathcal{X}}=(X,{\bf X})$ is a groupoid and
$\overline{\alpha}:G\times X\to{\bf X}$ is a map with

(3.3.4) $\displaystyle s(\overline{\alpha}(g,x))=x,\quad\overline{\alpha}({\rm
id},x)={\rm
id}_{x},\quad\overline{\alpha}(hg,x)=\overline{\alpha}(g,x)\circ\overline{\alpha}(h,t(\overline{\alpha}(g,x)))$

for all $g,h\in G,\;x\in X$. Then this induces an action of $G$ on
${\mathcal{X}}$ as in (ii) with

(3.3.5) $g*x:=t(\overline{\alpha}(g,x)),\qquad
g*m:=\overline{\alpha}(g^{-1},g*s(m))\,\circ\,m\,\circ\,\overline{\alpha}(g,t(m))$

for $x\in X$ and $m\in{\bf X}$. If ${\mathcal{X}}$ is a topological resp.
étale groupoid and ${\alpha}$ is continuous resp. étale, then this
construction yields an action of $G$ on the topological resp. étale category
${\mathcal{X}}$. Further, the induced action on $|{\mathcal{X}}|$ is trivial.

  * (iv)

Let ${\mathcal{B}}=(B,{\bf B})$ be a category that is nonsingular and locally
injective. Suppose that $G$ is a finite group that acts on the object set $B$
and preserves the equivalence relation $x\sim_{\mathcal{B}}y\Longrightarrow
g*x\sim_{\mathcal{B}}g*y$ for all $x,y,g$. Then this action extends uniquely