Smooth Homotopy of Infinite-Dimensional
C ∞ -Manifolds
Hiroshi Kihara

Author address:
Center for Mathematical Sciences, University of Aizu, Tsuruga,
Ikki-machi, Aizu-Wakamatsu City, Fukushima, 965-8580, Japan
Email address: (kihara@u-aizu.ac.jp)

Dedicated to My Parents

Contents
1. Introduction
1.1. Fundamental problems on C ∞ -manifolds
1.2. Main results on C ∞ -manifolds
1.3. Smooth homotopy theory of diffeological spaces
1.4. Notation and terminology
1.5. Organization of the paper
2. Diffeological spaces, arc-generated spaces, and C ∞ -manifolds
2.1. Categories D and C 0
2.2. Fully faithful embedding of C ∞ into D
2.3. Standard p-simplices and model structure on D
2.4. Quillen pairs (| |D , S D ) and ( e·, R)
3. Quillen equivalences between S, D, and C 0
3.1. Singular homology of a diffeological space
3.2. Proof of Theorem 1.5
3.3. Proof of Corollary 1.6
4. Smoothing of continuous maps
4.1. Enrichment of cartesian closed categories
4.2. Simplicial categories C 0 and D
4.3. Function complexes and homotopy function complexes for C 0 and D
4.4. Proof of Theorem 1.7
5. Smoothing of continuous principal bundles
5.1. C-partitions of unity
5.2. Principal bundles in C
5.3. Fiber bundles in C
5.4. Smoothing of principal bundles
6. Smoothing of continuous sections
6.1. Quillen equivalences between the overcategories of S, D, and C 0
6.2. Enrichment of overcategories
6.3. Simplicial categories C 0 /X and D/X
6.4. Function complexes and homotopy function complexes for C 0 /X andD/X
6.5. Proof of Theorem 1.8
e X)
e num
7. Dwyer-Kan equivalence between (PDG/X)num and (PC 0 G/
7.1. Enrichment of categories embedded into M
7.2. Enriched groupoid PMG/X
7.3. Smoothing of gauge transformations
7.4. Proof of Theorem 1.9
8. Diffeological polyhedra
8.1. Basic properties of two kinds of diffeological polyhedra
v

1
1
4
5
8
9
10
10
12
14
17
18
18
22
25
26
26
27
29
35
35
36
37
39
40
42
42
43
47
48
51
52
52
53
54
56
56
57

vi

HIROSHI KIHARA

8.2. D-homotopy equivalence between two kinds of diffeological polyhedra 60
9. Homotopy cofibrancy theorem
65
9.1. Diffeological spaces associated to a covering
65
9.2. Hurewicz cofibrations in D
71
9.3. Proof of Theorem 1.1077
10. Locally contractible diffeological spaces
79
11. Applications to C ∞ -manifolds
83
11.1. Proofs of Theorems 1.1-1.3
83
11.2. Classical atlases
84
11.3. Hereditary C ∞ -paracompactness
85
11.4. Hereditarily C ∞ -paracompact, semiclassical C ∞ -manifolds
89
Appendix A. Pathological diffeological spaces
100
104
Appendix B. Keller’s Cc∞ -theory and diffeological spaces
Appendix C. Smooth regularity and smooth paracompactness
106
Bibliography

110

Abstract
In this paper, we use homotopical algebra (or abstract homotopical methods)
to study smooth homotopical problems of infinite-dimensional C ∞ -manifolds inconvenient calculus. More precisely, we discuss the smoothing of maps, sections,principal bundles, and gauge transformations.
We first introduce the notion of hereditary C ∞ -paracompactness along with thesemiclassicality condition on a C ∞ -manifold, which enables us to use local convexity
in local arguments. Then, we prove that for C ∞ -manifolds M and N , the smooth
singular complex of the diffeological space C ∞ (M, N ) is weakly equivalent to theordinary singular complex of the topological space C 0 (M, N ) under the hereditary
C ∞ -paracompactness and semiclassicality conditions on M . We next generalize
this result to sections of fiber bundles over a C ∞ -manifold M under the same
conditions on M . Further, we establish the Dwyer-Kan equivalence between thesimplicial groupoid of smooth principal G-bundles over M and that of continuous
principal G-bundles over M for a Lie group G and a C ∞ -manifold M under thesame conditions on M , encoding the smoothing results for principal bundles andgauge transformations.
For the proofs, we fully faithfully embed the category C ∞ of C ∞ -manifolds into
the category D of diffeological spaces and develop the smooth homotopy theory ofdiffeological spaces via a homotopical algebraic study of the model category D andthe model category C 0 of arc-generated spaces, also known as ∆-generated spaces.
Then, the hereditary C ∞ -paracompactness and semiclassicality conditions on M
imply that M has the smooth homotopy type of a cofibrant object in D. This
result can be regarded as a smooth refinement of the results of Milnor, Palais,and Heisey, which give sufficient conditions under which an infinite-dimensional
topological manifold has the homotopy type of a CW -complex. We also show thatmost of the important C ∞ -manifolds introduced and studied by Kriegl, Michor,
and their coauthors are hereditarily C ∞ -paracompact and semiclassical, and hence,
results can be applied to them.

Received by the editor February 5, 2020.
2010 Mathematics Subject Classification. Primary 58B05; Secondary 58A40,18G55.
Key words and phrases. Smooth homotopy, C ∞ -manifolds, convenient calculus, diffeologicalspaces, model category.
I would like to show my greatest appreciation to Prof. Toshiro Watanabe who offered continuing support and constant encouragement.
vii

1. INTRODUCTION

1

1. Introduction
This paper aims to develop a smooth homotopy theory of diffeological spaces
and apply it to global analysis on infinite-dimensional C ∞ -manifolds by embedding
C ∞ -manifolds fully faithfully into the category of diffeological spaces.
In Section 1.1, we formulate fundamental problems on C ∞ -manifolds, and in
Section 1.2, we provide answers to these problems, as the main results on C ∞ manifolds. In Section 1.3, we outline the results obtained herein on smooth homotopy for diffeological spaces, to which most of this paper is devoted, and explain
how they yield the results in Section 1.2.
1.1. Fundamental problems on C ∞ -manifolds. Frölicher, Kriegl, and Michor [46] established the foundation of infinite-dimensional calculus, which is called
convenient calculus and is regarded as a prime candidate for the final theory of
infinite-dimensional calculus. However, it has no efficient approach for solving oneof the most critical problems: to investigate how many smooth maps exist between
the given infinite-dimensional C ∞ -manifolds M and N . Since the study of continuous maps between M and N is done by topological homotopy theory (or algebraic
topology), we formulate the problem as follows:(a) When do the smooth homotopy classes of smooth maps between M and N
bijectively correspond to the continuous homotopy classes of continuous
maps ?
The following two problems are also important; Problem (b) is a generalization of
Problem (a), and Problem (c) is closely related to Problems (a) and (b).
(b) Let p : E −→ M be a smooth fiber bundle. When do the vertical smoothhomotopy classes of smooth sections of E bijectively correspond to thevertical continuous homotopy classes of continuous sections ?
(c) Let G be a Lie group. When do the isomorphism classes of smooth principal G-bundles over M bijectively correspond to those of continuous principal G-bundles over M ? Let π : P −→ M be a smooth principal G-bundle.
When do the isotopy classes of smooth gauge transformations of P bijectively correspond to those of continuous gauge transformations of P ?
If all C ∞ -manifolds involved are finite-dimensional, then the correspondences in
Problems (a) and (b) are always bijective by the Steenrod approximation theorem([66, Section 6.7]), which is one of the most basic results in differential topology.
In more general settings, Problems (a), (b), and (c) were addressed in [47], [71],and [57, 70], respectively. Roughly speaking, as answers to these questions, it has
been shown that the correspondences in the questions are bijective, provided that
M is finite-dimensional. Precisely, Müller and Wockel [57, 70, 71] did not workin convenient calculus but in Keller’s Cc∞ -theory; moreover, to prove the smoothing result of continuous gauge transformations, Wockel imposed even compactness
condition on M along with additional conditions on G [70, Proposition 1.20]. However, no essential answer is known in the case where M is infinite-dimensional since
the existing approaches are essentially based on the finite dimensionality (or local
compactness) of M (cf. [71, Section 1] and [57, Introduction]).
In the rest of this subsection, we more precisely formulate the fundamental
problems mentioned above; we actually address the higher homotopical versions ofProblems (a)-(c) by observing that the involved relevant categories and functors
can be enriched over the category S of simplicial sets.

2

HIROSHI KIHARA

Throughout this paper, C ∞ -manifolds are ones in the sense of [46, Section 27]
unless stated otherwise, and C ∞ denotes the category of (separated) C ∞ -manifolds(see Section 2.2). Lie groups are defined as groups in C ∞ ([49, p. 75]). The underf of a C ∞ -manifold M is defined as the set M endowed
lying topological space M
with the final topology for the smooth curves ([46, 27.4]). Then, we have theunderlying topological space functor e· : C ∞ −→ C 0 , where C 0 is the category ofarc-generated spaces and continuous maps (see Section 2.1).
Since the category C ∞ is not closed under various categorical constructions, wefully faithfully embed C ∞ into the category D of diffeological spaces (see Section
e of a diffeological space X is2.2). Recall that the underlying topological space X
defined to be the set X endowed with the final topology for the diffeology DX .
Then, we can see that the fully faithful embedding C ∞ ,−→ D and the underlying
topological space functors for C ∞ and D form the commutative diagram
C∞

D
e·

C0

e·

consisting of functors that preserve finite products (Proposition 2.6). Because ofthis fact and the convenient properties of D and C 0 (Section 2.1), we use the categories D and C 0 as convenient categories of smooth spaces and topological spaces,
respectively.
Now, we shall discuss the S-enrichment of categories embedded into D and C 0 .
In [43], we constructed the standard simplices ∆p (p ≥ 0), which satisfy several
good properties, and used them to introduce the adjoint pair
| |D : S  D : S D
of the realization and singular functors; this is a diffeological analog of the adjointpair| | : S  C0 : Sof the topological realization and singular functors (see Section 2.3). Let M denote
one of the categories D and C 0 and set((| |D , S D ) for M = D,(| |M , S M ) =(| |, S)for M = C 0 .
Since M is cartesian closed, M is itself an M-category ([4, Proposition 6.2.6])
and hence an S-category via S M . Further, every category endowed with a faithful
functor into M has the canonical M-category and hence S-category structure (seeSection 7.1), and the underlying topological space functor e· : D −→ C 0 and its
restrictions to subcategories are S-functors (see Lemma 4.2 and Section 7.1). Using
these facts, we formulate higher homotopical versions of Problems (a)-(c).
Since e· : D −→ C 0 is an S-functor and C ∞ is an S-full subcategory of D via
the fully faithful embedding C ∞ ,−→ D, we have the S-functor
e· : C ∞ −→ C 0 ,which induces the natural inclusion of function complexes
f, N
e ).
e· : S D C ∞ (M, N ) = S D D(M, N ) ,−→ SC 0 (M

1. INTRODUCTION

3

Furthermore, since π0 (e·) is just the natural map from the smooth homotopy classesof smooth maps to the continuous homotopy classes of continuous maps (see Section 4.2), the following is a precise and higher homotopical formulation of Problem
(a).
f, N
e)
Problem A. Under what condition is the map e· : S D C ∞ (M, N ) ,−→ SC 0 (Ma weak equivalence of simplicial sets ?
Since the overcategory M/X has the canonical faithful functor into M for any
X ∈ M, M/X has the canonical M-category and hence S-category structure. The
hom-object M/X(D, E) of the M-category M/X is often denoted by Γ(X, E)
e is an S-functor for any
in the case where D = X. Since e· : D/X −→ C 0 /X∞
X ∈ D and C /M is an S-full subcategory of D/M via the fully faithful embedding
C ∞ /M ,−→ D/M , we have the S-functorf,e· : C ∞ /M −→ C 0 /M
which induces the natural inclusion of function complexesf, E).
ee· : S D Γ(M, E) ,−→ SΓ(MFurthermore, since π0 (e·) is just the natural map from the vertical smooth homotopy
classes of smooth sections to the vertical continuous homotopy classes of continuous
sections (Section 6.3), the following is a precise and higher homotopical formulationof Problem (b).
Problem B. Let p : E −→ B be a smooth fiber bundle of C ∞ -manifolds. Underf, E)
e a weak equivalence ofwhat condition is the map e· : S D Γ(M, E) ,−→ SΓ(Msimplicial sets ?
For an object X of M and a group G in M, PMG/X denotes the category of
principal G-bundles over X in M (see Definition 5.4 for the notions of smooth andcontinuous principal bundles referred to here). Since PMG/X has the canonicalfaithful functor into M, PMG/X is an M-category and hence an S-category.
Moreover, it is shown that the underlying topological space functor defines thesimplicial functore Xee· : PDG/X −→ PC 0 G/
(see Lemma 5.7 and Remark 5.8). If G is a Lie group, then the category PC ∞ G/Mof principal G-bundles over M in C ∞ is regarded as a D-category and hence an
S-category via the isomorphism PC ∞ G/M ∼= PDG/M (see Remark 5.5(3)). Therefore, we have the simplicial functore M
f.
e· : PC ∞ G/M −→ PC 0 G/
We thus consider the following problem. See Section 7.4 for the definition of a
Dwyer-Kan equivalence between simplicial categories.
Problem C Let G be a Lie group. Under what condition is the simplicial functore M
fe· : PC ∞ G/M −→ PC 0 G/
a Dwyer-Kan equivalence?

4

HIROSHI KIHARA

Now, we explain that Problem C is a precise formulation of a higher homotopical version of Problem (c). Observing that PMH/X is an M-groupoid, we
define the gauge group GauM (P ) of a principal H-bundle π : P −→ X in M
to be the automorphism group AutPMH/X (P ) (see Section 7.2). If PMH/X isregarded as an S-groupoid (or simplicial groupoid) via S M , then the automorphism group AutPMH/X (P ) is just S M GauM (P ). Thus, that the simplicial functore M
f between simplicial groupoids is a Dwyere· : PC ∞ G/M = PDG/M −→ PC 0 G/
Kan equivalence is equivalent to the following:ef admits a unique (up to iso• Every continuous principal G-bundle
over Mmorphism) smooth principal G-bundle structure over M .
• For any smooth principal G-bundle P over M , the natural inclusion
S D GauC ∞ (P ) = S D GauD (P ) ,−→ SGauC 0 (Pe)is a weak equivalence of simplicial sets.
(See Section 7.4).
1.2. Main results on C ∞ -manifolds. In this subsection, we state the main
results on C ∞ -manifolds, which constitute answers to Problems A, B, and C. To
this end, we begin by introducing the basic notions on C ∞ -manifolds.
A C ∞ -manifold M is called hereditarily C ∞ -paracompact if any open submanifold of M is C ∞ -paracompact (see Definition 5.1). A C ∞ -manifold M is called
semiclassical if M admits an atlas {(Uα , uα )}α∈A such that uα (Uα ) and uα (Uα ∩Uβ )are open in the model vector space Eα with respect to the locally convex topology
for any α, β ∈ A. (See Section 11.2 for the notion of a classical C ∞ -manifold.)
The following theorem answers Problem A.
Theorem 1.1. Let M and N be C ∞ -manifolds. If M is hereditarily C ∞ paracompact and semiclassical, then the natural inclusionf, Ne)
S D C ∞ (M, N ) ,−→ SC 0 (Mis a weak equivalence of simplicial sets.
The following theorem, which is a smoothing result of continuous sections,answers Problem B, generalizing Theorem 1.1.
Theorem 1.2. Let p : E −→ M be a smooth fiber bundle of C ∞ -manifolds. If
M is hereditarily C ∞ -paracompact and semiclassical, then the natural inclusionf, E)e
S D Γ(M, E) ,−→ SΓ(Mis a weak equivalence of simplicial sets.
Lastly, we provide an answer to Problem C.
Theorem 1.3. Let M be a C ∞ -manifold and G a Lie group. If M is hereditarily
C -paracompact and semiclassical, then the functor∞e M
fe· : PC ∞ G/M −→ PC 0 G/
is a Dwyer-Kan equivalence of simplicial groupoids.
Kriegl, Michor, and their coauthors introduced and studied many C ∞ -manifolds
which play vital roles in global analysis and differential geometry ([46, Chapter IX
and Section 47]). In this paper, we show that most of those C ∞ -manifolds are1. INTRODUCTION

5

hereditarily C ∞ -paracompact and semiclassical (see Section 11.4); therefore, Theorems 1.1-1.3 apply to them.
Remark 1.4. In spite of the importance of C ∞ -paracompactness in global analysis ([46, Chapter III]), few C ∞ -paracompact C ∞ -manifolds are known except finite dimensional C ∞ -manifolds and several kinds of convenient vector spaces. We
thus develop the basics of hereditary C ∞ -paracompactness (Appendix C and Section
11.3), and then establish the hereditary C ∞ -paracompactness of important C ∞ manifolds such as the C ∞ -manifolds C∞ (M, N ) and C ω (M, N ) of maps and theC ∞ -manifolds B(M, N ) and B ω (M, N ) of submanifolds (see Section 11.4); ourapproach also applies to paracompact Hilbert manifolds and direct limit manifolds
of finite dimensional C ∞ -manifolds. Note also that a C ∞ -manifold X is hereditarily C ∞ -paracompact if and only if every submanifold of X is C ∞ -paracompact(Corollary 11.11).
1.3. Smooth homotopy theory of diffeological spaces. Since C ∞ is a fullsubcategory of D (Proposition 2.6) and D is a model category which has various
convenient properties, we develop smooth homotopy theory of diffeological spaces
and prove Theorems 1.1-1.3 by applying the theory to C ∞ -manifolds. Precisely, inthis subsection, we outline the smooth homotopical results on diffeological spaces,and explain how Theorems 1.1-1.3 are deduced from them.
Our smooth homotopy theory primarily concerns simplicial categorical properties of objects and morphisms of D and C 0 (see Problems A-C and Theorems
1.1-1.3). However, the model categorical study of D and C 0 plays a crucial rolein many aspects of the theory. Thus, we begin by outlining the model categorical
relations between S, D, and C 0 .
In [43], a compactly generated model structure on the category D of diffeological spaces was constructed. In the study, the adjoint pair
| |D : S  D : S D
of the realization and singular functors, and the adjoint paire· : D  C 0 : Rof the underlying topological space functor and its right adjoint were introduced.
The composite of these adjoint pairs is just the adjoint pair
| | : S  C0 : Sof the topological realization and singular functors (see Section 2.3).
In the following theorem, we show that the model categories S, D, and C 0 are
Quillen equivalent via (| |D , S D ) and (˜·, R). See [34, 53] for model categories andQuillen equivalences, [42, 30, 53] for the model structure of S, and Remark 2.18
for the model structure of C 0 .
Theorem 1.5.
(1) | |D : S  D : S D is a pair of Quillen equivalences.
0(2) ˜· : D  C : R is a pair of Quillen equivalences.
Theorem 1.5 is the starting point of our homotopical study of diffeologicalspaces, which shows that the Quillen homotopy categories of D and C 0 are equivalent
via the total derived functors of ˜· and R; however, it does not guarantee thatS D D(X, Y ) is homotopically equivalent to SC 0 (X̃, Ỹ ) for X, Y ∈ D. In fact, thee Ye ) need not be a weak equivalence (see [8,natural inclusion S D (X, Y ) ,→ SC 0 (X,6

HIROSHI KIHARAExamples 3.12 and 3.20] and Appendix A). Thus, to establish a smoothing theorem
for continuous maps, we must give a sufficient condition under whiche Ye )
e· : S D D(X, Y ) ,−→ SC 0 (X,is a weak equivalence in S. For this, we introduce and study two subclasses of D.
One is the subclass WD defined by
WD = {A ∈ D | A has the D-homotopy type of a cofibrant diffeological space},
where the notion of a D-homotopy type (or smooth homotopy type) is defined using
the unit interval I (∼= ∆1 ) in the obvious manner (Section 2.3 and [43, Section 2.4]).
The class WD is a diffeological version of Milnor’s class W of topological spaces thathave the homotopy type of a CW -complex [56].
The other is the subclass VD defined bye is a weak equivalence in D ;VD = A ∈ D | id : A −→ RA
e is the unit of the adjoint pair (e·, R), which iswhere the map id : A −→ RA
set-theoretically the identity map (see Remark 2.3(2)).
For the description of the following corollary, we also introduce the subclass
WC 0 of C 0 defined by
WC 0 = {X ∈ C 0 | X has the homotopy type of a cofibrant arc-generated space}.
It should be noted that WC 0 is just the intersection W ∩ C 0 (see Remark 2.18 ande[43, Section 2.2]). Also note that the singular complex S D A is a subcomplex of S A
D(Lemma 3.2(1)) and let πp (A, a) denote the smooth homotopy group of a pointed
diffeological space (A, a) ([8], [38]).
e is in WC 0 .
Corollary 1.6.
(1) If A is in WD , then A(2) If A is in WD , then A is in VD .
(3) The following conditions are equivalent:(i) A is in VD ;e is a weak equivalence in S;(ii) The inclusion S D A ,−→ S A(iii) The natural homomorphisme a)πpD (A, a) −→ πp (A,is an isomorphism for any a ∈ A and any p ≥ 0.
To understand why the class WD is needed, see Remark 3.14. See also Appendix
A for diffeological spaces which are not in WD and VD .
We can now state the smoothing theorem for continuous maps.
Theorem 1.7. Let A and X be diffeological spaces. If A is in WD and X is in
VD , then the natural inclusion
S D D(A, X) ,−→ SC 0 (Ã, X̃)
is a weak equivalence in S.
Theorem 1.7 generalizes to the following smoothing theorem for continuous
sections. See Section 5 for the notion of a D-numerable F -bundle in D.

1. INTRODUCTION

7

Theorem 1.8. Let p : E −→ X be a D-numerable F -bundle in D. If X is in
WD and F is in VD , then the natural inclusione E)e
S D Γ(X, E) ,−→ SΓ(X,
is a weak equivalence in S.
Next, we state a smoothing theorem for principal bundles and gauge transformations. Let M denote one of the categories D and C 0 . (PMG/X)num denotes
the full subcategory of PMG/X consisting of M-numerable principal G-bundles
over X in M (Definition 5.4), which has the canonical M-full subcategory andhence S-full subcategory structure of PMG/X. For a diffeological space X ande Xe (see Section 1.1)
a diffeological group G, the S-functor e· : PDG/X −→ PC 0 G/
obviously restricts to the S-functor
e X)
e num .
e· : (PDG/X)num −→ (PC 0 G/
Theorem 1.9. Let X be a diffeological space and G a diffeological group. If X
is in WD and G is in VD , then the functor
e X)
e num
e· : (PDG/X)num −→ (PC 0 G/
is a Dwyer-Kan equivalence of simplicial groupoids.
To deduce Theorems 1.1-1.3 from Theorems 1.7-1.9, we must establish the following two theorems. See Definition 5.1 for a D-numerable covering and Definition
10.1 for a locally contractible diffeological space.
Theorem 1.10. Let X be a diffeological space and U = {Uα }α∈A a D-numerable
covering of X by subsets. If Uσ (:= ∩ Uα ) is in WD for any nonempty finite σ ⊂ A,α∈σthen X is in WD .
Theorem 1.11. Every locally contractible diffeological space is in VD .
Theorems 1.10-1.11 imply the following important facts:• Every hereditarily C ∞ -paracompact, semiclassical C ∞ -manifold is in WD(Theorem 11.1); see Section 1.2 for the definitions of hereditary C ∞ paracompactness and semiclassicality.
• Every C ∞ -manifold is in VD (Theorem 11.2).
By these facts, Theorems 1.7-1.9 easily imply Theorems 1.1-1.3 (see Section 11.1).
The smooth homotopical resutls outlined here are also applicable to the theoryof orbifolds [39], which we shall discuss in a subsequent study in this direction. The
following remark relates to the relationship between the smooth homotopy theory
discussed in this paper and topological homotopy theory.
Remark 1.12.
(1) Theorem 1.10 is a diffeological version of a theoremof tom Dieck [67, Theorem 4].
(2) Theorem 11.1 mentioned above and its variant (Proposition 11.4) can be
regarded as smooth refinements of the results of Palais, Heisey, and Milnoron the homotopy types of infinite-dimensional topological manifolds (seeRemark 11.19).
f of a hereditarily C ∞ -paracompact,(3) Since the underlying topological space M∞
semiclassical C -manifold M has the homotopy type of a CW -complex(Theorem 11.1 and Corollary 1.6(1)), Theorem 1.1 enables us to study
the homotopy of S D C ∞ (M, N ) via algebraic topological methods.

8

HIROSHI KIHARA

We end this subsection with the following remark on the relation between thesimplicial category structure and the model structure on M = D, C 0 .
Remark 1.13. The categories D and C 0 have simplicial category structuresand model structures, both of which define the homotopical notions on objects andmorphisms (see Section 4.3).
As mentioned in the beginning of this subsection, we are primarily concernedwith simplicial categorical properties of objects and morphisms of D and C 0 . However, the methods and results of model category theory are needed in many aspectsof our smooth homotopy theory. Since C 0 and its overcategory C 0 /X are simplicial
model categories (Corollaries 4.4 and 6.10), we can do homotopy theory in thesecategories without considering the differences between simplicial categorical notions
and model categorical notions (see Propositions 4.7 and 6.12 and their proofs).
However, since neither D nor its overcategory D/X is a simplicial model category(Corollaries 4.4, 6.10 and Remark 4.5), we need subtle arguments comparing thefunction complexes with the homotopy function complexes for these categories (seeTheorems 4.8 and 6.13 and their proofs).
1.4. Notation and terminology. We follow the notation and terminologyof [43], much of which is reviewed in Section 2 of this paper. Here, we recall someof the most important ones while introducing new notations.
The most important categories are the following:
The category D of diffeological spaces (Section 2.1),
The category C 0 of arc-generated spaces (Section 2.1),
The category S of simplicial sets.
In this paper, M denotes one of the categories D and C 0 . | |M : S  M : S M
denotes the adjoint pair of the realization and singular functors for M; explicitly,((| |D , S D ) for M = D,
M(| |M , S ) =(| |, S)for M = C 0(see Section 2.3).
The categories D and C 0 have the obvious underlying set functors. Thus, we use
the terminology of [24, Sections 8.7-8.8], which relates to initial and final structureswith respect to the underlying set functor, to deal with the categories D and C 0
simultaneously (see Proposition 2.2 and Remarks 2.3(1) and 2.4).
For a concrete category X, the following notation is used in this paper:(1) For X ∈ X, 1X (or 1) denotes the identity morphism of X.
(2) For X1 , X2 ∈ X having the same underlying set, id : X1 −→ X2 is the
set-theoretic identity map or the morphism whose underlying set-theoretic
map is the identity map.
Our basic reference for enriched category theory is [4, Chapter 6]. An Scategory and an S-functor are often called a simplicial category and a simplicial
functor respectively. However, by a diffeological category, we mean a category
object in D, not a D-category (see Section 9.1).
Our basic references for model category theory are [34, 53]. For a model
category E, the symbols Ec , Ef , and Ecf denote the full subcategories of E consistingof cofibrant objects, fibrant objects, and fibrant-cofibrant objects, respectively.
The symbol ∅ (resp. ∗) is used to denote an initial (resp. terminal) object of acategory.

1. INTRODUCTION

9For C 0 -homotopies h, k : X × I −→ Y with h(·, 1) = k(·, 0), the (vertical)
composite of h and k is defined in the obvious manner (see [49, p. 273]). On
the other hand, we always compose two D-homotopies h, k : X × I −→ Y withh(·, 1) = k(·, 0) smoothly using a cut-off function ([8, Definition 3.1]).
We end this subsection with the topological notation. For a subset S of a
topological space X, S̄ and S ◦ denote the closure and the interior of S respectively.
For a continuous function f : X −→ R, we define carr f and supp f bycarr f = {x ∈ X| f (x) 6= 0} and supp f = carr f(cf. [46, p. 165]).
1.5. Organization of the paper. This paper is organized as follows.
In Section 2, we recall the basic notions and results of diffeological spaces andarc-generated spaces from [43] and observe that the category C ∞ can be fully
faithfully embedded into D. We also see that | |D : S  D : S D and e· : D  C 0 : R
are Quillen pairs.
Sections 3 and 4 contain the main ideas of the smooth homotopy theory ofdiffeological spaces. In Section 3, we prove Theorem 1.5 and Corollary 1.6. In
Section 4, we study the simplicial category structures on C 0 and D. Then, we
compare function complexes and homotopy function complexes for M = C 0 , D,which enables us to prove Theorem 1.7 by replacing the function complexes withthe homotopy function complexes. Since D is not a simplicial model category(Corollary 4.4(2)), the arguments for D, which use Theorem 1.5, are much more
subtle than those for C 0 .
Sections 5-7 contain applications and generalizations of the results establishedin Section 4. In Section 5, we recall the basic facts on the classification theory of(smooth and continuous) principal bundles and use Theorem 1.7 to prove that under
e X)
e num
the assumption as in Theorem 1.9, the functor e· : (PDG/X)num −→ (PC 0 G/
induces the bijection on the isomorphism classes of objects (Theorem 5.17). In
Section 6, we compare function complexes and homotopy function complexes for
M/X (= C 0 /X, D/X), generalizing several results in Section 4. This leads to theproof of Theorem 1.8. In Section 7, we prove Theorem 1.9 using Theorem 1.8 alongwith Theorem 5.17.
Sections 8-10 are devoted to the proofs of Theorems 1.10 and 1.11, which enable
us to apply smooth homotopical results of diffeological spaces to C ∞ -manifolds. In
Section 8, we introduce and study the two kinds of notions of diffeological polyhedra for the proofs of Theorems 1.10 and 1.11. In Section 9, we first establish adiffeological version of [62, Proposition 4.1] (Proposition 9.5) using the results in
Section 8, and then introduce and study the notion of a Hurewicz cofibration in
D. Using the results of these studies, we prove Theorem 1.10. In Section 10, we
introduce the notion of a locally contractible diffeological space and prove Theorem
1.11.
In Section 11, we first establish two theorems (Theorems 11.1 and 11.2) from
Theorems 1.10 and 1.11. Then, Theorems 1.1-1.3 follow immediately from Theorems 1.7-1.9. We also precisely investigate semiclassicality and hereditary C ∞ paracompactness conditions, showing that many important C ∞ -manifolds satisfy
these two conditions.
The following chart indicates the interdependencies of the sections.

10

HIROSHI KIHARA

Section 2
Section 3
Section 4

Section 8

Section 5
Section 6

Sections 9 and 10

Section 7
Section 11
2. Diffeological spaces, arc-generated spaces, and C ∞ -manifolds
We recall the convenient properties of diffeological spaces and arc-generated
spaces, and then show that the category C ∞ of (separated) C ∞ -manifolds can be
fully faithfully embedded into the category D of diffeological spaces. We also makea review on the compactly generated model structure on D ([43]) and observe thatthe adjoint pairs | |D : S  D : S D and e· : D  C 0 : R are Quillen pairs.
2.1. Categories D and C 0 . In this subsection, we recall the definition of adiffeological space and summarize the basic properties of the categories D and C 0 ,and the underlying topological space functor e· : D −→ C 0 .
Let us begin with the definition of a diffeological space. A parametrization of
a set X is a (set-theoretic) map p : U −→ X, where U is an open subset of Rn forsome n.
Definition 2.1.
(1) A diffeological space is a set X together with a specified set DX of parametrizations of X satisfying the following conditions:(i) (Covering) Every constant parametrization p : U −→ X is in DX .
(ii) (Locality) Let p : U −→ X be a parametrization such that there
exists an open cover {Ui } of U satisfying p|Ui ∈ DX . Then, p is inDX .
(iii) (Smooth compatibility) Let p : U −→ X be in DX . Then, for everyn ≥ 0, every open set V of Rn and every smooth map F : V −→ U ,p ◦ F is in DX .
The set DX is called the diffeology of X, and its elements are called plots.
(2) Let X = (X, DX ) and Y = (Y, DY ) be diffeological spaces, and let f :X −→ Y be a (set-theoretic) map. We say that f is smooth if for any
p ∈ DX , f ◦ p ∈ DY .
The convenient properties of D are summarized in the following proposition.
Recall that a topological space X is called arc-generated if its topology is final
for the continuous curves from R to X and that C 0 denotes the category of arcgenerated spaces and continuous maps. See [24, pp. 230-233] for initial and final
structures with respect to the underlying set functor.

2. DIFFEOLOGICAL SPACES, ARC-GENERATED SPACES, AND C ∞ -MANIFOLDS

11

Proposition 2.2.
(1) The category D has initial and final structureswith respect to the underlying set functor. In particular, D is complete
and cocomplete.
(2) The category D is cartesian closed.
(3) The underlying set functor D −→ Set is factored as the underlying topological space functor e· : D −→ C 0 followed by the underlying set functor C 0 −→ Set. Further, the functor e· : D −→ C 0 has a right adjoint
R : C 0 −→ D.
Proof. See [7, p. 90], [38, pp. 35-36], and [43, Propositions 2.1 and 2.10].



The following remark relates to Proposition 2.2
Remark 2.3.
(1) Let X be a concrete category (i.e., a category equippedwith a faithful functor to Set); the faithful functor X −→ Set is called
the underlying set functor. See [24, Section 8.8] for the notions of an Xembedding, an X-subspace, an X-quotient map, and an X-quotient space.
D-subspaces and D-quotient spaces are usually called diffeological subspaces and quotient diffeological spaces, respectively.
(2) For Proposition 2.2(3), recall that the underlying topological space Ã of adiffeological space A = (A, DA ) is defined to be the set A endowed withthe final topology for DA and that R assigns to an arc-generated space X
the set X endowed with the diffeology
DRX = {continuous parametrizations of X}.
e ofThen, we can easily see that ˜· ◦ R = IdC 0 and that the unit A −→ RA
the adjoint pair (e·, R) is set-theoretically the identity map.
(3) Iglesias-Zemmour and many authors following him used the categoricalconvenience of D to deal with various differential-geometric and globalanalytic problems (see, eg, [37, 38, 39]).
The following remark explains that C 0 is the most suitable category as a target
category of the underlying topological space functor for diffeological spaces.
Remark 2.4. The category C 0 has the following good properties:(1) C 0 has initial and final structures with respect to the underlying set functor.
In particular, C 0 is complete and cocomplete ([43, Proposition 2.6]).
(2) C 0 is cartesian closed ([43, Proposition 2.9]).
(3) A topological space X is in C 0 if and only if X is the underlying topologicalspace of some diffeological space (Remark 2.3(2)).
(4) The underlying topological space functor e· : D −→ C 0 preserves finite
products ([43, Proposition 2.13]).
The category T of topological spaces does not have property (2), (3), or (4). The
most important property is property (2), which is essential for the proof of property(4). However, other well-known categories of topological spaces having property (2)usually consist of topological spaces satisfying some separation axiom ([52, Chapter5], [25]), and hence cannot be the target category of the underlying topological spacefunctor for D.
The above good properties of C 0 are efficiently used in this paper. Properties(2) and (4) lead us to Lemmas 4.2, 6.8, and 5.7, which are the starting points of

12

HIROSHI KIHARA

our smoothing theorems for maps, sections, and principal bundles respectively (seealso Remark 5.8).
The categories D and C 0 share properties (1) and (2), which often enables us
to deal with D and C 0 simultaneously (see Sections 5.3, 6, and 7).
We end this subsection by introducing the separation condition for arc-generated
spaces. An arc-generated space X is called separated if the diagonal is closed in
the product X ×C 0 X. Separatedness is stronger than T1 -axiom and weaker than
Hausdorff property. For a group G in C 0 , separatedness is equivalent to T1 -axiom.
We can also observe the following closure properties of separatedness:• If arc-generated spaces X and Y are separated, then X ×C 0 Y is also
separated.
• If an arc-generated space A admits a continuous injection into a separated
arc-generated space, then A is also separated.
2.2. Fully faithful embedding of C ∞ into D. In this subsection, we recall
the notion of a C ∞ -manifold (in convenient calculus) from [46] and observe thatthe category C ∞ of (separated) C ∞ -manifolds can be fully faithfully embedded into
the category D.
First, we make a review on the local calculus. Let E be a locally convex vector
space. A set-theoretic curve c : R −→ E is called smooth if all derivatives exist
and are continuous - this is a concept without problems. The final topology for theset of smooth curves into E is called the c∞ -topology of E, which is finer than theoriginal locally convex topology; for a Fréchet space, the c∞ -topology coincides withthe original locally convex topology ([46, Theorem 4.11(1)]). A set-theoretic map
f : U −→ V between c∞ -open sets of locally convex vector spaces are defined to besmooth or C ∞ if f maps smooth curves in U to smooth curves in V . In the finite
dimensional case, this gives the usual notion of smooth mappings ([46, Corollary3.14]). Smooth maps are obviously continuous for the c∞ -topologies; however, asmooth map need not be continuous for the original locally convex topologies ([46,Corollary 2.11]).
Next, we recall the notion of a C ∞ -manifold. In convenient calculus, C ∞ manifolds are defined by gluing c∞ -open sets of convenient vector spaces, which arelocally convex vector spaces satisfying a week completeness condition ([46, Theorem
2.14]). The precise definition of a C ∞ -manifold is as follows:u
A chart (U, u) on a set M is a bijection M ⊃ U −→ u(U ) ⊂ EU from a∞subset U of M onto a c -open subset of a convenient vector space EU . A family{(Uα , uα )}α∈A of charts on M is called an atlas for M if M = ∪ Uα and all chartchangings
uαβ := uα ◦ u−1
β : uβ (Uα ∩ Uβ ) −→ uα (Uα ∩ Uβ )are smooth maps between c∞ -open subsets. Two atlas are called equivalent if theirunion is again an atlas for M . An equivalence class of atlas is called a C ∞ -structure
on M . A C ∞ -manifold M is a set together with a C ∞ -structure on it.
Smooth maps between C ∞ -manifolds M and N are defined using charts in theobvious manner. Thus, the category C ∞ -mf d of C ∞ -manifolds and smooth maps
are defined. We can easily see that a set-theoretic map f : M −→ N is smooth if
and only if f preserves smooth curves.
f of a C ∞ -manifold M is defined to be the
The underlying topological space M
set M endowed with the final topology for the smooth curves ([46, 27.4]).

2. DIFFEOLOGICAL SPACES, ARC-GENERATED SPACES, AND C ∞ -MANIFOLDS

13

Now we introduce a fully faithful embedding of C ∞ -mf d into D. Define thefunctor I : C ∞ -mf d −→ D by assigning to a C ∞ -manifold M the set M endowedwith the diffeology
DIM = { C ∞ -parametrizations of M }.
Lemma 2.5. The functor I : C ∞ -mf d −→ D is a fully faithful functor which
f of a C ∞ preserves finite products. Further, the underlying topological space M
gmanifold M is just the underlying topological space IM of the diffeological space
IM .
Proof. Recall that a (set-theoretic) map f : M −→ N between C ∞ -manifolds
is smooth if and only if f preserves smooth curves and note that the smooth curvesof a C ∞ -manifold M are just the plots of IM with source R. Then, we see that I
is fully faithful.
We have the equality C ∞ (U, M × N ) = C ∞ (U, M ) × C ∞ (U, N ) for an open set
U of Rn and M, N ∈ C ∞ -mf d, from which we see that I preserves finite products.
f = IMg , it suffices to show that the ordinary topology of an
To prove that M
nopen set U of R is the final topology for the smooth curves, which is easily shownby Special Curve Lemma [46, p. 18] (alternatively, see [46, Theorem 4.11(1)]). 
From now on, we require that the underlying topological spaces of C ∞ manifolds are separated as arc-generated spaces (see the end of Section 2.1);
the category of (separated) C ∞ -manifolds are denoted by C ∞ .
From Lemma 2.5, we obtain the following result on the functor I : C ∞ −→ D,
which is the restriction of I : C ∞ -mf d −→ D to C ∞ .
Proposition 2.6.
(1) The category C ∞ is closed under finite products
and the functor I : C ∞ −→ D is a fully faithful functor which preserves
finite products.
(2) The functor I : C ∞ −→ D restricts to the fully faithful functor from thecategory of Lie groups to that of diffeological groups.
(3) The functor I : C ∞ −→ D is compatible with the underlying topologicalspace functors. In other words, the following diagram commutes:
I

C∞

D
e·

e·

C

0

Proof. (1) We see from Lemma 2.5 and property (4) in Remark 2.4 that˜· : C ∞ -mf d −→ C 0 preserves finite products, and hence that C ∞ is closed underfinite products (see the end of Section 2.1). The rest of the statement follows
immediately from Lemma 2.5.
(2) Since Lie groups and diffeological groups are defined to be groups in C ∞and D respectively, the result is immediate from Part 1.
(3) The result follows immediately from Lemma 2.5.
The diffeological space IM associated to a C ∞ -manifold M is often denoted
simply by M if there is no confusion in context.

14

HIROSHI KIHARA

Remark 2.7.
(1) It is obvious that finite dimensional C ∞ -manifolds embed fully faithfully into the category D via the functor I. It was shown by
Losik [48] that Fréchet manifolds embed fully faithfully into D via I.
(2) Though the definition of I applies to C ∞ -manifolds in classical differential
calculi, the functor I need not be full. In fact, I : Cc∞ −→ D is not full,
where Cc∞ denotes the category of C ∞ -manifolds in Keller’s Cc∞ -theory(see Appendix B).
The following remark relates to separation conditions for C ∞ -manifolds.
Remark 2.8. In [46, 27.4], Kriegl and Michor proposed the following three
separation conditions for C ∞ -manifolds:
f is separated as an arc-generated space.
(i) M
f is Hausdorff as a topological space.
(ii) M(iii) M is smoothly Hausdorff (i.e., the smooth functions in C ∞ (M, R) separate
points in M ).
(The implications (iii) =⇒ (ii) =⇒ (i) hold obviously. Use the fact that ˜· : C ∞ mf d −→ C 0 preserves finite products to see the equivalence of separation condition(i) and condition (2) in [46, 27.4].) They honestly wrote “it is not so clear which
separation property should be required for a manifold” and temporarily required thatC ∞ -manifolds are smoothly Hausdorff. Whichever separation condition is selected,
all the essential results in this paper remain true. However, we choose separation
condition (i), which seems most natural for our categorical setting; it also simplifies
some results and definitions (see Remarks 5.5(3)-(4), the definition of the functor
J : Cc∞conv −→ C ∞ in Appendix B, and [44]).
We end this subsection with a remark concerning the semiclassicality condition
on a C ∞ -manifold (see Section 1.2).
Remark 2.9. The c∞ -topology of a locally convex vector space can be strictly
finer than the original locally convex topology ([46, Section 4]). Thus, we need thenotion of a semiclassical C ∞ -manifold to use local convexity in local arguments.
2.3. Standard p-simplices and model structure on D. We recall thebasics of the model structure on D introduced in [43].
The principal part of our construction of a model structure on D is the construction of good diffeologies on the setsX
∆p = {(x0 , . . . , xp ) ∈ Rp+1 |xi = 1, xi ≥ 0} (p ≥ 0)iwhich enable us to define weak equivalences, fibrations, and cofibrations and toverify the model axioms (see Definition 2.13 and Theorem 2.14). The required
properties of the diffeologies on ∆p (p ≥ 0) are expressed in the following four
axioms:Axiom 1. The underlying topological space of ∆p is the topological standard
p-simplex for p ≥ 0.
Recall that f : ∆p −→ ∆q is an affine map if f preserves convex combinations.
Axiom 2. Any affine map f : ∆p −→ ∆q is smooth.

2. DIFFEOLOGICAL SPACES, ARC-GENERATED SPACES, AND C ∞ -MANIFOLDS

15

For K ∈ S, the simplex category ∆ ↓ K is defined to be the full subcategoryof the overcategory S ↓ K consisting of maps σ : ∆[n] → K. By Axiom 2, we canconsider the diagram ∆ ↓ K −→ D sending σ : ∆[n] −→ K to ∆n . Thus, we define
the realization functor| |D : S −→ D
by |K|D = colim ∆n .
∆↓Kp˙Consider the smooth map |∆[p]|
D ,−→ |∆[p]|D = ∆ induced by the inclusion
˙of the boundary ∆[p]
into ∆[p].

Axiom 3. The canonical smooth injection
˙
∆[p]

D

,−→ ∆pis a D-embedding.
The D-homotopical notions, especially the notion of a D-deformation retract,
are defined in the same manner as in the category of topological spaces by using
the unit interval I = [0, 1] endowed with a diffeology via the canonical bijection
with ∆1 ([43, Section 2.4]). The k th horn of ∆p is a diffeological subspace of ∆pdefined byΛpk = {(x0 , . . . , xp ) ∈ ∆p | xi = 0 for some i 6= k}.
Axiom 4. The k th horn Λpk is a D-deformation retract of ∆p for p ≥ 1 and0 ≤ k ≤ p.
P
For a subset A of the affine p-space Ap = {(x0 , . . . , xp ) ∈ Rp+1 |xi =1}, Asub denotes the set A endowed with the sub-diffeology of Ap (∼= Rp ). The
diffeological space ∆psub , used in [32] to study diffeological spaces by homotopicalmeans, satisfies neither Axiom 3 nor 4 for p ≥ 2 ([43, Proposition A.2]). Thus, wemust construct a new diffeology on ∆p , at least for p ≥ 2.
Let (i) denote the vertex (0, . . . , 1 , . . . , 0) of ∆p , and let di denote the affine(i)map from ∆p−1 to ∆p , defined byid ((k)) =(k)
for k < i,(k + 1) for k ≥ i.

Definition 2.10. We define the standard p-simplices ∆p (p ≥ 0) inductively.
Set ∆p = ∆psub for p ≤ 1. Suppose that the diffeologies on ∆k (k < p) are defined.
We define the map
ϕi : ∆p−1 × [0, 1) −→∆p

by ϕi (x, t) = (1 − t)(i) + tdi (x), and endow ∆p with the final structure for the maps
ϕ0 , . . . , ϕp .
The following result is established in [43, Propositions 3.2, 5.1, 7.1, and 8.1].
Proposition 2.11. The standard p-simplices ∆p (p ≥ 0) in Definition 2.10
satisfy Axioms 1-4.
Without explicit mention, the symbol ∆p denotes the standard p-simplex defined in Definition 2.10 and a subset of ∆p is endowed with the sub-diffeology of ∆p .
Since the diffeology of ∆p is the sub-diffeology of Ap for p ≤ 1, the D-homotopical

16

HIROSHI KIHARA

notions, especially the notion of a D-deformation retract, coincide with the ordinary smooth homotopical notions in the theory of diffeological spaces (see [38, p.
108] and [43, Remark 2.14]).
By Axiom 2, we can define the singular complex S D X of a diffeological space
X to have smooth maps σ : ∆p −→ X as p-simplices, thereby defining the singularfunctor S D : D −→ S. We have the following result ([43, Proposition 9.1 andLemma 9.5]).
Proposition 2.12.
(1) | |D : S  D : S D is an adjoint pair.
(2) The composite of the two adjoint pairs| |D : S  D : S D and ˜· : D  C 0 : R
is just the adjoint pair
| | : S  C0 : Sof the topological realization and singular functors.
Weak equivalences, fibrations, and cofibrations in D are defined as follows.
Definition 2.13. Define a map f : X −→ Y in D to be(1) a weak equivalence if S D f : S D X −→ S D Y is a weak equivalence in the
category of simplicial sets,(2) a fibration if the map f has the right lifting property with respect to theinclusions Λpk ,−→ ∆p for all p > 0 and 0 ≤ k ≤ p, and(3) a cofibration if the map f has the left lifting property with respect to allmaps that are both fibrations and weak equivalences.
Define the sets I and J of morphisms of D by
˙ p ,−→ ∆p | p ≥ 0},
I = {∆
J= {Λpk ,−→ ∆p | p > 0, 0 ≤ k ≤ p}.

Then, we have the following theorem ([43, Theorem 1.3]). See [53, Definition
15.2.1] for a compactly generated model category.
Theorem 2.14. With Definition 2.13, D is a compactly generated model category whose object is always fibrant. I and J are the sets of generating cofibrations
and generating trivial cofibrations respectively.
Remark 2.15. The proofs of Proposition 2.12 and Theorem 2.14 are constructed using only the convenient properties of the category D (Proposition 2.2)
and Axioms 1-4 for the standard simplices.
The following theorem, which is [43, Theorem 1.4], shows that weak equivalences in D are just smooth maps inducing isomorphisms on smooth homotopy
groups. See [8, Section 3.1] or [38, Chapter 5] for the smooth homotopy groups
πpD (X, x) of a pointed diffeological space (X, x). Note that S D X is always a Kancomplex (Theorem 2.14) and see [30, p. 25] for the homotopy groups πp (K, x) of apointed Kan complex (K, x).
Theorem 2.16. Let (X, x) be a pointed diffeological space. Then, there exists
a natural bijection
ΘX : πpD (X, x) −→ πp (S D X, x) for p ≥ 0,that is an isomorphism of groups for p > 0.

2. DIFFEOLOGICAL SPACES, ARC-GENERATED SPACES, AND C ∞ -MANIFOLDS

17

The standard simplices ∆p define not only the model structure on D but alsoa simplicial category structure on D via Proposition 2.12(1); the relation between
these two structures on D is one of our greatest concerns (see Section 4). Furthermore, the standard simplices ∆psub are also used for constructing smooth maps andsmooth homotopies (see Section 8). Thus, we end this subsection with the following
results on the standard simplices.
A subset A of ∆p endowed with the sub-diffeology of ∆psub is denoted by Asub ,which is compatible with the notation introduced above.
Lemma 2.17.
(1) The horn Λpk , and hence, the standard p-simplex ∆p is
D-contractible.
(2) The set {∆psub }p≥0 satisfies Axioms 1 and 2 for the standard simplices but
satisfies neither Axiom 3 nor 4.
(3) The map id : ∆p −→ ∆psub is smooth, which restricts to the diffeomorphism
id : ∆p − skp−2 ∆p −→ (∆p − skp−2 ∆p )sub , where skp−2 ∆p denotes the∼=(p − 2)-skeleton of ∆p .
Proof. See [43, Remark 9.3, Lemma A.1, Proposition A.2, and Lemmas 3.1
and 4.2].

2.4. Quillen pairs (| |D , S D ) and ( e·, R). In this subsection, we see that theadjoint pairs (| |D , S D ) and (e·, R) are Quillen pairs whose composite is a pair of
Quillen equivalences.
We begin with the following remark which relates to the model structure on
C0.
Remark 2.18. The category C 0 is a compactly generated model category whose
object is always fibrant (cf. [34, Section 7.10]); this is shown in the same manneras in the cases of the category T of topological spaces ([18, Section 8]) and thatof compactly generated Hausdorff spaces ([30, Proposition 9.2 in Chapter I]). We
can easily see that the adjoint pair I : C 0  T : α ([43, Lemma 2.7]) is a pair of
Quillen equivalences.
For the adjoint pairs (| |D , S D ) and (˜·, R), we have the following result.
Lemma 2.19. | |D : S  D : S D and e· : D  C 0 : R are Quillen pairs.
Proof. Since S D preserves both fibrations and trivial fibrations ([43, Lemma9.6]), (| |D , S D ) is a Quillen pair. Since R also preserves both fibrations and trivial
fibrations (Proposition 2.2(3) and [43, Lemma 9.8]), (e·, R) is a Quillen pair.
The following lemma states that the composite of the adjoint pairs (| |D , S D )and (˜·, R) is a pair of Quillen equivalences (see Proposition 2.12(2)).
Lemma 2.20. | | : S  C 0 : S is a pair of Quillen equivalences.
Proof. This result is shown in the same manner as in the case of | | : S 
B : S, where B is the category T of topological spaces or the category K of compactly generated Hausdorff spaces; see [51, Theorem 16.1] and [30, Theorem 11.4
in Chapter I] for the cases of T and K respectively.
Remark 2.21. As mentioned in the proof of Lemma 2.20, two proofs of the factthat (| |, S) is a pair of Quillen equivalences are known ([51, Theorem 16.1], [30,18

HIROSHI KIHARA

Theorem 11.4 in Chapter I]). We prove Part 1 of Theorem 1.5 in a way analogous
to the proof of [51, Theorem 16.1] (see Section 3).
The other proof is based on Quillen’s result that the topological realization ofa Kan fibration is a Serre fibration ([30, Theorem 10.10 in Chapter I]). For therealization functor | |D : S −→ D, the analogous result does not seem to hold; notethat | |D does not preserve finite products (in fact, |∆[1] × ∆[1]|D is not the product|∆[1]|D × |∆[1]|D but a diffeological space obtained by patching together two copiesof ∆2 ; see the argument in the proof of [43, Proposition A.2(1)]). Thus, it does notseem that Theorem 1.5(1) can be proved according to the idea of the proof of [30,
Theorem 11.4 in Chapter I].
3. Quillen equivalences between S, D, and C 0
In this section, we develop the theory of singular homology for diffeologicalspaces and use it to establish the Quillen equivalences between S, D, and C 0 (Theorem 1.5) and the basic properties of the subclasses WD and VD (Corollary 1.6).
3.1. Singular homology of a diffeological space. In this subsection, we
introduce the singular homology of a diffeological pair (i.e., a pair of a diffeologicalspace and its diffeological subspace) and establish the results used in the proof of
Theorem 1.5.
To define the homology H∗ (X, A) of a diffeological pair (X, A), we begin byrecalling the homology groups of a simplicial pair (i.e., a pair of a simplicial set
and its simplicial subset). For a category A, sA denotes the category of simplicial
objects in A ([51, Section 2]). Let Z-mod denote the category of Z-modules, and let
Kom≥0 (Z-mod) denote the category of non-negatively graded chain complexes ofZ-modules. The free Z-module functor Z· : Set −→ Z-mod extends to the functor
Z· : S = sSet −→ sZ-mod. The compositeZ·S −→ sZ-mod −→ Kom≥0 (Z-mod)is also denoted by Z·, where the functor sZ-mod −→ Kom≥0 (Z-mod) is defined byassigning to a simplicial Z-module M the chain complex
∂

∂

1
0
· · · −→
M1 −→
M0 −→ 0Pnwith ∂n = i=0 (−1)i di . Then, the homology H∗ (K, L) of a simplicial pair (K, L)
is defined byH∗ (K, L) = H∗ (Z(K)/Z(L)).
The homology of a simplicial pair (K, ∅) is usually referred to as the homology ofa simplicial set K, written H∗ (K).
Using the homology of a simplicial pair, we define the singular homology
H∗ (X, A) of a diffeological pair (X, A) by

H∗ (X, A) = H∗ (S D X, S D A).
The singular homology of a diffeological pair (X, ∅) is usually referred to as the
singular homology of a diffeological space X, written H∗ (X).
The singular homology for diffeological pairs has the following desirable properties. For a subset S of a topological space Z, S and S ◦ denote the closure andthe interior of S respectively.
Proposition 3.1.
(1) (Exactness) For a diffeological pair (X, A), there
exists a natural exact sequence3. QUILLEN EQUIVALENCES BETWEEN S, D, AND C 0

19

···
∂

Hp (A)Hp (X)

Hp−1 (A)···Hp (X, A)∂(2) (Excision) If (X, A) is a diffeological pair and U is a diffeological subspaceof X such that U ⊂ A◦ , then the inclusion (X − U, A − U ) ,−→ (X, A)
induces an isomorphism H∗ (X − U, A − U ) −−
→ H∗ (X, A).
∼=(3) (Homotopy) Let f, g : (X, A) −→ (Y, B) be maps of diffeological pairs. If
f 'D g, then the induced maps f∗ , g∗ : H∗ (X, A) −→ H∗ (Y, B) coincide.
(4) (Dimension) Let ∗ be a terminal object of D. Then(Z p=0
Hp (∗) =0 p 6= 0.
(5) (Additivity) If X is the coproduct of diffeological spaces {Xi }, then the
inclusions Xi −→ X induce an isomorphism
M
H∗ (Xi ) −→ H∗ (X).
∼i=

For the proof of Proposition 3.1, we need the following lemma.
Lemma 3.2.
(1) Let X be a diffeological space. Then the singular complexe
S D X is a subcomplex of the topological singular complex S X.
(2) If f : ∆r −→ ∆p × I is an affine map (i.e., a map preserving convex
combinations), then f is smooth.
Proof. Parts 1 and 2 follow immediately from Proposition 2.11 (Axioms 1
and 2, respectively).
Proof of Proposition 3.1. Parts 1, 4, and 5 are easily verified. For thee (Lemma
proof of Parts 2 and 3, we note that ZS D X is a chain subcomplex of ZS X
e
3.2(1)), and verify that the relevant chain map and chain homotopies on ZS X
restrict to ones on ZS D X.
(2) To verify the excision axiom for the topological singular homology, we use
e and the chain homotopy T on ZS X
e connecting 1 e
the chain map Sd on ZS X
ZS Xand Sd, where Sd and T are defined using the barycentric subdivision of ∆p andthe triangulation of ∆p × I connecting the trivial triangulation of ∆p and thebarycentric subdivision of ∆p , respectively ([5, Section 17 in Chapter IV]). Thus,
Sd and T restrict to a chain map and a chain homotopy on ZS D X respectively (seeProposition 2.11(Axiom 2) and Lemma 3.2(2)).
e −→ ZS Ye , we use the chain homotopy D on
(3) To show that f˜] ' g̃] : ZS X
e defined using the prism decomposition of ∆p × I ([31, Section 11]), which
ZS X
restricts to a chain homotopy on ZS D X (see Lemma 3.2(2)).
The Mayer-Vietoris exact sequence and the homology exact sequence of a triple
are also deduced from Proposition 3.1 via the usual arguments (cf. [63, p. 37]);

20

HIROSHI KIHARA

these exact sequences can be also derived from the consideration of the singularchain complexes.
˚ p denotes the set ∆p \∆˙ p endowed with the subdiffeology of ∆p
Recall that ∆([43, Definition 4.1]). For a Z-module M and n ≥ 0, let M [n] denote the graded
module with M [n]n = M and M [n]i = 0 (i 6= n).
Lemma 3.3. Let b denote the barycenter of ∆p . Then˚p, ∆˚ p − {b}) ∼H∗ (∆= Z[p].
e ∗ (X) of a diffeological space X in the
Proof. Define the reduced homology Hsame manner as the reduced homology of a topological space (see, eg, [63, p.
˚ p is D-contractible (Lemma 2.17(3)), we have only to show that38]). Since ∆pe˚H∗ (∆ − {b}) ∼= Z[p − 1] (see Proposition 3.1(1)).
˚ p − {b} contains a diffeological subspace which is diffeomorphic to S p−1
Since ∆
as a D-deformation retract, we can use Proposition 3.1 and the Mayer-Vietorise ∗ (S p−1 ) ∼e ∗ (∆
˚ p − {b}) ∼
exact sequence to see that H= Z[p − 1].
=H
Let us introduce the notions of a CW -complex in D and its cellular homology.
Definition 3.4. A CW -complex X in D is a diffeological space X which isthe colimit of a sequence in D
i3

i2

i1

i0

∅ = X −1 ,−→ X 0 ,−→ X 1 ,−→ X 2 ,−→ · · ·such that each in fits into a pushout diagram of the form
` ˙n
` n
∆λ
∆λ
λ∈Λn

λ∈Λn

in

X n−1

X n.

˙ p ,−→
Recall that D is a cofibrantly generated model category having I = {∆∆ | p ≥ 0} as the set of generating cofibrations. Thus, a CW -complex in D is a
special type of I-cell complex, and hence a cofibrant object in D. Let us see thatthe realizations of simplicial sets form an important class of CW -complexes in D.
pLemma 3.5. Let K be a simplicial set. Then, the realization |K|D is a CW complex in D having one n-cell for each non-degenerate n-simplex of K.
Proof. Let K n denote the n-skeleton of K. Then, K is the colimit of thesequence
i0

i1

i2

i3

∅ = K −1 ,−→ K 0 ,−−−→ K 1 ,−−−→ K 2 ,−−−→ · · ·and each in fits into the pushout diagram
` ˙
`
∆[n]λ
∆[n]λ
λ∈N KnK n−1

λ∈N KninKn ,where N Kn is the set of non-degenerate n-simplices of K (see [30, p. 8]). Applying
the realization functor | |D , we obtain the presentation of the CW -structure of |K|D
(see Propositions 2.12(1) and 2.11(Axiom 3)).


3. QUILLEN EQUIVALENCES BETWEEN S, D, AND C 0

21

For a CW -complex X in D, define the cellular chain complex (C(X), ∂C ) byCn (X) =Hn (X n , X n−1 ),∂C =the connecting homomorphism coming from thehomology exact sequence of the triple (X n , X n−1 , X n−2 ).
2(See [63, p. 39] for the verification of ∂C= 0.) The homology H∗ C(X) is called
the cellular homology of X.

Lemma 3.6. Let X be a CW -complex in D.
(1) The nth component Cn (X) is a free Z-module on the set of n-cells of X.
(2) H∗ C(X) is isomorphic to H∗ (X).
Proof. Express X as in Definition 3.4, and let bλ denote the barycenter of ∆nλ .
The point of X n corresponding to bλ is also denoted by bλ . From [43, Proposition
6.2, Lemmas 6.3(2) and 7.2], we observe that` n∆λ − {bλ }X n − {bλ |λ ∈ Λn }=X n−1 `∪λ∈Λn

˙ n λ∈Λ∆nλ

X n−1 .

'

By Proposition 3.1 and Lemma 3.3, we can prove the result in the same manner asin the topological case (see the proof of [63, Theorem 5.4.2]).
Proposition 3.7. For a simplicial set K, there exists a natural isomorphismof graded modules
H∗ (K) −→H∗ (|K|D ).
∼=Proof. Since (| |D , S D ) is an adjoint pair (Proposition 2.12(1)), we have thenatural map iK : K −→ S D |K|D , which induces the homomorphismiK

H∗ (K) −−−−∗−→ H∗ (S D |K|D ) = H∗ (|K|D ).
With the help of Lemma 3.6, we can show that iK∗ is an isomorphism by anargument similar to that in the proof of [51, Proposition 16.2(i)].

We have proven all the results on the singular homology and the realization ofa simplicial set which are used in the proof of Theorem 1.5(1). The following areadditional remarks.
Remark 3.8.
(1) Let π be an abelian group. For a diffeological pair
(X, A), the singular homology H∗ (X, A; π) with coefficients in π is defined byH∗ (X, A; π) = H∗ (ZS D X/ZS D A ⊗ π).
Then, Proposition 3.1 generalizes to the case of an arbitrary coefficientgroup π (see [5, p. 183] and the proof of Proposition 3.1).
For a diffeological pair (X, A), the singular cohomology H ∗ (X, A; π)with coefficients in π is defined byH ∗ (X, A; π) = H ∗ Hom(ZS D X/ZS D A, π).
Then, it has desirable properties analogous to Proposition 3.1 (see [5, pp.
285-286] and the proof of Proposition 3.1).

22

HIROSHI KIHARA(2) The universal coefficient theorems obviously hold for the singular homology
and cohomology of diffeological spaces (cf. [5, pp. 281-285]). Since theEilenberg-Zilber theorem holds for simplicial sets ([51, Corollary 29.6]),
the Künneth theorems hold for singular homology and cohomology of diffeological spaces, and hence, the cross products on homology and cohomology, and the cup product on cohomology are defined and satisfy the same
formulas as those in the topological case.
(3) Since the functor S D preserves fibrations (Lemma 2.19), the Serre spectral
sequence works for fibrations in D (see [51, Section 32]).
Remark 3.9. Hector [32] defined the singular complex of a diffeological space X
Dusing ∆psub (p ≥ 0); his singular complex and its homology are denoted by Ssub
(X)
0and H∗ (X) respectively. Iglesias-Zemmour defined the complex C? (X) of reducedgroups of cubic chains for a diffeological space X, and called its homology H∗ (X)the cubic homology of X ([38, pp. 182-183]).
Similarly to the topological case, we can see that H∗0 (∆p ) ∼= H∗ (∆p ) ∼= Z[0], andDDhence that ZS (X), ZSsub (X), and C? (X) are naturally chain homotopy equivalent(see [22]). Thus, our singular homology, Hector’s singular homology, and the cubic
homology are naturally isomorphic, which justifies the comment in the footnote of[38, p. 183].
D
It should be noted that the canonical natural chain homotopy equivalence ZSsub(X) ,−→DZS (X) exists (see [43, Lemma 3.1 and Remark A.5]).
3.2. Proof of Theorem 1.5. In this subsection, we give a proof of Theorem
1.5. Since the composite of the Quillen pairs
S

| |D
SD

D

e·

C0

R

is just the pair of Quillen equivalences | | : S  C 0 : S (Lemmas 2.19-2.20 and
Proposition 2.12), we have only to prove Theorem 1.5(1) (see [35, Corollary 1.3.15]).
We can reduce the proof of Theorem 1.5(1) by the following lemma on the
Quillen pair (| |D , S D ).
Lemma 3.10. The following are equivalent:(i) (| |D , S D ) is a pair of Quillen equivalences.
(ii) For K ∈ S and X ∈ D, the natural maps
iK : K −→ S D |K|D and pX : |S D X|D −→ X
are weak equivalences in S and D respectively.
(iii) For K ∈ S, the natural map iK : K −→ S D |K|D is a weak equivalence inS.
Proof. Note that every object of S is cofibrant and that every object of D isfibrant. Then the implications (i) =⇒ (ii) =⇒ (iii) are obvious.
(iii) =⇒ (i) Let X be a diffeological space. For a map f : K −→ S D X, consider

3. QUILLEN EQUIVALENCES BETWEEN S, D, AND C 0

23

the commutative diagramK

fiS D XiKS D |K|DSDX

S D |f |DSD f ˆ

S D |S D X|D

1S D XS D pXS D X,
where f ˆ : |K|D −→ X is the left adjunct of f : K −→ S D X. Since iK is a weakequivalence, we have the equivalences
f is a weak equivalence ⇔ S D f ˆ is a weak equivalence⇔ f ˆ is a weak equivalence,
which show that (| |D , S D ) is a pair of Quillen equivalences.



Thus, we have only to show that for K ∈ S, the natural map iK : K −→S D |K|D is a weak equivalence in S. Since | |D and S D preserve coproducts, we
may assume that K is connected.
Let us further reduce the proof of Theorem 1.5(1) to simpler cases.
Lemma 3.11. Consider the following conditions:(i) f : K −→ K 0 is a weak equivalence in S.
(ii) |f |D : |K|D −→ |K 0 |D is a weak equivalence in D.
(iii) S D |f |D : S D |K|D −→ S D |K 0 |D is a weak equivalence in S.
Then, the implications (i) =⇒ (ii) =⇒ (iii) hold.
Proof. Since every object of S is cofibrant, the implication (i) =⇒ (ii) holds
by Lemma 2.19 and [34, Corollary 7.7.2]. The implication (ii) =⇒ (iii) is immediatefrom the definition of a weak equivalence in D.
For a connected simplicial set K, consider weak equivalences in S
K ,→ L ←- M,
where L is a fibrant approximation of K and M is a minimal subcomplex of L ([51,§ 9]). Then, by Lemma 3.11, we have only to show that for a Kan complex K with
only one vertex, iK : K −→ S D |K|D is a weak equivalence in S.
A simplicial map $ : L −→ K is called a covering if $ is a fiber bundle
with discrete fiber. For a Kan complex K with only one vertex, a Kan fibration$ : K̃ −→ K with K̃ 1-connected is constructed in a natural way ([51, Definition
16.4]). Since the fiber of $ is a minimal complex, $ is a covering in our sense (seee −→ K the[25, Corollary 10.8 in Chapter I]). Thus, we call the fibration $ : K
universal covering of K.
A smooth map p : E −→ B is called a fiber bundle if any b ∈ B has an open
neighborhood U such that p−1 (U ) ∼= U × F over U for some F ∈ D. A smooth
map p : E −→ B is called a covering if p is a fiber bundle with discrete fiber.
Lemma 3.12. Let K be a Kan complex with only one vertex, and let $ : K̃ −→K be the universal covering of K.
(1) |$|D : |K̃|D −→ |K|D is a covering.

24

HIROSHI KIHARA(2) S D |$|D : S D |K̃|D −→ S D |K|D is a covering.
Proof. (1) We begin by showing the following three facts on |$|D : |K̃|D −→|K|D .
Fact 1: |$|D is a D-quotient map. Recall from Section 2.3 that the realization|L|D of a simplicial set L is defined by |L|D = colim ∆n . Then, we have the∆↓Lcommutative diagram in D
`

∆n

|K̃|D

ob(∆↓K̃)
|$|D

`

∆n

|K|D

ob(∆↓K)

consisting of the canonical surjective morphisms, where`∆n denotes the co-ob(∆↓L)product of the standard simplices indexed by the objects of ∆ ↓ L. Since thehorizontal arrows and the left vertical arrow are D-quotient maps, |$|D is also aD-quotient map.
Fact 2: Each g ∈ π1 (K) acts on |K̃|D as a diffeomorphism over |K|D . This fact
is easily seen since the action of π1 (K) on |K̃|D is induced from the obvious actionof π1 (K) on K̃ via the realization functor.
Fact 3: |$|D is a topological covering with fiber π1 (K). The fact follows from
Proposition 2.12(2) and [25, Theorem 4.2 in Chapter III].
Let us prove the result using Facts 1-3. For x ∈ |K|D , choose an open neighborhood U which is topologically contractible ([25, 1.9 in Chapter III]). Then|$|D : |$|−1
D (U ) −→ U
is a D-quotient map (Fact 1 and [43, Lemma 2.4(2)]) and |$|−1
D (U ) is isomorphic`
toUg , where each Ug is an open diffeological subspace of |K̃|D which isg∈π1 (K)topologically isomorphic to U (Fact 3). Since each g ∈ π1 (K) acts on |$|−1
D (U ) asa diffeomorphism over U (Fact 2), each Ug is diffeomorphic to U via |$|D .
(2) Let σ : ∆n −→ |K|D be a singular simplex, and consider the following
lifting problem
|K̃|D
|$|D

∆n

σ

|K|D .

By Proposition 2.11(Axiom 1) and Fact 3 in the proof of Part 1, there is a continuous
solution τ : ∆n −→ |K̃|D and {τ · g}g∈π1 (K) is the set of all continuous solutions.
Part 1 shows that the elements τ · g are smooth, which completes the proof.
Consider the morphism between homotopy exact sequences induced by themorphism of coverings

3. QUILLEN EQUIVALENCES BETWEEN S, D, AND C 0

π1 (K)π1 (K)

=

iK̃

K̃

25

S D |K̃|D
S D |$|D$iKK

S D |K|D .

For the proof of Theorem 1.5(1), we then have only to show that for a 1-connected
Kan complex K, iK : K −→ S D |K|D is a weak equivalence in S. Thus, thefollowing lemma and Proposition 3.7 complete the proof of Theorem 1.5(1) via theWhitehead theorem ([51, Theorem 13.9]).
Lemma 3.13. If K is a 1-connected Kan complex, then S D |K|D is also a 1connected Kan complex.
Proof. By replacing K with its minimal subcomplex, we may assume that K
has only one non-degenerate simplex of dimension ≤ 1 (see Lemma 3.11). Then
|K|D is a diffeological space obtained from the 0-simplex ∗ by attaching simplices ofdimension ≥ 2 (see Lemma 3.5). Since π1 (S D |K|D ) ∼= π1D (|K|D ) (Theorem 2.16),
D
we can easily see that π1 (S |K|D ) = 0 by [43, Lemma 9.9], the transversality
theorem ([6, (14.7)]), and the argument in the proof of Lemma 3.6.

3.3. Proof of Corollary 1.6. In this subsection, we give a proof of Corollary
1.6.
Proof of Corollary 1.6. (1) Since A is in WD , there are a cofibrant diff0 is afeological space A0 and a D-homotopy equivalence f : A0 −→ A. Then, A0f0 −→ Ae is a C -homotopy
cofibrant arc-generated space (Lemma 2.19) and fe : A
equivalence (see [43, Proposition 2.13 and Lemma 2.12]).
(2) We prove the result in two steps.
Step 1: The case where A is cofibrant. Since e· : D  C 0 : R is a pair of Quillen
e is a weak equivaequivalences (Theorem 1.5(2)) and Cf0 = C 0 holds, id : A −→ RA
lence in D.
Step 2: The case where A is in WD . Let A0 and f : A0 −→ A be as in the proof ofPart 1. Since R preserves products and id : I −→ RI˜ is smooth (Proposition 2.2(3)),
f0 −→ Ae defines the D-homotopy equivalence
the C 0 -homotopy equivalence fe : A0ef −→ RA.
e Thus, we have the commutative diagram in D
Rf : RA
f
'

A0
id

e0
RA

A
id

Rfe
'

e
RA.

e is a weak equivalence by Step 1 and [43, Lemma 9.4(2)].
Hence, id : A −→ RA(3) (i) ⇐⇒ (ii) From the definition of a weak equivalence in D (Definition 2.13)
and the equality S D R = S (Proposition 2.12(2)), we see that (i) is equivalent to
(ii).
e are
(ii) ⇐⇒ (iii) Recall from Theorem 2.14 and Remark 2.18 that S D A and S A
Kan complexes. Since there are natural isomorphisms
πpD (A, a) −→ πp (S D A, a),
∼
=

26

HIROSHI KIHARAe a) −e a)
πp (A,→ πp (S A,∼=(Theorem 2.16), the equivalence (ii) ⇐⇒ (iii) is obvious (see, eg, [42, Section3]).

We end this subsection with a remark on the subclass WD .
Remark 3.14. In principle, the subclass Ecf of fibrant-cofibrant objects of amodel category E is regarded as a subclass of objects behaving best from a modelcategorical viewpoint. However, for the category D, we adopt not Dc (= Dcf ) but
WD as a subclass of smooth homotopically good objects. The reasons are as follows:• We are primarily concerned with D-homotopical (or simplicial categorical)
properties of diffeological spaces, not with model categorical properties ofdiffeological spaces (see Remark 1.13).
• We cannot expect that good diffeological spaces such as finite dimensional
C ∞ -manifolds are cofibrant; in fact, cofibrant objects are far from seamlesssince they are sequential I-cell complexes or their retracts (see Theorem
2.14 for the set I). However, we can show that a large class of C ∞ manifolds is contained in WD (see Section 11).
Note that the notion of the subclass WD is a hybrid of model categorical andsimplicial categorical notions.
4. Smoothing of continuous maps
We are concerned with function complexes S D D(A, B) and SC 0 (X, Y ), andtheir relation (Theorem 1.7). However, model category theory mainly deals withhomotopy function complexes ([34, Chapter 17]). This section is devoted to thestudy of the relation between function complexes and homotopy function complexes
for the category M (= C 0 , D), which leads us to the proof of the smoothing theorem
for continuous maps (Theorem 1.7). More precisely, it enables us to reduce the proofof Theorem 1.7 to the proof of the weak equivalence between the homotopy function
complexes of D and C 0 .
We begin by clarifying significant differences between the simplicial categories
C 0 and D, which explains why subtle arguments are needed to obtain results forthe category D.
4.1. Enrichment of cartesian closed categories. In this paper, we often
deal with the categories C 0 and D as categories enriched over themselves and over
S. In this subsection, we thus recall the basics of enrichment of a cartesian closed
category over itself and establish a result on base change, which is used in this andlater sections.
Let V be a cartesian closed category. Then, V itself is an V-category ([4,Proposition 6.2.6]). Further, V is both a tensored and cotensored V-category whose
tensor and cotensor are given by
V ⊗X =V ×X

andY V = V(V, Y )([4, Proposition 6.5.3]); in other words, the pairs· × X : V  V : V(X, ·) and V(·, Y ) : V  V ∗ : V(·, Y )are V-adjoint pairs for X, Y ∈ V, where V ∗ is the dual of V (see [4, Propositions
6.2.2 and 6.7.4]).

4. SMOOTHING OF CONTINUOUS MAPS

27

The following result on base change is used in the next subsection to clarify
differences between the simplicial categories C 0 and D. See [4, Section 6.4] for thebasics of base change.
Proposition 4.1. Let L : U  V : R be an adjoint pair between cartesianclosed categories and regard V as a U-category via R. Suppose that L preserves
terminal objects. Then, the following are equivalent:(i) L preserves finite products.
(ii) The adjoint pair (L, R) can be enriched over U.
(iii) V is a tensored U-category.
(iv) V is a cotensored U-category.
Proof. We prove the implications (i) ⇒ (ii) ⇒ (iii) ⇒ (i); we can prove theimplications (ii) ⇒ (iv) ⇒ (i) similarly. Note that V is regarded as a U-category
with hom-object RV(X, Y ).
(i) ⇒ (ii) Noticing that R is a right adjoint, we can see that the functor R : V → U
is enriched over U in a canonical manner. Using condition (i), we also see that thefunctor L : U −→ V and the natural bijection U(U, RX) ∼= V(LU, X) are enrichedover U in a canonical manner.
(ii) ⇒ (iii) The pair · × X : V  V : V(X, ·) is a V-(and hence U-)adjoint pair. By
composing (L, R) with (· × X, V(X, ·)), we thus obtain the U-adjoint pair
L · ×X : U  V : RV(X, ·),which shows that V is a tensored U-category ([4, Proposition 6.7.4]).
(iii) ⇒ (i) Let U ⊗ X denote the tensor of U ∈ U and X ∈ V. Then, we have theisomorphism in U
U(U, RV(X, Y )) ∼= RV(U ⊗ X, Y ).
Since L preserves terminal objects, this isomorphism implies the bijection
U(U, RV(X, Y )) ∼= V(U ⊗ X, Y ),which shows that U ⊗ X ∼= LU × X. Thus, we see from [4, Proposition 6.5.4] that
LU × LU 0 ∼
= L(U × U 0 ).
4.2. Simplicial categories C 0 and D. The categories S, D, and C 0 are cartesian closed and the composite of the adjoint pairs
S

| |D
S

D

D

e·

C0

R

is just the adjoint pair | | : S  C 0 : S (Proposition 2.2, Remark 2.4, and Proposition 2.12). Thus, D and C 0 are S-categories with function complexes S D D(X, Y )and SC 0 (X, Y ), respectively. In this subsection, we give a precise definition of thee B)e dealt with in Theorem 1.7, and discuss themap e· : S D D(A, B) ,−→ SC 0 (A,differences between the simplicial categories C 0 and D.
First, we prove the following result.
Lemma 4.2. The adjoint paire· : D  C 0 : R
can be enriched over S.

28

HIROSHI KIHARA

Proof. Since e· : D −→ C 0 preserves finite products (Remark 2.4), the adjoint
pair e· : D  C 0 : R can be enriched over D (see Proposition 4.1). Thus, we obtain
the result using the equality S D ◦ R = S.
The simplicial functor e· : D −→ C 0 defines the natural inclusion between function complexese B),ee· : S D D(A, B) −→ SC 0 (A,which is the map that we are interested in (see Theorem 1.7). Note that π0 (e·) is
just the natural map
e B]
e C0
[A, B]D −→ [A,([43, Section 2.4]).
Next, we show the following basic result on the simplicial categories C 0 and D.
Proposition 4.3.
(1) The S-category C 0 is both tensored and cotensored.
(2) The S-category D is neither tensored nor cotensored.
Proof. (1) Consider the adjoint pair | | : S  C 0 : S. Since C 0 is cartesianclosed, the topological realization functor | | : S −→ C 0 preserves finite products (seethe argument in [25, 3.5 in Chapter III]). Thus, the result follows from Proposition
4.1.
(2) Consider the adjoint pair | |D : S  D : S D . Since | |D : S −→ D does notpreserve finite products (Remark 2.21), the result follows from Proposition 4.1. 
Let E be both a simplicial category and a model category. If E satisfies axioms
M6 and M7 in [34, Definition 9.1.6], then E is called a simplicial model category;axiom M6 in [34, Definition 9.1.6] requires that the simplicial category E is tensoredand cotensored.
Recall the model structures on C 0 and D from Sections 2.3 and 2.4.
Corollary 4.4.
(1) C 0 is a simplicial model category.
(2) D is not a simplicial model category.
Proof. (1) By Proposition 4.3(1), C 0 satisfies axiom M6. Axiom M7 is easily
verified using [30, Proposition 3.13 in Chapter II].
(2) By Proposition 4.3(2), D does not satisfy axiom M6.
Remark 4.5. It is natural to develop smooth homotopy theory using the notions
and results of model category theory. Then, it seems to be an inevitable feature ofhomotopical algebra of smooth spaces that one must deal with a category of smooth
spaces which is not a simplicial model category. In fact, if a convenient category
A of smooth spaces is regarded as a simplicial category via a singular functor S A
having a left adjoint, then A does not seem to be a tensored or cotensored simplicial
category (see Remark 2.21 and Proposition 4.1).
Remark 4.6. This remark relates to the notation for hom-objects of enrichedcategories.
Suppose that a category A is enriched over a cartesian closed category V. The
hom-object of A is denoted by A(A, A0 ). In the case of V = M (= D, C 0 ), thehom-object A(A, A0 ) is just the hom-set A(A, A0 ) endowed with the canonical Mspace structure, and hence this notation is no problem. However, in the case of
V = S, the hom-object (or function complex) A(A, A0 ) is a simplicial set whose 0th
component is the hom-set A(A, A0 ), and hence we often write MapA (A, A0 ) for the4. SMOOTHING OF CONTINUOUS MAPS

29

function complex of A. Further if the S-category structure of A is induced froman M-category structure on A via the singular functor S M : M −→ S, then weusually write S M A(A, A0 ) for the function complex of A.
4.3. Function complexes and homotopy function complexes for C 0
and D. In this subsection, we show that under the cofibrancy condition on thesource, the function complexes S D D(A, X) and SC 0 (A, X) are homotopy equivalent
to the homotopy function complexes mapD (A, X) and mapC 0 (A, X) respectively.
After briefly discussing the case of C 0 (Proposition 4.7), we establish the result for
D (Theorem 4.8).
First, recall the following basic properties of the categories S, C 0 , and D:• The category S is a simplicial model category whose objects are cofibrant([34, Example 9.1.13]).
• The category C 0 is a simplicial model category whose objects are fibrant(Remark 2.18 and Corollary 4.4(1)).
• The category D is both a simplicial category and a model category, but
it is not a simplicial model category (Corollary 4.4(2)). All objects of D
are fibrant.
Next, let us review the basics of function complexes in a simplicial category
and homotopy function complexes in a model category.
Let E be a simplicial category. Then, the function complex MapE (A, X) is
assigned to A, X ∈ E. The E-homotopy set [A, X]E is defined by[A, X]E = π0 MapE (A, X).
Let us call a 1-simplex of MapE (A, X) an E-homotopy. Then, the E-homotopy set[A, X]E is just the quotient set of E(A, X) by the equivalence relation 'E generatedby E-homotopies. We simply write ' for 'E if there is no confusion in context.
The homotopy category π0 E of E is defined to be the category in which the objects
are the same as those of E and the hom-sets are the E-homotopy sets. These Ehomotopical notions specialize to the ordinary ones in the case of E = M (= C 0 , D)(see [43, Section 2.4]).
Next, let E be a model category. Then the homotopy function complex mapE (A, X)
is assigned to A, X ∈ E. For simplicity, we assume that X is fibrant. Then,
mapE (A, X) can be defined bymapE (A, X) = E(A, X),
where A is a cosimplicial resolution of A, i.e., a cofibrant approximation to theconstant cosimplicial object cA at A in the Reedy model category E ∆ ([34, p. 318]).
We further assume that A is cofibrant. Then the set π0 (mapE (A, X)) is isomorphic
to the classical homotopy set πE (A, X) = E(A, X)/ 'c` ([34, Proposition 17.7.1]),
and hence, to the hom-set Ho E(A, X) of the Quillen homotopy category Ho E
of E ([34, Theorem 8.3.9]); see [34, Definition 7.3.2(5) and Theorem 7.4.9] for the
relation 'c` on E(A, X). (Throughout this paper, we refer to the homotopy relationin [34, Definition 7.3.2(5)] as the classical homotopy relation and write 'c` for itin order to distinguish it from the E-homotopy relation defined in the case where E
is also a simplicial category.) The classical homotopy category π Ecf of E is defined
to be the category whose objects and hom-sets are fibrant-cofibrant objects of E
and classical homotopy sets respectively. In the case of E = C 0 or D, we write π Ec
for π Ecf , since every object is fibrant.

30

HIROSHI KIHARA

Last, let E be a simplicial model category. For any object A, A∗ denotes the
cosimplicial object in E whose nth component is A ⊗ ∆[n]. If A is cofibrant, thecanonical map A∗ −→ cA in E ∆ is a cosimplicial resolution for A ([34, Proposition
16.1.3]), and hence the homotopy function complex mapE (A, X) coincides with thefunction complex MapE (A, X) for fibrant X.
For the category C 0 , we obtain the following result.
Proposition 4.7. Let A be a cofibrant arc-generated space and X an arcgenerated space. Then, the equality
mapC 0 (A, X)= SC 0 (A, X),

and hence the equality
πC 0 (A, X) = [A, X]C 0
holds.
Proof. Since C 0 is a simplicial model category (Corollary 4.4(1)) and every
object of C 0 is fibrant, the result follows from the comment above.
The following theorem is a diffeological analogue of Proposition 4.7.
Theorem 4.8. Let A be a cofibrant diffeological space and X a diffeologicalspace. Then there exists a homotopy equivalence
mapD (A, X) ' S D D(A, X)
which is natural up to homotopy. Hence, the equality
πD (A, X) = [A, X]Dholds.
Recall that D is not a simplicial model category (Corollary 4.4(2)). For the
proof of Theorem 4.8, we thus need the following two results along with Theorem
1.5(1).
Proposition 4.9. Let i : A ,−→ B be a trivial cofibration in D. Then A is a
D-deformation retract of B.
Proof. We prove the result in two steps.
Step 1: The case where i is a sequential relative J -cell complex. By the definitionof a sequential relative J -cell complex [53, Definition 15.1.2], there is a sequence
i

i

i

i

in+1

1
2
3
n
A = B0 ,−−−
→ B1 ,−−−→ B2 ,−−−→ · · · ,−−−
→ Bn ,−−−→ · · ·in D such that i coincides with the canonical map A −→ lim Bn and each in fits−→ninto a pushout diagram of the form
` pλ
Λkλ

`

λ

Bn−1

∆pλ

λ

in

Bn .

We construct a D-deformation R : B×I −→ B onto A as follows; see Section 1.4
for the (smooth) composite of D-homotopies h : X ×[a, b] −→ Y , k : X ×[b, c] −→ Ywith h(·, b) = k(·, b).

4. SMOOTHING OF CONTINUOUS MAPS

31

• R is defined to be the constant homotopy of 1A on A × I.
• R is defined to be the constant homotopy of 1B1 on B1 × [0, 12 ] and a
deformation onto A on B1 × [ 12 , 1] ([43, Lemma 9.7]).
• Suppose that R is defined on Bn−1 × I. Then R is defined to be theconstant homotopy of 1Bn on Bn × [0, 21n ], a deformation onto Bn−1 on1], and the composite
Bn × [ 21n , 2n−1




1
R( · , n−1
)×1
1
1
R
2
−−−−−→ Bn−1 × n−1 , 1 −−−→ Bn−1 ,−→ Bn
Bn × n−1 , 1 −−−−−−
2
2
1
on Bn × [ 2n−1
, 1].
Since B × I = (lim Bn ) × I = lim (Bn × I) (Proposition 2.2(2)), R : B × I −→ B
−→

−→

n

n

is the desired smooth deformation.
Step 2: The case where i is a general trivial cofibration. The trivial cofibration i
has the functional factorization
i0

A

A0
p0

i

B,
where i0 is a sequential relative J -cell complex and p0 is a (trivial) fibration (see[43, the proof of Theorem 1.3]). Thus, for the commutative solid arrow diagram
i0

A0

`

p0

A
i

B

1

B,

the dotted arrows ` exists, making the diagram commute. Hence, we have theretract diagram
A

1

i

B

A
i

`

1

A

0

A0

i

B.

p0

Since i0 is a sequential relative J -cell complex, there are a retract r0 : A0 −→ A anda homotopy relative to A R0 : A0 × I −→ A0 connecting 1A0 to i0 r0 (Step 1). Definer0`

the map r : B −→ A to be the composite B −−−→ A0 −−−→ A. Then, it is easily
verified that r is a retraction of B onto A. Define the homotopy R : B × I −→ B
to be the composite
`×1

R0

p0

B × I ,−−−→ A0 × I −−−→ A0 −−−→ B.
We can easily see that R|A×I is the constant homotopy of i and that R is a homotopy
connecting 1B to ir, which complete the proof.
Lemma 4.10. Let f : A −→ B be a morphism of D. Then, the following are
equivalent:(i) f : A −→ B is a D-homotopy equivalence.

32

HIROSHI KIHARA(ii) f ] : D(B, X) −→ D(A, X) is a D-homotopy equivalence for any X ∈ D,
where f ] is defined by precomposition with f .
(iii) S D f ] : S D D(B, X) −→ S D D(A, X) is a homotopy equivalence of Kancomplexes for any X ∈ D.
Proof. (i) =⇒ (ii) From Proposition 2.2(2), we see that the induced map[C, f ] ]D : [C, D(B, X)]D −→ [C, D(A, X)]D
is bijective for any C ∈ D, and hence that f ] : D(B, X) −→ D(A, X) is a Dhomotopy equivalence.
(ii) =⇒ (iii) See [43, Lemma 9.4].
(iii) =⇒ (i) Applying the functor π0 to the homotopy equivalence S D f ] : S D D(B, X) −→S D D(A, X), we see thatf ] : [B, X]D −→ [A, X]Dis bijective for X ∈ D, which implies that f : A −→ B is a D-homotopy equivalence.
Proof of Theorem 4.8. We prove the result in two steps.
Step 1: Suppose that A = |K|D for some simplicial set K. We establish a homotopy
equivalence
mapD (|K|D , X) ' S D D(|K|D , X).
Since | |D : S  D : S D is a Quillen pair (Lemma 2.19), the cosimplicial resolution
K ∗ −→ cK of K induces the cosimplicial resolution |K ∗ |D −→ |cK|D = c |K|D of|K|D ([34, Proposition 16.2.1(1)]). Hence, we have the equality
mapD (|K|D , X) = D(|K ∗ |D , X).
Thus, we have only to establish a homotopy equivalence
D(|K ∗ |D , X) ' S D D(|K|D , X).
Consider the bisimplicial set
S D D(|K ∗ |D , X) = D(∆∗ × |K ∗ |D , X)
and note that

∼ D(|K ∗ | , X),
D(∆0 × |K ∗ |D , X) =D∗
0
D(∆ × K D , X) ∼= S D D(|K|D , X)

hold.
The canonical map ι : X = D(∗, X) −→ D(∆p , X) is a weak equivalence between fibrant objects (see [43, Remark 9.3] and Lemma 4.10). Thus, the induced
map D(|K ∗ |D , X) −→ D(|K ∗ |D , D(∆p , X)) ∼= D(∆p × |K ∗ |D , X) is a weak equivalence ([34, Theorem 17.6.3(1)]), and hence the horizontal face and degeneracy
operators of D(∆∗ × |K ∗ |D , X) are weak equivalences.
Since | |D : S  D : S D is a Quillen pair and K × (0) ,→ K × ∆[q] is a trivial
cofibration in S, |K|D = |K × (0)|D ,−→ |K × ∆[q]|D is a trivial cofibration in D,
and hence a D-homotopy equivalence (see Proposition 4.9). From this, we see that|proj|D : |K × ∆[q]|D −→ |K|D is also a D-homotopy equivalence. Thus, the induced map S D D(|K|D , X) −→ S D D(|K × ∆[q]|D , X) = D(∆∗ × |K × ∆[q]|D , X)
is a weak equivalence (Lemma 4.10), and hence the vertical face and degeneracy
operators of D(∆∗ × |K ∗ |D , X) are weak equivalences.

4. SMOOTHING OF CONTINUOUS MAPS

33

From these results, we have the diagram consisting of weak equivalences
D(|K ∗ |D , X)

diag D(∆∗ × |K ∗ |D , X)

S D D(|K|D , X)([34, Corollary 15.11.12]), which shows that D(|K ∗ |D , X) ' S D D(|K|D , X).
Step 2: We deal with the case where A is a general cofibrant object of D. We havethe homotopy equivalence
mapD (|S D A|D , X) ' S D D(|S D A|D , X)
(Step 1). Since the canonical map pA : |S D A|D −→ A is a weak equivalence(Theorem 1.5(1) and Lemma 3.10), mapD (|S D A|D , X) ' mapD (A, X) holds by[34, Theorem 17.6.3(2)]. Thus, we have only to show that the weak equivalencepA : S D A D −→ A is a D-homotopy equivalence (see Lemma 4.10).
By the Whitehead theorem ([34, Theorem 7.5.10]), there is a map qA : A −→S D A D such that pA qA 'c` 1A and qA pA 'c` 1|S D A|D . Thus there exist a cylinder
object Cyl(A) of A and a classical homotopy h : Cyl(A) −→ A such that thecompositei0 +i1h
A q A −−−→ Cyl(A) −−−→ A
is pA qA + 1A . Factor the map pA as
i0A

SDA D

p0A

( S D A D )0A

where i0A is a trivial cofibration and p0A is a fibration (see Theorem 2.14) and consider
the solid arrow commutative diagram
A

qA

SDA D

i0A

( S D A D )0

H

i0

Cyl(A)p0AA.

hSince i0 is a trivial cofibration and p0A is a fibration, the dotted arrow H exists,00= 1A . On the:= H ◦ i1 satisfies that p0A qA
making the diagram commute. Then qA
other hand, we have the commutative diagram in the classical homotopy category
π Dc( S D A D )0
i0A

SDA D

p0A

0
qA

A

( S D A D )0
qA

pA

1|S D A|i0A

SDA D .

D

0 0 0Thus, we see that i0A 'c` qA
pA iA . Noticing that πD (|K|D , Z) ∼= [|K|D , Z]D for K ∈0 0 0
S and Z ∈ D (Step 1), we have i0A 'D qA
pA iA . Since i0A has a D-homotopy inverse0
0 0
rA(Proposition 4.9), we obtain 1(|S D A|D )0 'D qA
pA . Therefore, p0A : (|S D A|D )0 −→A, and hence pA : |S D A|D −→ A is a D-homotopy equivalence.


We derive a few corollaries of Theorem 4.8 using Theorem 1.5(2).
Let M denote one of the categories D and C 0 . Let Mc (resp. WM ) denotethe full subcategory of M consisting of cofibrant objects (resp. objects having theM-homotopy type of a cofibrant object). Regarding Mc and WM as simplicial
full subcategories of M, we consider the homotopy categories π0 Mc and π0 WM

34

HIROSHI KIHARA(see the beginning of this subsection). Then, we see that the canonical functor
π0 Mc −→ π0 WM is an equivalence of categories. Recall the equivalence π Mc −→Ho M between the classical homotopy category and the Quillen homotopy category([34, Theorem 8.3.9]).
Corollary 4.11. The following commutative diagram of equivalences of categories exists:
π0 WD

π0 Dc

π Dc

Ho D

π0 e·

π0 e·

πe·

Le·

π0 WC 0π0 C 0 c

π C0c

Ho C 0 .

The horizontal functors are just inclusions on the object level. The vertical functors
are induced by the underlying topological space functor e· : D −→ C 0 .
Proof. By Proposition 4.7 and Theorem 4.8, we have the isomorphisms of
categories π0 C 0 c −→ π C 0 c and π0 Dc −→ π Dc . Thus, we have the commutative
diagram of functors as in the statement, whose horizontal functors are equivalencesof categories. Since the right vertical arrow L e·, which is the total left derived
functor of e·, is an equivalence of categories by Theorem 1.5(2), the other vertical
functors are also equivalences of categories.

Corollary 4.12. Let A and B be diffeological spaces in WD . Then, the following are equivalent:(i) f : A −→ B is a D-homotopy equivalence.
(ii) f : A −→ B is a weak equivalence in D.
e −→ Be is a C 0 -homotopy equivalence.
(iii) fe : A
e −→ Be is a weak equivalence in C 0 .
(iv) fe : A
Proof. We prove the result in two steps.
Step 1: The case where A and B are cofibrant. The result is immediate from
Corollary 4.11.
Step 2: The case where A and B are in WD . For a smooth map g : X −→ Y , theimplications
g is a D-homotopy equivalence

g is a weak equivalence in D

ge is a C 0 -homotopy equivalence

ge is a weak equivalence in C 0

hold (see [43, Lemma 9.4 and Section 2.4]). We thus choose D-homotopy equivalences A0 −→ A and B −→ B 0 with A0 and B 0 cofibrant and consider the compositefA0 −−−−→ A −−−→ B −−−−→ B 0 .
'D'DThen, the result follows from Step 1.



The following result is a diffeological version of [56, Proposition 3].
Corollary 4.13. Let A and B be diffeological spaces in WD . Then the productA × B is also in WD .

5. SMOOTHING OF CONTINUOUS PRINCIPAL BUNDLES

35

Proof. We prove the result in two steps.
Step 1: We show that |K × L|D is D-homotopy equivalent to |K|D × |L|D for
K, L ∈ S.
We haveS D D(|M |D , X) ' MapS (M, S D X)
for M ∈ S and X ∈ D (see Theorem 4.8, Lemma 2.19, [34, Proposition 17.4.16],
and the comment before Proposition 4.7). Thus, by the cartesian closedness of S
and D, we have
S D D(|K × L|D , X) '
∼
=
'
'
∼
=

MapS (K × L, S D X)
MapS (L, MapS (K, S D X))
MapS (L, S D D(|K|D , X))
S D D(|L|D , D(|K|D , X))
S D D(|K|D × |L|D , X),

which implies that |K × L|D 'D |K|D × |L|D .
Step 2: From Step 1 and Corollary 4.12, we have the composite of D-homotopy
equivalencespA ×pBS D A × S D B D −→ S D A D × S D B D −−−−−→ A × B,
which completes the proof.



4.4. Proof of Theorem 1.7. We end this section by proving Theorem 1.7.
Proof of Theorem 1.7. Since A is in WD , we may assume that A is a cofibrant object (see Lemma 4.10 and Corollary 4.12). Note that the left Quillen
functor e· : D −→ C 0 defines the map
e X)ee· : mapD (A, X) −→ mapC 0 (A,([34, Proposition 16.2.1(1)]). By Proposition 4.7 and Theorem 4.8, we have only
to show that this map is a weak equivalence. Since e· : D  C 0 : R is a Quillen pair
(Lemma 2.19), we have the isomorphism
e ∼
e X)
e
mapD (A, RX)= mapC 0 (A,e is a weak(see [34, Proposition 17.4.16]). Since X is in VD , id : X −→ RXequivalence between fibrant objects, and henceemapD (A, X) −→ mapD (A, RX)
is a weak equivalence ([34, Theorem 17.6.3(1)]), which completes the proof.



5. Smoothing of continuous principal bundles
In this section, we give precise definitions of smooth and continuous principal
bundles, and then establish a smoothing theorem for principal bundles (Theorem
5.17) using Theorem 1.7. Throughout this section, C denotes one of the categories
C ∞ , D, C 0 , and T (see Remarks 2.4 and 2.18 for the category T ).

36

HIROSHI KIHARA

5.1. C-partitions of unity. In this subsection, we introduce the notion of a
C-partition of unity and the related notions.
Consider the faithful functors
(5.1)

I

e·

I

C ∞ ,−→ D −→ C 0 ,−→ T

and recall from Proposition 2.6 and the definition of C 0 that the two functors
denoted by I are fully faithful. Equip the category C with the faithful functor
U : C → T , which is the composite of functors in (5.1). The underlying topologicalspace U X of X ∈ C is often denoted by X if there is no confusion in context.
For an object X of C, a C-subspace of X is defined to be a subset A of X
endowed with the initial structure for the inclusion A ,−→ X with respect to the
obvious underlying set functor C −→ Set (see Remark 2.3(1)). If C = D, C 0 , or T ,or if A is an open set of U X, then A admits a unique C-subspace structure.
Note that R is canonically viewed as an object of C in a way compatible withthe faithful functors in (5.1). A morphism of C with target R is often called a
C-function.
Definition 5.1. Let X be an object of C.
(1) A set {ϕi : X −→ R} of C-functions is called a C-partition of unity
if {U ϕi : U X −→ U R} is a partition of unity in the usual sense (i.e.,{carr ϕi }, and hence {supp ϕi }Pis a locally finite covering of X, ϕi (x) ≥ 0
for any i and any x ∈ X, and i ϕi = 1).
(2) A covering {Ui } of X is called C-numerable if there exists a C-partition ofunity {ϕi : X −→ R} subordinate to {Ui } (i.e., supp ϕi ⊂ Ui ).
(3) X is called C-paracompact if any open covering of X is C-numerable. X iscalled hereditarily C-paracompact if any open C-subspace is C-paracompact.
Remark 5.2.
(1) In the definition of a C-numerable covering, {Ui } isoften assumed to be an open covering. However, the definition and many
important results apply to a covering by subsets (see, eg, [62] and [67]).
(2) For a C ∞ -manifold M , C ∞ -paracompactness and D-paracompactness areequivalent (see Proposition 2.6). Thus, both C ∞ -paracompactness and Dparacompactness are often referred to as smooth paracompactness.
Both C 0 -paracompactness and T -paracompactness imply (ordinary)paracompactness, and the converse also holds under Hausdorff condition.
Remark 5.3. An object X of C is called C-regular if for any x ∈ X and any
open neighborhood U of x, there exists a C-function f : X −→ R such that f (x) = 1
and carr(f ) ⊂ U (cf. [46, Convention 14.1]). If X is C-paracompact and U X is a
T1 -space, then X is C-regular.
For a C ∞ -manifold M , C ∞ -regularity and D-regularity are equivalent, andhence these are often referred to as smooth regularity. The notion of smooth regularity is needed in Section 11. If C is C 0 or T , then C-regularity is just complete
regularity.
We can easily see that C-regularity is inherited by every C-subspace whose underlying topology is the subspace topology (especially by every open C-subspace).
For the basics of smooth paracompactness and smooth regularity, refer to [46,Sections 14 and 16], in which these notions are defined for a Hausdorff topological0space X with a sheaf AX of subalgebras of the sheaf FX
of algebras of continuous functions satisfying the following condition: The algebra AX (U ) of sections

5. SMOOTHING OF CONTINUOUS PRINCIPAL BUNDLES

37

is closed under composites with elements of C ∞ (R, R) for any open set U . (Thisis a corrected version of the notion introduced in [46, Convention 14.1]. We callsuch a space a smoothly ringed space.) We can assign to an object X of C withU X Hausdorff the smoothly ringed space (U X, C(·, R)). Note that separation axiom T1 is equivalent to Hausdorff condition under C-regularity (and hence under
C-paracompactness).
5.2. Principal bundles in C. Let us introduce the notion of a principal
bundle in C.
Definition 5.4. Let B be an object of C and G a group in C ([49, p. 75]).
(1) Let CG denote the category of right G-objects of C (i.e., objects of C
endowed with a right G-action). Regard B as an object of CG with trivial
G-action.
An object π : E −→ B of the overcategory CG/B is called a principal G-bundle if there exists an open cover {Ui } of B such that E|Ui :=π −1 (Ui ) ∼= Ui × G in CG/Ui .
Let PCG/B denote the full subcategory of CG/B consisting of principal G-bundles.
(2) A principal G-bundle π : E −→ B is called C-numerable if it has a trivialization open cover {Ui } which is C-numerable.
Let (PCG/B)num denote the full subcategory of PCG/B consisting of
C-numerable principal G-bundles.
We see that a morphism f : A −→ B of C induces the pullback functors
f ∗ : PCG/B −→ PCG/A and f ∗ : (PCG/B)num −→ (PCG/A)num .
Remark 5.5.
(1) Unlike D, C 0 , and T , the category C ∞ is not completeor cocomplete. However, we can easily see that all the finite limits neededin Definition 5.4 and the comment after it exist in C ∞ .
(2) The well-known description of a principal bundle using an open cover andtransition functions applies to C (= C ∞ , D, C 0 , T ). Thus, the categories
PCG/B and (PCG/B)num are essentially small.
(3) Let B be a C ∞ -manifold and G a Lie group. Then, the fully faithful embedding C ∞ ,−→ D induces an isomorphism of categories PC ∞ G/B −→PDG/B, which restricts to an isomorphism (PC ∞ G/B)num −→ (PDG/B)num .
To prove this, we have only to show that the total space E of a principal
G-bundle over B in D is necessarily separated (see Proposition 2.6). First,
consider the commutative solid arrow diagram in D
E
diag

E×E

E×E

B

B

diag

B × B.

Then, there exists a (unique) dotted arrow from E to the fiber product
E ×B E, making the diagram commute. Since B is separated, E ×B E isa closed subset of E × E. Note that a local trivialization E|U ∼= U ×G

38

HIROSHI KIHARA

defines the local trivialization E ×B E|U ∼= U ×G×G. Then, we can easily
construct a retract diagram in D
E ×B E|U ,−→ E|U × E|U −→ E ×B E|U ,which shows that the topology of E ×B E is the subspace topology of E ×E.
Thus, we have only to show that E is closed in E ×B E, which is easily
seen from the separatedness of G using the local trivializations E ×B E|U ∼=U × G × G.
(4) Even if we select separation condition (ii) in Remark 2.8, the result in Part
3 remains true. However, if we select separation condition (iii) in Remark
2.8, the situation is different. We can show that PC ∞ G/B ,−→ PDG/B isa fully faithful functor which restricts to an isomorphism (PC ∞ G/B)num −→(PDG/B)num and that PC ∞ G/B −→ PDG/B is an isomorphism under the assumption that B is C ∞ -regular. But, we cannot prove thatPC ∞ G/B ,−→ PDG/B is an isomorphism without any additional assumption.
The following lemma is simple but important.
Lemma 5.6. The category PCG/B is a groupoid.
Proof. We observe that the natural isomorphism of monoids
CG/V (V × G, V × G) ∼= C(V, G)
exists, and hence that PCG/V (V × G, V × G) (= CG/V (V × G, V × G)) is a group.
From this, we can easily see that PCG/B is a groupoid.

The following lemma and remark explain that the underlying topological objecteof a principal G-bundle in D should be considered as a principal G-bundle
in C 0 .
Lemma 5.7. Let B be a diffeological space and G a diffeological group. Then,the functor e· : D −→ C 0 induces a functore B,e
PDG/B −→ PC 0 G/
which restricts to a functor
e B)
e num .
(PDG/B)num −→ (PC 0 G/
Proof. For an object π : E −→ B of CG/B, a local trivialization E|U ∼= U ×G
is identified with a pullback diagram in C
j

U ×G

π

proj

U

E

i

B

such that i is an open C-embedding and j is G-equivariant.
Recall from Remark 2.4 that the functor e· : D −→ C 0 preserves finite products.
e is a group in C 0 and that e· induces the functor e· : DG/B −→Then, we see that G
0e e
C G/B. Further, recall that the underlying set functors D −→ Set and C 0 −→ Set
create limits (Proposition 2.2(1) and [43, Proposition 2.6]) and note that Lemma
2.3 in [43] remains true for the concrete category C 0 and that e· transforms open
D-embeddings to open C 0 -embeddings (cf. [7, Lemma 3.18]). Then, we see that e·

5. SMOOTHING OF CONTINUOUS PRINCIPAL BUNDLES

39transforms local trivializations of an object π : E −→ B of DG/B to ones of thee −→ Be of C 0 G/
e B,
e which completes the proof.
object πe:E
We call a group in C 0 an arc-generated group. In the proof of Lemma 5.7, it is
e is an arc-generated group.
also shown that if G is a diffeological group, then G
Remark 5.8. Lemma 5.7 does not hold if we replace C 0 with T . In fact,e·Isince U : D −→ T , which is the composite D −→ C 0 ,−→ T , does not preserve
finite products, the underlying topological space U G of a diffeological group G is notnecessarily a topological group (see [46, 4.16-4.26] and Proposition 2.6). However,suppose that one of the following conditions is satisfied:(i) U G is locally compact.
(ii) U G and U X satisfy the first axiom of countability.
∼ PT U G/U X,
e X
e =Then, U G is a topological group and the isomorphism PC 0 G/
and hence the functor U : PDG/X −→ PT U G/U X exists (see [43, Lemma 2.8
and Proposition 2.9] and observe that every arc-generated space is locally arcwise
connected).
5.3. Fiber bundles in C. We discuss the smoothing of continuous sectionsof fiber bundles in the next section. Thus, we give a precise definition of a fiber
bundle in C and study fiber bundles, especially in M (= D, C 0 ) (see Section 1.4).
Definition 5.9. An object p : E −→ B of the overcategory C/B is called afiber bundle if there exist an object F of C and an open cover {Ui } of B such that
E|Ui := p−1 (Ui ) ∼= Ui × F in C/Ui ; the object F is called the fiber of p : E −→ B
and p is often called an F -bundle.
An F -bundle p : E −→ B is called C-numerable if it has a trivialization open
cover {Ui } which is C-numerable.
Lemma 5.10. Let p : E −→ B be a fiber bundle in C (with nonempty fiber).
Then, p is a C-quotient map whose underlying continuous map U p is a topologicalquotient map.
Proof. For any x ∈ B, there exist an open neighborhood U and a C-sections on U . Thus, we can easily see that p is a C-quotient map and that U p is a
T -quotient map.
In the rest of this subsection, we restrict ourselves to the case of C = M(= D, C 0 ).
Lemma 5.11. The underlying topological space functor e· : D −→ C 0 sends an
F -bundle in D to an Fe-bundle in C 0 .
Proof. For an object p : E −→ B of C/B, a local trivialization E|U ∼=U ×F
is identified with a pullback diagram in C
j

U ×F

p

proj

U

E

i

B

such that i is an open C-embedding. Thus, we can prove the result by an argument
similar to that in the proof of Lemma 5.7.


40

HIROSHI KIHARA

Since Lemma 5.11 is an analogue of Lemma 5.7, statements analogous to thoseof Remark 5.8 hold; the precise statements are left to the reader.
We endow the automorphism group AutM (F ) of an object F of M with theinitial structure for the two maps
AutM (F )

incl
inv

M(F, F ),where incl(ϕ) = ϕ and inv(ϕ) = ϕ−1 .
Lemma 5.12. Let M denote one of the categories D and C 0 and let F be an
object of M.
(1) There exists a bijection, natural in A,∼ AutM/A (A × F ).
M(A, AutM (F )) =Hence, AutM (F ) is a group in M.
(2) The isomorphism classes of principal AutM (F )-bundles over X in M
bijectively correspond to those of F -bundles over X in M.
Proof. (1) From the definition of AutM (F ), we can easily`observe that a map
ϕ : A −→ AutM (F ) is a morphism of M if and only if ϕ̂ :=ϕ(a) : A × F −→a∈A

A × F is an isomorphism of M/A. This implies the first statement, from which thesecond statement follows.
(2) Recall the notion of a fiber bundle with structure group G and the description of such a bundle using transition functions (see, eg, [1, Chapter 2]), and notethat they also apply to the category M. Since an F -bundle in M is regarded as
an F -bundle with structure group AutM (F ) (Part 1), the result holds.

We can construct an equivalence of categories which induces the bijection between the sets of isomorphism classes in Lemma 5.12(2); see Lemma 7.4(1) and see
also [11, Theorem 6.1] for the case of M = D.
We end this subsection with the following remark.
Remark 5.13. Though we are mainly concerned with numerable locally trivial
bundles (Definitions 5.4 and 5.9), we also mention other important objects of D/B.
Iglesias-Zemmour introduced weaker notions of a fiber bundle and a principal
bundle in D, which are defined by local triviality of the pullback along any plot ([38,Chapter 8]). However, no analogue exists in C ∞ , C 0 , or T and in addition, suchbundles are difficult to deal with homotopically, as shown in [11, Section 3] using
results of [72, 12]. Thus, we deal with locally trivial bundles under numerability
conditions (see Section 5.4).
Christensen-Wu [9, 10] introduced the tangent bundle T B of a diffeologicalspace B as a vector space in D/B. But, T B need not be even a fiber bundle in theweaker sense.
5.4. Smoothing of principal bundles. In this subsection, we establish asmoothing theorem for principal bundles (Theorem 5.17). We begin by making abrief review on the classification of principal bundles in M (= D, C 0 ).
Milnor [55] constructed the universal principal bundle π : EG −→ BG for a
topological group G (see [36, Chapter 4]). Since the construction also works inthe category C 0 , we have the universal principal bundle π : EG −→ BG for an
arc-generated group G.

5. SMOOTHING OF CONTINUOUS PRINCIPAL BUNDLES

41

Magnot-Watts [50] and Christensen-Wu [11] constructed the universal principal bundles for a diffeological group G, which have the same underlying set andsleightly different diffeologies. Since Christensen-Wu’s one π : EG −→ BG classifies D-numerable principal G-bundles without any assumption on the base, we
adopt theirs as the universal principal bundle for G.
For an essentially small category A, KA denotes the set of isomorphism classesof objects of A.
Proposition 5.14.
(1) Let X be an arc-generated space and G an arcgenerated group. Then, there exists a natural bijection
K(PC 0 G/X)num −→[X, BG]C 0 .
∼=(2) Let X be a diffeological space and G a diffeological group. Then, there
exists a natural bijection
K(PDG/X)num −→[X, BG]D .
∼=g −→ BG
g is a universal(3) Let G be a diffeological group. Then, πe : EGeprincipal G-bundle.
Proof.
(1) The relevant arguments in [36, Chapter 4] also apply to thearc-generated case.
(2) See [11, Theorem 5.10].
g −→ BG
g is a C 0 -numerable principal G-bundlee(3) By Lemma 5.7, πe : EGin0
g is contractible,
C . Since EG is D-contractible ([11, Corollary 5.5]), EG
which implies the universality of πe by the arc-generated version of [14,Theorem 7.5].

Recall the model structures of D and C 0 from Sections 2.3-2.4 and the subclass
VD from Section 1.3.
Corollary 5.15.
(1) Let M be one of the categories D and C 0 . Then,every fiber bundle in M is a fibration with respect to the standard model
structure of M.
(2) Let π : E −→ B be an F -bundle in D. If two of F , E, and B are in VD ,then so is the third.
Proof. (1) The case of M = D. We have only to show that for an F -bundle
π : E −→ B in D, S D π : S D E −→ S D B is an S D F -bundle
in S (see [43, LemmaP
9.6(1)]). The affine space Ap = {(x0 , . . . , xp ) ∈ Rp+1 | xi = 1} is smoothly paracompact ([6, Theorem 7.3]). Since the inclusion ∆p ,−→ Ap is a smooth map anda closed topological embedding (Lemma 2.17(3) and Proposition 2.11), ∆p is alsosmoothly paracompact. Thus, every F -bundle over ∆p is necessarily D-numerable,
and hence trivial (see Lemmas 2.17(1) and 5.12(2), Proposition 5.14(2)). From
this, we see that the pullback of S D p along any map ∆[p] −→ S D B is isomorphic
to ∆[p] × S D F over ∆[p].
The case of M = C 0 . We have only to show that for an F -bundle π : E −→ B in C 0 ,Sπ : SE −→ SB is an SF -bundle in S. Since the topological standard p-simplex∆ptop is paracompact and contractible, an argument similar to that in the case of
M = D applies.

42

HIROSHI KIHARA

e −→ Be is an Fe-bundle (Lemma 5.11). Then we have the(2) Recall that πe:Emorphism of simplicial fiber bundles
SDE

e
SE

SD π

Se
π

SDB

e
SB

(Part 1), from which we can easily see the result.



Corollary 5.16. Let G be a diffeological group. Then, G is in VD if and only
if BG is in VD .
Proof. Consider the universal principal G-bundle π : EG −→ BG and recallg −→ BG
g is the universal principal G-bundleethat πe : EG(Proposition 5.14(3)).
DgSince S EG ' S EG ' ∗ (see the proof of Proposition 5.14(3)), the result easily
follows from Corollary 5.15.
Now, we can prove the following theorem.
Theorem 5.17. Let X be a diffeological space and G a diffeological group. If
X is in WD and G is in VD , then the functor e· : D −→ C 0 induces the bijection
e X)
e num .
K(PDG/X)num −→K(PC 0 G/
∼
=

e X)
e num
Proof. By Proposition 5.14, the natural map K(PDG/X)num −→ K(PC 0 G/
is identified with
e BG]
g C0 .
[X, BG]D −→ [X,Thus, we see that this map is bijective from Theorem 1.7 and Corollary 5.16.



Remark 5.18. In addition to the assumptions in Theorem 5.17, assume that
X is D-paracompact. Then, we have the natural bijectione X).
eK(PDG/X) −−−∼−−→ K(PC 0 G/=

6. Smoothing of continuous sections
In this section, we prove the smoothing theorem for continuous sections (Theorem 1.8). For this, we first establish Quillen equivalences between the overcategories
of S, D, and C 0 . We then study the relation between function complexes and homotopy function complexes for the overcategory M/X (= C 0 /X, D/X), generalizing
several results in Section 4.
6.1. Quillen equivalences between the overcategories of S, D, and C 0 .
In this subsection, we establish Quillen equivalences between the overcategories ofS, D, and C 0 , generalizing Theorem 1.5.
We begin with an observation on adjoint pairs. Let F : A  B : G be an
adjoint pair. Suppose that A has pullbacks. Then, the functor
F : A/A −→ B/F A

6. SMOOTHING OF CONTINUOUS SECTIONS

43

has a right adjoint for any A ∈ A. In fact, the right adjoint G0 : B/F A −→ A/A is
defined by assigning to an object q : Z −→ F A the object G0 q : G0 Z −→ A which
is defined by the pullback diagram in A
G0 Z

GZ

G0 q

Gq
iA

A

GF A,

where iA is the unit of the adjoint pair (F, G).
From this, we see that the functorse| |D : S/K −→ D/|K|D and e· : D/X −→ C 0 /X
have the right adjoints (S D ·)0 and R0 respectively.
Recall that the overcategory of a model category has a canonical model structure ([34, Theorem 7.6.5(2)]) and note that if F : A  B : G is a Quillen pairbetween model categories A and B, then F : A/A  B/F A : G0 is also a Quillenpair.
We have the following result, which generalizes Theorem 1.5.
Proposition 6.1.
(1) | |D : S/K  D/|K|D : (S D ·)0 is a pair of Quillen
equivalences for any simplicial set K.
(2) The Quillen paire : R0e· : D/X  C 0 /X
is a pair of Quillen equivalences if and only if X is in VD .
Proof. (1) Note that every object of S/K is cofibrant and that E −→ |K|D
is fibrant in D/|K|D if and only if E −→ |K|D is a fibration in D. Then, we caneasily see from Theorem 1.5(1) and [34, Theorem 13.1.13] that (| |D , (S D ·)0 ) is apair of Quillen equivalences.
(2) Note that f : A −→ X is cofibrant in D/X if and only if A is cofibrant in
e is fibrant in C 0 /X
e if and only if E −→ X
e is a fibration in C 0 .
D and that E −→ X
e of the adjoint
By the definition, X is in VD if and only if the unit id : X −→ RXpair (e·, R) is a weak equivalence. Thus, the result follows from Theorem 1.5(2) andthe fact that D is right proper (see Theorem 2.14 and [34, Corollary 13.1.3]). 
The composite of the two pairs of Quillen equivalences
S/K

| |D
D

(S ·)

0

D/ |K|De·
R0

C 0 / |K|

is just the pair of Quillen equivalences| | : S/K  C 0 /|K| : (S·)0(see Proposition 2.12).
6.2. Enrichment of overcategories. To further investigate the categories
C 0 /X and D/X, we have to establish basic results on the enrichment of overcategories of cartesian closed categories.

44

HIROSHI KIHARA

Let V be a cartesian closed category with pullbacks. For objects p : E −→ X,
p0 : E 0 −→ X of V/X, define the object V/X(E, E 0 ) of V by the pullback diagram
in V
V/X(E, E 0 )V(E, E 0 )

(6.1)

p0]

∗

V(E, X),

where the lower horizontal arrow is the map from the terminal object ∗ to V(E, X)
corresponding to p. The object V/X(X, E) is often denoted by Γ(X, E). The
underlying set of V/X(E, E 0 ), defined to be the set V(∗, V/X(E, E 0 )), is just thehom-set V/X(E, E 0 ) of the category V/X.
Proposition 6.2. Let V be a cartesian closed category with pullbacks and X
an object of V. Then, V/X is a tensored and cotensored V-category with hom-object
V/X(E, E 0 ).
Proof. We prove the results in three steps.
Step 1. V/X is a V-category. From the definition of V/X(E, E 0 ), we can observethat the natural bijections(6.2)

V(V, V/X(E, E 0 )) ∼= V/X(V × E, E 0 )∼ V/V × X(V × E, V × E 0 )=exist. Using these, we can define the composition morphism V/X(E, E 0 )×V/X(E 0 , E 00 ) −→V/X(E, E 00 ) and the unit morphism ∗ −→ V/X(E, E) satisfying associativity axiom and unit axiom ([4, Definition 6.2.1]).
Step 2. V/X is tensored. Let V and E be objects of V and V/X, respectively. To
see that V × E is the tensor of V and E, we have to see that the natural bijection
V(V, V/X(E, E 0 )) ∼= V/X(V × E, E 0 )in (6.2) is an isomorphism in V. This is easily seen by replacing V with U × V .
Step 3. V/X is cotensored. For an object V of V and an object p : E −→ X of
V/X, define the object pV : E V −→ X of V/X by the pullback diagram in V
EV
pV

X

V(V, E)p]

V(V, X),

where the lower horizontal arrow is the map induced by the canonical map V −→ ∗.
To see that E V is the cotensor of V and E, we have to prove that
V/X(D, E V ) ∼= V(V, V/X(D, E))
holds in V. By the definition of E V , the natural bijection
V/X(D, E V ) ∼= V/X(V × D, E)
exists. By replacing D with U × D, we see that this bijection is an isomorphism inV, which along with Step 2, implies the desired natural isomorphism.
6. SMOOTHING OF CONTINUOUS SECTIONS

45

Corollary 6.3. Let V be a cartesian closed category with pullbacks and X an
object of V.
(1) Let E be an object of V/X. Then,· × E : V  V/X : V/X(E, ·) and E • : V  (V/X)∗ : V/X(·, E)
are V-adjoint pairs.
(2) X × · : V  V/X : Γ(X, ·) is a V-adjoint pair.
(3) Let f : Z −→ X be a morphism of V. Then,f] : V/Z  V/X : f ∗is a V-adjoint pair, where f] is defined by assigning to p : D −→ Z thepfobject D −−−→ Z −−−→ X of V/X and f ∗ is the pullback functor alongf.
Proof. (1) The result is immediate from Proposition 6.2 (see [4, Proposition
6.7.4]).
(2) The first V-adjoint pair in Part 1 specializes to the V-adjoint pair (X ×·, Γ(X, ·)) by setting E = X.
(3) It is obvious that (f] , f ∗ ) is an adjoint pair. Using (6.2), we can easily seethat f] and f ∗ are V-functors and that the natural isomorphism in V
∼ V/X(f] D, E)
V/Z(D, f ∗ E) =exists.



From Corollary 6.3(1), we have the natural isomorphisms in V
(6.3)
V(V, V/X(D, E)) ∼= V/X(V × D, E) ∼= V/X(D, E V ).
We also have the following natural isomorphism.
Corollary 6.4. The natural isomorphism in V
V(V, V/X(E, E 0 )) ∼= V/V × X(V × E, V × E 0 )exists.
Proof. Recall from (6.2) the natural bijection V(V, V/X(E, E 0 )) ∼= V/V ×
X(V × E, V × E 0 ). By replacing V with U × V , we see that this bijection is an
isomorphism in V.
Remark 6.5. Suppose that V is a locally cartesian closed category. Then, V/X
is a cartesian closed category with internal hom VX (E, E 0 ), and hence a tensoredand cotensored V/X-category with hom-object VX (E, E 0 ) ([4, Proposition 6.5.3]).
Thus, we see from Corollary 6.3 and [4, Proposition 6.7.4] that
V/X(E, E 0 ) = Γ(X, VX (E, E 0 )) and E V = VX (X × V, E)
hold. Recall the local cartesian closedness of D and S from [2, 16] and [41, Theorem
1.42].
Given an adjoint pair L : U  V : R between cartesian closed categories, we
can regard V/X as a U-category via R. We thus consider when L : U/U  V/LU :R0 can be enriched over U (see Section 6.1 for R0 ) and when V/X is tensoredand cotensored as a U-category. The following two results answer these questions,generalizing Proposition 4.1.

46

HIROSHI KIHARA

Lemma 6.6. Let L : U  V : R be an adjoint pair between cartesian closed
categories with pullbacks. Suppose that L preserves terminal objects. Then, thefollowing are equivalent:(i) L preserves finite products.
(ii) The adjoint pair L : U/U  V/LU : R0 can be enriched over U for any
U ∈ U.
(iii) The adjoint pair L : U/U  V/LU : R0 can be enriched over U for someU ∈ U with U(∗, U ) 6= ∅.
Proof. By the assumption, U/U and V/LU are enriched over U; U/U (D, D0 )and RV/LU (E, E 0 ) are hom-objects of the U-categories U/U and V/LU respectively.
(i) ⇒ (ii) Note that L preserves finite products and that R is a right adjoint. Then,
we can observe from (6.2) that the adjoint pair L : U/U  V/LU : R0 is enrichedover U in a canonical manner for any U ∈ U.
(ii) ⇒ (iii) Obvious.
(iii) ⇒ (i) Choose an object U of U for which L : U/U  V/LU : R0 is enrichedover U and which admits a morphism u : ∗ −→ U , and consider the composite ofU-adjoint pairs
U = U/∗

u]
∗

L

U/U

u

R

V/LU

0

c]
c∗

V/∗ = V,

where c : LU −→ ∗ is the canonical map to the terminal object ∗ (see Corollary
6.3(3)). Since this composite is just (L, R), Proposition 4.1 shows that (iii) ⇒(i).
Proposition 6.7. Let L : U  V : R be an adjoint pair between cartesianclosed categories. Suppose that V has pullbacks and that L preserves terminal objects. Then, the following are equivalent:(i) L preserves finite products.
(ii) V/X is both a tensored and cotensored U-category for any X ∈ V.
(iii) V/X is a tensored or cotensored U-category for some X ∈ V with V(∗, X) 6=∅.
Proof. Note that V/X is a U-category with hom-object RV/X(E, E 0 ).
(i) ⇒ (ii) For E ∈ V/X, consider the composites of U-adjoint pairs
U

U

L
R
L
R

V

V

·×E
V/X(E,·)E•

V/X,(V/X)∗

V/X(·,E)(Proposition 4.1 and Corollary 6.3(1)), which show that V/X is both a tensoredand cotensored U-category (see [4, Proposition 6.7.4]).
(ii) ⇒ (iii) Obvious.
(iii) ⇒ (i) We prove the implication in the tensored case; the proof in the cotensoredcase is similar.
Let X be an object of V which admits a morphism x : ∗ −→ X. Supposethat V/X is a tensored U-category and write U ⊗ E for the tensor of U ∈ U and6. SMOOTHING OF CONTINUOUS SECTIONS

47

E ∈ V/X. For E ∈ V/X, we then consider the composite of U-adjoint pairs
U

·⊗E

V/X

RV/X(E,·)c]c∗V/∗ = V,

where c : X −→ ∗ is the canonical map to the terminal object ∗ (see [4, Proposition
6.7.4] and Corollary 6.3(3)). Note that RV/X(E, ·) ◦ c∗ = RV(E, ·) and that every
object V of V can be regarded as an object of V/X by endowing it with thex
structure morphism V −→ ∗ −→ X. Then, we see that V is a tensored U-category([4, Proposition 6.7.4]), and hence that L preserves finite products (see Proposition
4.1).

6.3. Simplicial categories C 0 /X and D/X. In this subsection, we establish
the basic properties of the S-categories C 0 /X and D/X and discuss the differences
between these simplicial categories.
For M = C 0 , D, the category M/X is an M-category (Proposition 6.2), andhence an S-category with function complex S M M/X(E, E 0 ) (see Section 1.4).
The following lemma is a generalization of Lemma 4.2.
Lemma 6.8. Let X be a diffeological space. Then, the adjoint paire : R0e· : D/X  C 0 /X
can be enriched over S.
Proof. Since e· : D  C 0 : R is an adjoint pair whose left adjoint preserves
finite products (Remark 2.4), (e·, R0 ) can be enriched over D (see Lemma 6.6). Thus,
we obtain the result by applying the functor S D (see Proposition 2.12).

Since M/X is a simplicial category, M/X-homotopies are defined as 1-simplices
of function complexes S M M/X(E, E 0 ) (see Section 4.3). We can easily see from
(6.3) that an M/X-homotopy is just an M-homotopy over X (or vertical Mhomotopy), and hence that the M/X-homotopy set [E, E 0 ]M/X is just the set ofvertical M-homotopy classes of morphisms from E to E 0 over X.
e introduced in Lemma 6.8 defines the
The simplicial functor e· : D/X −→ C 0 /X
natural inclusion
e D,
e E),
e
e· : S D D/X(D, E) ,−→ SC 0 /X(
which specializes to the natural inclusione E).
ee· : S D Γ(X, E) ,−→ SΓ(X,This is the map which are dealt with in Theorem 1.8. Note that π0 (e·) is just thenatural map from the vertical smooth homotopy classes of smooth sections to thevertical continuous homotopy classes of continuous sections.
Next, we give the following basic results for C 0 /X and D/X, generalizing Proposition 4.3 and Corollary 4.4.
Proposition 6.9.
(1) The S-category C 0 /X is both tensored and cotensored for any X ∈ C 0 .
(2) The S-category D/X is neither tensored nor cotensored for any nonempty
X ∈ D.
Proof. The results follows from Propositions 4.3, 4.1, and 6.7.



48

HIROSHI KIHARA

Note that the tensor K ⊗ E and the cotensor E K of the S-category C 0 /X aregiven by
(6.4)

K ⊗ E = |K| × E and E K = E |K|(see the proofs of Propositions 6.2 and 6.7).
Corollary 6.10.
(1) C 0 /X is a simplicial model category for any X ∈0C .
(2) D/X is not a simplicial model category for any nonempty X ∈ D.
Proof. (1) By Proposition 6.9(1), we have only to verify that C 0 /X satisfies
axiom M7 in [34, Definition 9.1.6]. Since the tensor K ⊗ E is given by K ⊗ E =|K| × E (see (6.4)), axiom M7 follows from Corollary 4.4(1) and [34, Proposition
9.3.7].
(2) The result follows from Proposition 6.9(2).
Remark 6.11. Note that S/K is a tensored and cotensored S-category whose
tensor L ⊗ X is given by L ⊗ X = L × X (see Proposition 6.2). Then, we can showthat S/K is a simplicial model category for any K ∈ S by an argument similar tothat in the proof of Corollary 6.10(1).
6.4. Function complexes and homotopy function complexes for C 0 /X
and D/X. In this subsection, we study the relationship between function complexes S M M/X(D, E) and homotopy function complexes mapM/X (D, E) for M/X(= C 0 /X and D/X), which leads us to the proof of Theorem 1.8.
First, we deal with the case of M = C 0 .
Proposition 6.12. Let f : D −→ X and p : E −→ X be objects of C 0 /X.
Suppose that D is cofibrant in C 0 and that p is a fibration in C 0 . Then, the equality
mapC 0 /X (D, E) = SC 0 /X(D, E)
holds. In particular, for a cofibrant object X and a fibration p : E −→ X in C 0 , theequality
mapC 0 /X (X, E) = SΓ(X, E)
holds.
Proof. Note that h : Z −→ X is cofibrant in C 0 /X if and only if Z is cofibrantin C 0 and that h : Z −→ X is fibrant in C 0 /X if and only if h is a fibration in C 0 .
Then, the result follows from Corollary 6.10(1) and the comment before Proposition
4.7.

Next, we establish a diffeological version of Proposition 6.12. Since D/X is nota simplicial model category unlike C 0 /X (Corollary 6.10), we must impose stronger
assumptions on X and E.
Theorem 6.13. Let K be a simplicial set and p : E −→ |K|D be a D-numerable
F -bundle in D. Then, there exists a homotopy equivalence
mapD/|K|D (|K|D , E) ' S D Γ(|K|D , E)
which is natural up to homotopy.

6. SMOOTHING OF CONTINUOUS SECTIONS

49

Remark 6.14. We can prove that every cofibrant object of D is a D-paracompact.
Thus, the D-numerability condition on p : E −→ |K|D in Theorem 6.13 is actually
unnecessary.
For the proof of Theorem 6.13, we need a few lemmas. First, recall the pullf∗

back functor V/Z ←− V/X from Corollary 6.3(3) and note that the morphismf∗

V/Z(f ∗ D, f ∗ E) ←− V/X(D, E) between the hom-objects specializes to the morf∗

phism Γ(Z, f ∗ E) ←→ Γ(X, E).
Lemma 6.15. Let p : E −→ X be a D-numerable F -bundle in D and f : Z −→X be a D-homotopy equivalence. Then, the induced mapf∗Γ(Z, f ∗ E) ←− Γ(X, E)
is a D-homotopy equivalence.
Proof. Choose a D-homotopy inverse g of f and consider the pullback diagramsg∗ f ∗ E

f ∗E
g

X

Z

E
f

X

and the induced morphisms of Dg∗f∗Γ(X, g ∗ f ∗ E) ←− Γ(Z, f ∗ E) ←− Γ(X, E).
Next, choose a D-homotopy H : X × I −→ X connecting 1X to f ◦ g and anΦisomorphism E × I ←−H ∗ E over X × I such that Φ0 := Φ|X×(0) = 1E (see [11,∼=Corollary 4.15]). Since Γ(X × I, E × I) ∼= D(I, Γ(X, E)) (Corollary 6.4), we havethe commutative diagram
D(I, Γ(X, E)) ∼= Γ(X × I, E × I)
(i0 ] ,i1 ] )Φ]∼=Γ(X × I, H ∗ E)

H∗

Γ(X, E)(1Γ(X,E) ,Φ1] ◦g ∗ ◦f ∗ )Γ(X, E) × Γ(X, E),
which shows that Φ1] ◦g ∗ ◦f ∗ 'D 1Γ(X,E) , and hence that f ∗ has a left D-homotopy
inverse. Since Φ1] is an isomorphism, g ∗ has a right D-homotopy inverse. Since g ∗also has a left D-homotopy inverse, g ∗ , and hence f ∗ is a D-homotopy equivalence.

Recall from the proof of Proposition 6.2 the definition of the cotensor in the
V-category V/X.
Lemma 6.16. If p : E −→ X be an F -bundle in D, then the cotensor pA :E A −→ X is a D(A, F )-bundle in D for any A ∈ D.

50

HIROSHI KIHARA

Proof. We can identify an local trivialization E|U ∼= U × F with a pullback
diagram in D
U ×F
(6.5)

E
p

proj

U

X

(see the proof of Lemma 5.11). Since D(A, ·) preserves limits (Proposition 2.2), we
obtain from pullback diagram (6.5) the pullback diagrams in D
U × D(A, F )D(A, U ) × D(A, F )D(A, E)
p]

proj

proj

D(A, U )U

D(A, X),

which give a local trivialization of pA (see [49, Exercise 8 on page 72]).

Remark 6.17. Lemmas 6.15 and 6.16 remain true even if the category D is
replaced by C 0 , since the arguments in their proofs also apply to the arc-generated
context. The arc-generated version of Lemma 6.15 is also needed in the proof of
Theorem 1.8.
Proof of Theorem 6.13. Recall the Quillen pair
| |D : S/K  D/|K|D : (S D ·)0(Proposition 6.1(1)). Since S/K is a simplicial model category (Remark 6.11), thecanonical morphism K ∗ (:= K × ∆[∗]) −→ cK is a cosimplicial resolution of theterminal object K of S/K, and hence |K ∗ |D −→ c|K|D is a cosimplicial resolutionof the terminal object |K|D of D/|K|D (see [34, Proposition 16.2.1]). Hence, we
have
mapD/|K|D (|K|D , E) = D/|K|D (|K ∗ |D , E)(see Corollary 5.15(1)). Thus, we have only to establish a homotopy equivalence
D/|K|D (|K ∗ |D , E) ' S D Γ(|K|D , E).
Consider the bisimplicial set
S D D/|K|D (|K ∗ |D , E) = D/|K|D (∆∗ × |K ∗ |D , E)
and note that
D/|K|D (∆0 × |K ∗ |D , E) ∼= D/|K|D (|K ∗ |D , E),
D/|K|D (∆∗ × |K 0 |D , E) ∼= S D Γ(|K|D , E)
hold (see (6.3)).
p
We can observe from (6.3) that the canonical map E −→ E ∆ is a vertical
D-homotopy equivalence, and hence, a weak equivalence between fibrant objects of
D/|K|D (see Lemma 6.16 and Corollary 5.15(1)). Thus, the mapp

D/|K|D (|K ∗ |D , E) −→ D/|K|D (|K ∗ |D , E ∆ ) = D/|K|D (∆p × |K ∗ |D , E)
is a weak equivalence (see [34, Theorem 17.6.3(1)]), and hence the horizontal faceand degeneracy operators of D/|K|D (∆∗ × |K ∗ |D , E) are weak equivalences.

6. SMOOTHING OF CONTINUOUS SECTIONS

51

Since the map |proj|D : |K × ∆[q]|D −→ |K|D is a D-homotopy equivalence(Lemma 2.19, [34, Corollary 7.7.2] and Corollary 4.12), we have the homotopy
equivalences
D/|K|D (∆∗ × |K × ∆[q]|D , E) ∼= S D D/|K|D (|K × ∆[q]|D , E)∗∼
= S D Γ(|K × ∆[q]|D , |proj|D E)' S D Γ(|K|D , E)
(Corollary 6.3(3) and Lemma 6.15), from which we see that the vertical face anddegeneracy operators of D/|K|D (∆∗ × |K ∗ |D , E) are weak equivalences.
From these results, we have the diagram consisting of weak equivalences
S D Γ(|K|D , E) −→ diag D/|K|D (∆∗ × |K ∗ |D , E) ←− D/|K|D (|K ∗ |D , E)([34, Corollary 15.11.12]), which completes the proof.



6.5. Proof of Theorem 1.8. In this subsection, we give a proof of Theorem
1.8.
Proof of Theorem 1.8. Note that pX : |S D X|D −→ X is a D-homotopy
D0eequivalence and hence pf
X : |S X| −→ X is a C -homotopy equivalence (Theorem
1.5(1) and Corollary 4.12). Use the local triviality of E to see that the canonical∗e0D∗ E −→ p
map p]f
X E is an isomorphism in C /|S X| (cf. the proof of LemmaX5.11). Then, by Lemma 6.15 and Remark 6.17, we may assume that X = |K|D
for some K ∈ S. Thus, we consider the cosimplicial resolution |K ∗ |D −→ c|K|Dof the terminal object |K|D of D/|K|D (see the proof of Theorem 6.13). Using theQuillen paire· : D/|K|D  C 0 /|K| : R0
(Proposition 6.1(2)), we have the isomorphism between the homotopy function
complexese −eD/|K|D (|K ∗ |D , R0 E)→ C 0 /|K|(|K ∗ |, E)
∼
=

e −→
(see Lemma 5.11, Corollary 5.15(1), and [34, Proposition 17.4.16]). Since R0 E
0ee
|K|D is a fibration in D with fiber RF , id : E −→ R E is a weak equivalence
between fibrant objects in D/|K|D (see [43, Lemma 9.6(1)]). Thus, we see that thenatural mape
D/|K|D (|K ∗ |D , E) −→ D/|K|D (|K ∗ |D , R0 E),
and hence the natural mape
D/|K|D (|K ∗ |D , E) −→ C 0 /|K|(|K ∗ |, E)
is a weak equivalence, which shows thate
S D Γ(|K|D , E) −→ SΓ(|K|, E)
is a weak equivalence (see Proposition 6.12 and Theorem 6.13).

52

HIROSHI KIHARA

e X)
e num
7. Dwyer-Kan equivalence between (PDG/X)num and (PC 0 G/
In this section, we establish the Dwyer-Kan equivalence between (PDG/X)num
e X)
e num under the conditions that X ∈ WD and that G ∈ VD (Theoremand (PC 0 G/
1.9). After introducing the notion of a groupoid enriched over a cartesian closed
category, we use the smoothing theorem for sections (Theorem 1.8) to prove asmoothing result for gauge transformations, which along with Theorem 5.17 implies
Theorem 1.9.
Recall that M denotes one of the categories C 0 and D (see Section 1.4).
7.1. Enrichment of categories embedded into M. Let A and B be Mcategories and I : A ,−→ B an embedding (i.e., a faithful functor). We say that
I : A ,−→ B is an M-embedding if the inclusion A(A, A0 ) ,−→ B(IA, IA0 ) is an
M-embedding for any A, A0 ∈ A (see Remark 2.3(1)).
If a category A has an embedding A ,−→ B into an M-category B, then A
has the canonical M-category structure for which A ,−→ B is an M-embedding(see Proposition 2.2 and Remark 2.4). Thus, we have the commutative diagram ofM-embeddings
MG
(7.1)

PMG/X

MG/X

M

M/X
for every X ∈ M and every group G in M (see Definition 5.4 for the categories in(7.1)). Note that the hom-object M/X(E, E 0 ) of M/X introduced here coincides
with that introduced by pullback diagram (6.1). Since every M-category is alsoan S-category via the singular functor S M : M −→ S (Proposition 2.12), we canregard (7.1) as a commutative diagram of S-categories. By Definition 5.4, PMG/X
is an M-full subcategory and hence an S-full subcategory of MG/X.
Assume given a commutative diagram of categories
A

F

A0

B

Φ

B0

in which the vertical arrows are embeddings and the lower horizontal arrow is an
M-functor between M-categories. Then, F is an M-functor with respect to thecanonical M-subcategory structures of A and A0 . Recall that if C 0 is regarded asa D-category via R : C 0 −→ D, then e· : D −→ C 0 is a D-functor (see the proof of
Lemma 4.2). Then, we see that the D-functor e· : D −→ C 0 restricts to the D-(ande · : DG −→ C 0 G,
e e· : DG/X −→ C 0 G/
e X,
e and
hence S-)functors e· : D/X −→ C 0 /X,e0eee· : PDG/X −→ PC G/X.
Remark 7.1. Since C ∞ is a full subcategory of D (Proposition 2.6), the categories in (7.1) are also D-categories and hence S-categories even if M and X
are replaced with C ∞ and M respectively. Especially, C ∞ , C ∞ /M , and PC ∞ G/M
are D-(and hence S-)full subcategories of D, D/M , and PDG/M , respectively, and

e X)
e NUM
7. DWYER-KAN EQUIVALENCE BETWEEN (PDG/X)NUM AND (PC 0 G/

53

their hom-objects are denoted by C ∞ (N, N 0 ), C ∞ /M (E, E 0 ), and PC ∞ G/M (P, P 0 ),respectively (see Remark 4.6).
7.2. Enriched groupoid PMG/X. In this subsection, we introduce the notion of a groupoid enriched over a cartesian closed category and show that PMG/X
is a groupoid enriched over M. For a principal G-bundle π : P −→ X in M, thegauge group GauM (P ) is introduced as the automorphism group of P in the Mgroupoid PMG/X.
Definition 7.2. Let V be a cartesian closed category. A V-category A is calleda V-groupoid if for any pair A, A0 ∈ A, there exists a morphism of V·−1 : A(A, A0 ) −→ A(A0 , A)such that the following diagram commutes:
A(A0 , A) × A(A, A0 )(·−1 ,1A(A,A0 ) )A(A, A0 )(1A(A,A0 ) , ·−1 )A(A, A0 ) × A(A0 , A)

composition

A(A0 , A0 )composition
unit

∗

unit

A(A, A).

For an object A of a V-groupoid A, the automorphism group AutA (A) of A isdefined by
AutA (A) = A(A, A),
which is a group in V. S-groupoids are just simplicial groupoids in the sense of[17].
Lemma 7.3. The M-category PMG/X is an M-groupoid, and hence an Sgroupoid via the singular functor S M : M −→ S.
Proof. Since PMG/X is a groupoid (Lemma 5.6), we have the inverse-assigning
function·−1 : PMG/X(E, E 0 ) −→ PMG/X(E 0 , E)
0for E, E ∈ PMG/X. Thus, we have only to show that the map ·−1 is a morphismof M.
Observe that the natural bijections(7.2)

M(A, PMG/X(E, E 0 )) = M(A, MG/X(E, E 0 ))∼ MG/X(A × E, E 0 )=∼= MG/A × X(A × E, A × E 0 )= PMG/A × X(A × E, A × E 0 )exist (see Section 7.1). Then, we can easily see that ·−1 is a morphism of M.

Let π : P −→ X be a principal G-bundle in M. Then, the group GauM (P ) ofgauge transformations is defined by
GauM (P ) = AutPMG/X (P ).
By the definition, we have
GauM (P ) = PMG/X(P, P )= MG/X(P, P )(see Section 7.1). When PMG/X is regarded as an S-groupoid, the automorphismgroup of P in PMG/X is just S M GauM (P ).

54

HIROSHI KIHARA

7.3. Smoothing of gauge transformations. In this subsection, we formulate the gauge group GauM (P ) as a space of sections, and apply Theorem 1.8 toestablish a smoothing result of gauge transformations. For this, we begin by refining well-known constructions (e.g., [46, Section 37]) in the convenient categoryM.
Let π : P −→ X be a principal G-bundle in M and F an object of M endowedwith a left G-action. Then, the object P ×G F −→ X of M/X is defined by
P ×G F = P × F/ ∼,where (ug, f ) ∼ (u, gf ) for every u ∈ P , g ∈ G, and f ∈ F . Note that a left G-object
F of M can be regarded as a right G-object of M by f · g := g −1 · f (f ∈ F, g ∈ G).
Lemma 7.4. Let π : P −→ X be a principal G-bundle in M and F a left
G-object of M.
(1) The canonical map P ×G F −→ X is an F -bundle in M.
(2) There exists a natural isomorphism in M
MG(P, F ) ∼= Γ(X, P ×G F ).
Proof. (1) Consider a commutative diagram in M
U ×G×F

j×1

j×1

U ×F

P ×F
P ×G Fi

U

X

such that the outer rectangle is the local trivialization of the G × F -bundle P × F
induced by a local trivialization of the principal G-bundle P (see the proofs ofLemmas 5.7 and 5.11) and the middle horizontal arrow j × 1 is induced from theupper horizontal arrow j × 1 via the quotient space functor ·/G : MG −→ M.
Since j × 1 is an open M-embedding, j × 1 is also an open M-embedding. Further,since the lower square is a pullback diagram in Set, it is a pullback diagram in M,
and hence a local trivialization of P ×G F .
(2) Define the map
Φ : MG(P, F ) −→ Γ(X, P ×G F )by assigning to an equivariant map ϕ the section [1P , ϕ] obtained from the equivariant map (1P , ϕ) : P −→ P × F via the quotient space functor ·/G : MG −→ M.
The map Φ is then a bijection. In fact, the inverse of Φ is defined by assigning toa section s the equivariant map ϕs making the following diagram commute:P(1P ,ϕs )

P ×F

π

X

s

P ×G F.

Next, we show that the natural bijection Φ is an isomorphism in M. Replacing
π : P −→ X with 1 × π : A × P −→ A × X, we have the natural bijection
MG(A × P, F ) −→ Γ(A × X, (A × P ) ×G F ).

e X)
e NUM
7. DWYER-KAN EQUIVALENCE BETWEEN (PDG/X)NUM AND (PC 0 G/

55

Observe that the natural bijections(7.3)

MG(A × P, F ) ∼= M(A, MG(P, F )),Γ(A × X, (A × P ) ×G F ) ∼= M(A, Γ(X, P ×G F ))exist (see [43, Lemma 2.5 and its proof] and (6.2) for the second bijection). Then,
we have the natural bijection
M(A, MG(P, F )) ∼= M(A, Γ(X, P ×G F )),which shows that Φ : MG(P, F ) −→ Γ(X, P ×G F ) is an natural isomorphism inM.
Let π : P −→ X be a principal G-bundle in M. Then, the (non-principal)
G-bundle P [G, conj] −→ X is defined by
P [G, conj] = P ×G (G, conj)= P × G/(ug, h) ∼ (u, ghg −1 ).
Lemma 7.5. Let π : P −→ X be a principal G-bundle in M.
(1) P [G, conj] −→ X is a group in M/X. Thus, Γ(X, P [G, conj]) is a groupin M.
(2) GauM (P ) is isomorphic to Γ(X, P [G, conj]) as a group in M.
Proof. (1) Define the multiplicationµ

P [G, conj] ×X P [G, conj] −→ P [G, conj]
by([u, g1 ], [u, g2 ]) 7−→ [u, g1 g2 ]and verify that µ is well-defined and associative. Then, we can easily find the unit
map e : X −→ P [G, conj] and the inverse map ν : P [G, conj] −→ P [G, conj] whichmakes (P [G, conj], µ) a group in M/X. Since Γ(X, ·) is a right adjoint (Corollary
6.3(2)), Γ(X, P [G, conj]) is a group in M.
(2) Note that MG(P, (G, conj)) is a group in M with respect to the obvious
multiplicative structure (see (7.3)). Next, observe that the natural isomorphism inLemma 7.4(2) specializes to the natural isomorphism of groups in M
∼ Γ(X, P [G, conj]).
MG(P, (G, conj)) =Then, we have only to show that there exists a natural isomorphism of groups inMΨ : MG(P, (G, conj)) −→ GauM (P ).
Define the map Ψ by assigning to an equivariant map ϕ the automorphism ϕ̂ : P −→P such that ϕ̂(u) = uϕ(u). We can easily see that Ψ is a group isomorphism. Next,
let us see that Ψ is an isomorphism of groups in M. By replacing π : P −→ X
with 1 × π : A × P −→ A × X, we have the group isomorphismΨ : MG(A × P, (G, conj)) −→ GauM (A × P ).
Using (7.3) and (7.2), we have the natural isomorphism of groups
M(A, MG(P, (G, conj))) ∼= M(A, GauM (P )),which shows that Ψ is an isomorphism of groups in M and completes the proof. 

56

HIROSHI KIHARA

Proposition 7.6. Let π : P −→ X be a D-numerable principal G-bundle in
D. If X is in WD and G is in VD , then the natural inclusion
S D GauD (P ) ,−→ SGauC 0 (Pe)is a weak equivalence.
Proof. Since P [G, conj] = P ×G (G, conj) = P × G/ ∼, we havee ∼ = Pe[G,e conj](P [G, conj])e = (P × G/ ∼)e = Pe × G/
(see [43, Lemma 2.11 and Proposition 2.13]). Thus, the result follows from Lemmas
7.4 and 7.5 and Theorem 1.8.

7.4. Proof of Theorem 1.9. In this subsection, we give a proof of Theorem
1.9.
PMG/X is an S-category with function complex S M PMG/X(P, P 0 ) and theunderlying topological space functor e· defines the S-functore Xee· : PDG/X −→ PC 0 G/
(see Section 7.1).
Let us recall the notion of a Dwyer-Kan equivalence of simplicial categories ([3,p. 2044]). First recall the homotopy category π0 A of a simplicial category A from
Section 4.3. A simplicial functor F : A −→ B is called a Dwyer-Kan equivalence if
the following two conditions are satisfied:(i) π0 F : π0 A −→ π0 B is an equivalence of categories.
(ii) F : MapA (A, A0 ) −→ MapB (F A, F A0 ) is a weak equivalence in S for any
A, A0 ∈ A.
If A is a simplicial groupoid, then the isomorphism classes of objects of π0 A
are just those of objects of A and the following holds:(
MapM (A, A) = AutM (A) if A ∼= A0
0 ∼MapM (A, A ) =∅if A  A0 .
Thus, a simplicial functor F : A −→ B between simplicial groupoids is a Dwyer-Kan
equivalence if and only if it satisfies the following conditions:
0(i) F induces the bijection between the isomorphism classes of objects of A
and those of objects of B.
0(ii) F : AutA (A) −→ AutB (F A) is a weak equivalence in S for any A ∈ A.
e X)
e num are simpliProof of Theorem 1.9. Since (PDG/X)num and (PC 0 G/
cial groupoids (Lemma 7.3), we have only to show that the simplicial functor
e X)
e num satisfies conditions (i)0 and (ii)0 , which aree· : (PDG/X)num −→ (PC 0 G/
verified in Theorem 5.17 and Proposition 7.6.

8. Diffeological polyhedra
Polyhedra, which are a special type of CW -complexes, not only behave well assources but also play an important role as targets (recall the notion of a barycentricmap).
In the diffeological context, we must introduce two kinds of notions of a diffeological polyhedron, which correspond to the two kinds of standard simplices{∆p }p≥0 and {∆psub }p≥0 . Diffeological polyhedra corresponding to {∆p }p≥0 , which8. DIFFEOLOGICAL POLYHEDRA

57

are called D-polyhedra, form an important class of cofibrant objects, and hence
behave well as sources. Diffeological polyhedra corresponding to {∆psub }p≥0 are
more directly related to constructions of smooth maps and smooth homotopies andespecially provide targets of barycentric maps (Lemmas 8.3 and 8.4). These twokinds of diffeological polyhedra are not diffeomorphic but naturally D-homotopy
equivalent (Theorem 8.6).
8.1. Basic properties of two kinds of diffeological polyhedra. In thissubsection, we introduce and study two kinds of diffeological polyhedra.
Let us begin by recalling the definition of a simplicial complex. A simplicial
complex K consists of a set K of vertices and a set ΣK of finite nonempty subsets
of K called simplices such that(i) Any set consisting of exactly one vertex is a simplex.
(ii) Any nonempty subset of a simplex is a simplex.
A simplicial map between simplicial complexes is defined in an obvious manner.
The category of simplicial complexes and simplicial maps is denoted by SC. The
notion of a subcomplex of a simplicial complex K, especially that of the p-skeleton
K p of K, is also defined in an obvious manner ([64, Section 1 in Chapter 3]).
For a simplicial complex K, the set |K| is defined by




X
|K| = x : K −→ [0, 1] {v ∈ K | x(v) > 0} ∈ ΣK andx(v) = 1 .
v∈KFor a vertex v of K, xv ∈ |K| is defined by(
1 if v 0 = v,
0xv (v ) =0 if v 0 6= v.
We usually identify v ∈ K with xv ∈ |K|. For a p-simplex σ = {v0 , . . . , vp } of K,
the closed p-simplex |σ| of |K| is defined by|σ| = {x ∈ |K| | x(v) > 0 ⇒ v ∈ σ},which is just the convex full of σ = {v0 , . . . , vp }.
The D-polyhedron |K|D of a simplicial complex K is defined to be the set|K| endowed with the final diffeology for the inclusions {|σ| ,−→ |K| | σ ∈ ΣK },where the closed p-simplex |σ| is regarded as a diffeological space via an affine
identification with ∆p . Thus, we have the functor | |D : SC −→ D (see Proposition
2.11). The underlying topological space of |K|D is just the topological polyhedronof K (Proposition 2.11 and [43, Lemma 2.11]), which is also denoted by |K| ([64,p. 111]). Since the D-polyhedron |K|D can be identified with the realization of K
viewed as a simplicial set by choosing a well-ordering on the vertices ([51, Example1.4]), |K|D is a cofibrant object.
The D-polyhedron |L|D of a subcomplex L of K is called a D-subpolyhedron
of |K|D . The following is a basic result on D-subpolyhedra.
Lemma 8.1. Let K be a simplicial complex.
(1) The inclusion |L|D ,−→ |K|D is a cofibration and a D-embedding for any
subcomplex L of K.

58

HIROSHI KIHARA(2) Let {Li } be a set of subcomplexes of K. Then, L := ∪ Li = (∪ Li , ∪ ΣLi )is also a subcomplex of K and the canonical mapa|Li |D −→ |L|D
is a D-quotient map.
Proof.
(1) By choosing a well-ordering on the vertices of K, we canregard the inclusion L ,−→ K as a monomorphism (i.e., a cofibration)in S. Thus, |L|D ,−→ |K|D is a cofibration in D (Lemma 2.19) and a
D-embedding ([43, Proposition 9.2]).
(2) The result is immediate from the definition of a D-polyhedron.
For a simplicial complex K, we can introduce another diffeology on |K| byusing {∆psub }p≥0 instead of {∆p }p≥0 (see Section 2.3); the resulting diffeologicalspace is denoted by |K|0D , whose underlying topological space is also the topologicalpolyhedron of K (see Lemma 2.17(2) and [43, Lemma 2.11]).
Let ∆(p) denote the simplicial complex defined by ∆(p) = {0, · · · , p} andΣ∆(p) = { nonempty subsets of ∆(p)}. Then, |∆(p)|D = ∆p and |∆(p)|0D = ∆psub
hold obviously. The simplicial complex ∆(1) is often denoted by I.
Regard a p-simplex σ of a simplicial complex K as a subcomplex of K which isisomorphic to ∆(p). Then, |σ|D (resp. |σ|0D ) is just the closed p-simplex |σ| viewed
as a diffeological space via an affine identification with ∆p (resp. ∆psub ).
Remark 8.2. Let K be a simplicial complex and ι : L ,−→ K the inclusion of
a subcomplex L into K. Then, |ι|D : |L|D ,−→ |K|D is a D-embedding (see Lemma
8.1(1)). On the other hand, |ι|0D : |L|0D ,−→ |K|0D need not be a D-embedding;consider the subcomplex L of K = ∆(2) defined byΣL = {{(0)}, {(1)}, {(2)}, {(0), (1)}, {(0), (2)}}and see [43, the proof of Proposition A.2(1)]. However, if L is a p-simplex σ viewed|ι|0D
as a subcomplex of K, then the map ∆psub ∼= |σ|0D ,−−→ |K|0D is a D-embedding.
This result is shown as follows. Fix an isomorphism ∆(p) ∼= σ. Since a retractionιρ : K −→ ∆(p) to ∆(p) ∼
σ
,−
−
−
→
K
is
defined
by
ρ(v)
=
(0)
for v 6∈ σ, we obtain=
the retract diagram

0

|σ|D

|ι|0D

0

|K|D

|ρ|0D

0

|σ|D ,

1which implies that |ι|0D is a D-embedding.
For a covering U = {Uα }α∈A of a diffeological space X by subsets, we define
the nerve N U to be the simplicial complex such that N U = {α ∈ A | Uα 6= ∅} and{α0 , . . . , αp } ∈ ΣN U if and only if Uα0 ∩ · · · ∩ Uαp 6= ∅.
Lemma 8.3. Let U = {Uα }α∈A be a covering of a diffeological space X bysubsets and {ϕα }α∈A be a smooth partition
P of unity subordinate to U. Then themap ϕ : X −→ |N U|0D defined by ϕ(x) =ϕα (x)α is smooth.
α∈A

8. DIFFEOLOGICAL POLYHEDRA

59

Proof. For a given x ∈ X, σ := {α | x ∈ supp ϕα } is a finite subset of A suchthat x ∈ Uσ := ∩ Uα . Noticing that {supp ϕα | α ∈ A} is locally finite, we defineα∈σthe open neighborhood Vσ of x by
Vσ = Uσ − ∪ supp ϕα .
α6∈σThen, there exists a smooth map φσ : Vσ −→ |σ|0D making the diagram
0

|σ|D
φσ

Vσ

X

ϕ

0

|N U|D

commute. Hence, ϕ is smooth.



Lemma 8.4. Let X be a diffeological space, and K a simplicial complex. Let
0f, g : X −→ |K|D be smooth
Pmaps.`Suppose that there exists a set {φi : Xi −→ X}of smooth maps such that
φi : Xi −→ X is a D-quotient map and f (φi (Xi ))and g(φi (Xi )) are contained in the same closed simplex of |K|0D for any i. Then,f is D-homotopic to g.
Proof. By the assumption, we can define the set-theoretic map h : X × I −→|K|0D by h(x, t) = (1 − t)f (x) + tg(x). Since f ◦ φi and g ◦ φi factor through a Dp
∼ 0 ,−→ |K|0 corresponding to some simplex σ of K (Remark
embeddingDP∆sub = |σ|`D8.2) andφi × 1 : (Xi × I) −→ X × I is a D-quotient map ([43, Lemma 2.5]
and Proposition 2.2(2)), h is a smooth homotopy between f and g.
Lemmas 8.3 and 8.4 along with the following remark explain why we need not
only {∆p }p≥0 but also {∆psub }p≥0 .
Remark 8.5. We show that a barycentric map to |N U|D and a linear homotopy
between smooth maps to |K|D need not be smooth. For this, we need the following
fact:
FACT. Let c : (−ε, ε) −→ ∆2sub be a smooth curve. If the images c((−ε, 0)) and c((0, ε)) are contained in the open 1-simplices(0, 1) and (0, 2) respectively, then c is not smooth as a curve into
∆2 .
(See [43, the proof of Proposition A.2(1)]).
(1) (A non-smooth barycentric map to |N U|D ) Lemma 8.3 does not hold if| |0D qis replaced by | q|D . In fact, choose positive numbers r0 and r such20
3 <r <r <23
2 and consider the open covering U = {U0 , U1 , U2 }of X = ∆ defined by Ui = {x ∈ ∆2 | d(x, (i)) < r}, where d(·, ·) is thestandard distance function of R3 . We construct smooth functions fi :∆2 −→ [0, 1] with supp fi = {x ∈ ∆2 | d(x, (i)) ≤ r0 } (see [6, p. 64]), andthat

define the partition of unity {ϕ0 , ϕ1 , ϕ2 } subordinate to U by
ϕi = fi /(f0 + f1 + f2 ).
Let P be the point of ∆2 such that d(P, (1)) = d(P, (2)) = r0 and define
the smooth curve ` : (−ε, ε) −→ ∆2 by `(t) = P + t(0, −1, 1). Since`ϕthe composite (−ε, ε) −−−−→ ∆2 −−−→ |N U|D = ∆2 is not smooth, the60

HIROSHI KIHARA

barycentric map ϕ : ∆2 −→ |N U|D is not smooth. (See Fact and Fig.
8.1.)
(2) (A non-smooth linear homotopy between smooth maps to |K|D ) The essenceof the proof of Lemma 8.4 is the following:
Under the assumptions of Lemma 8.4, the linear homotopy h :X × I −→ |K|0D between f and g is smooth.
This does not hold if | |0D is replaced by | |D . In fact, consider the linear
embeddings f, g : [0, 1] −→ ∆2 depicted in Fig. 8.2 (see Proposition 2.11
for the smoothness of f and g). Then, the linear homotopy
h : [0, 1] × I −→ ∆2
is not smooth (see Fact).
We, however, show that Lemma 8.4 remains true even if | |0D is replaced by | |D (see Corollary 8.8).
(0)

(0)

g

f
(1)
(2) (1)(2)
Fig 8.1 Supports of ϕ0 , ϕ1 , and ϕ2 . Fig 8.2 The linear embeddings f and g.
8.2. D-homotopy equivalence between two kinds of diffeological polyhedra. Since id : ∆p −→ ∆psub is smooth, id : |K|D −→ |K|0D is a smooth map
which is a homeomorphism with respect to the underlying topologies (see Lemma
2.17). The rest of this section is devoted to proving the following theorem.
Theorem 8.6. Let K be a simplicial complex. Then the (set-theoretic) identity0

id : |K|D −→ |K|D
is a D-homotopy equivalence.
For the proof of Theorem 8.6, we recall the local diffeological structures of ∆pand ∆psub from [43, Section 4]. Let I = {i0 , . . . , ik } be a subset of the ordered set{0, . . . , p}; we always assume that the elements i0 , . . . , ik are arranged such thati0 < · · · < ik . Then, (i0 , . . . , ik ) denotes the open k-simplex defined by(i0 , . . . , ik ) = {(x0 , . . . , xp ) ∈ ∆p | xi = 0 if and only if i 6∈ {i0 , . . . , ik }},and UI denotes the diffeological subspace of ∆p defined by
UI = {(x0 , . . . , xp ) ∈ ∆p | xi > 0 for i ∈ I}.
UI is an open neighborhood of the open simplex (i0 , . . . , ik ) of ∆p . For I = {i}, UI
is just the ith half-open simplex ∆pî ([43, Definition 4.1]).

8. DIFFEOLOGICAL POLYHEDRA

61

Proposition 8.7. Let I = {i0 , . . . , ik } be a subset of the ordered set {0, . . . , p}and define the map
˚ k × ∆p−kΦI : UI −→ ∆0̂byΦI (x0 , . . . , xp ) = (xi0 (0) + · · · + xik (k), (xi0 +· · ·+xik )(0)+xj1 (1)+· · ·+xjp−k (p−k)),x i0 + · · · + x ik

where {j1 , . . . , jp−k } = {0, . . . , p}\{i0 , . . . , ik }.
˚ k × ∆p−k is a diffeomorphism.
(1) ΦI : UI −→ ∆0̂˚ k × ∆p−k is a diffeomorphism.
(2) ΦI : UI sub −→ ∆0̂ sub
Proof. See [43, Proposition 4.5, Remark 4.7(2), and Lemma 4.2].

In the proof of Theorem 8.6, I denotes a subset {i0 , . . . , ik } of the ordered set{0, . . . , p}; the unit interval is denoted by [0, 1] to avoid confusion. For a subset
B of a topological space A, B and B ◦ denote the closure and the interior of B
respectively. We write h : f 'D g for a D-homotopy h : X × [0, 1] −→ Y connecting
f to g.
Proof of Theorem 8.6. Suppose that there exists a smooth map ψK : |K|D −→|K|D satisfying the following conditions:• ψK preserves each closed simplex.
• ψK is D-homotopic to 1|K|D .
0• ψK is smooth as a map from |K|D to |K|D .
0

0

Then ψK : |K|D −→ |K|D is a D-homotopy inverse of id : |K|D −→ |K|D (seeLemma 8.4). We thus construct such a smooth map ψK in five steps.
Step 1: Constructions of λp : ∆p0̂ −→ ∆p0̂ and Λp : λp 'D 1∆p . For a positive0̂
number  < 1, we choose a monotone increasing smooth function λ : [0, 1) −→ [0, 1)
such that(0 (0 ≤ t ≤ 2 )
λ(t) =
and 0 < λ(t) < t on ( , ),2t (t ≥ )and define Λ : [0, 1) × [0, 1] −→ [0, 1) to be the linear homotopy connecting λ to1[0,1) ; explicitly,
Λ(t, s) = (1 − s)λ(t) + st.
pThen, the smooth maps λ : ∆p0̂ −→ ∆p0̂ and Λp : ∆p0̂ × [0, 1] −→ ∆p0̂ are defined by
the commutative diagrams∆p−1 × [0, 1)1×λ∆p−1 × [0, 1)

λp∆p0̂ ,

ϕ0

∆p0̂

ϕ0

∆p−1 × [0, 1) × [0, 1]

1×Λ

∆p−1 × [0, 1)

Λp

∆p0̂

ϕ0

ϕ0 ×1

∆p0̂ × [0, 1](see Definition 2.10 and [43, Lemmas 4.3 and 2.5]). The map Λp is a D-homotopy
connecting λp to 1∆p and is given by0̂Λp (x, s) = (1 − s)λp (x) + sx.
Step 2: Notations. We introduce some notations, which are used only in this proof.

62

HIROSHI KIHARA

Let f : A −→ A be a continuous self-map of a topological space A. Then, we
setcarr f = {x ∈ A | f (x) 6= x} and supp f = carr f .
In the case where A is an open set of ∆p , we also set
carrk f = {x ∈ A | f (x) 6= x, f (x) ∈ skk ∆p } and suppk f = carrk f(see [43, Definition 4.1] for the k-skeleton skk ∆p ).
Recalling that I = {i0 , . . . , ik }, we next define the open neighborhood VI of theopen k-simplex (i0 , . . . , ik ) in ∆p by
VI = {(x0 , . . . , xp ) ∈ ∆p | xi > xj for i ∈ I and j 6∈ I},
which is obviously contained in UI .
Step 3: Constructions of ψkp : ∆p −→ ∆p and hpk : ψkp 'D 1∆p . First, we construct
sequences of smooth maps
ψp

p
ψk−1

ψp

ψp

1
0
k
→ ∆p −−−→ ∆p (p ≥ 0)
· · · −→ ∆p −−−→ ∆p −−−→ · · · −−−

satisfying the following conditions:(i) ψkp is equivariant with respect to the obvious action of the symmetricgroup Sp+1 on ∆p and satisfies the inclusion relationasupp ψkp ⊂V{i0 ...ik } .
i0 <···<ik(ii)ψkp preserves each closed simplex of ∆p and coincides with ψkp−1 on eachp−1closed (p − 1)-simplex under an affine identification with ∆ .
(iii) ψkp = 1∆p on the k-skeleton skk ∆p , and hence ψkp = 1∆p for k ≥ p.
(iv) (supp0 ψ0p )◦ ∪ · · · ∪ (suppk ψkp )◦ is a neighborhood of skk ∆p .
By condition (iii), we may assume that k < p. The maps ψkp are constructedby induction on k. To construct ψ0p : ∆p −→ ∆p , choose a positive number 0 < 21
and consider the map λp : ∆p0̂ −→ ∆p0̂ for  = 0 (Step 1). Since supp λp ⊂ V{0} ,ψ0p is defined by ψ0p |V{0} = λp and condition (i). The maps ψ0p (p ≥ 0) obviously
satisfy conditions (i)-(iv).
Suppose that the maps ψdp are constructed for d < k. Let us construct ψkp :p∆ −→ ∆p . First, choose a smooth function φk : ∆k −→ [0, 1] satisfying thefollowing conditions:
˙ k.
• φk ≡ 0 near ∆k ◦k• (supp0 ψ0 ) ∪ · · · ∪ (suppk−1 ψk−1)◦ ∪ (φk )−1 (1)◦ = ∆k .
k
• φ is Sk+1 -invariant.
(Use [43, Lemma 3.1] and [6, Remark 7.4].) Set mk = min min {yi | (y0 , . . . , yk ) ∈
0≤i≤k
p−k
p−k
1
p−k
: ∆0̂ −→ ∆0̂ be the map defined in Step
supp φ } and k = 2 mk , and let λ˚ k × ∆p−k via Φ{0,...,k}1 for  = k . Recalling that U{0,...,k} is diffeomorphic to ∆0̂˚ k ×∆p−k ) defined(Proposition 8.7(1)), we consider the self-map λpk on U{0,...,k} (∼
=∆
0̂
k

by

(y, z) 7−→ (y, φk (y)λp−k (z) + (1 − φk (y))z).
Since λpk (y, z) = (y, Λp−k (z, 1 − φk (y))), λpk is smooth. Note that
supp λpk ∼= {(y, z) | y ∈ supp φk , z1 + · · · + zp−k ≤ k },8. DIFFEOLOGICAL POLYHEDRA(suppk λpk )◦ ∼= {(y, z) | y ∈ (φk )−1 (1)◦ , z1 + · · · + zp−k <

63

k
}.
2

Recalling the definition of Φ{0,...,k} (Proposition 8.7) and the choice of k , we seethat if x ∈ supp λpk , then (y, z) = Φ{0,...,k} (x) satisfies the inequalities
xj = zj−k ≤ k =1
1
mk ≤ yi < xi
2
2

for i ≤ k and k < j, and hence that supp λpk ⊂ V{0,...,k} . Thus, ψkp is definedby ψkp |V{0,...,k} = λpk |V{0,...,k} and condition (i). We can easily see that the maps ψkp
satisfy conditions (i)-(iv).
Next, we construct the D-homotopy hpk connecting ψkp to 1∆p . Define the mappΛk : U{0,...,k} × [0, 1] −→ U{0,...,k} by(y, z, s) 7→ (y, φk (y)Λp−k (z, s) + (1 − φk (y))z)˚ k × ∆p−k (Proposition 8.7(1)). Since theunder the identification U{0,··· ,k} ∼
= ∆
0̂

∆p−k
-component of Λpk is just Λp−k (z, φk (y)s + (1 − φk (y))), it is a smooth map,0̂
which restricts to a D-homotopy connecting ψkp |V{0,...,k} to 1V{0,...,k} . Thus, this Dhomotopy extends to the desired D-homotopy hpk : ∆p ×[0, 1] −→ ∆p in the obvious
manner. Then, hpk is an Sp+1 -equivariant D-homotopy that preserves each closed
simplex of ∆p and coincides with hkp−1 on each closed (p − 1)-simplex under anaffine identification with ∆p−1 . Further, hpk coincides with the constant homotopy
at 1∆p on the k-skeleton skk ∆p .
Step 4: Constructions of ψK : |K|D −→ |K|D and hK : ψK 'D 1|K|D . By condition(ii) on the maps ψkp (Step 3), {ψkp }p≥0 defines the smooth map ψk : |K|D −→ |K|D .
Thus, the smooth map ψK : |K|D −→ |K|D is defined by ψK = ψ0 ◦ ψ1 ◦ ψ2 ◦ · · · ;note that ψK = ψ0 ◦ ψ1 ◦ · · · ◦ ψp−1 on a closed p-simplex of |K|D .
Let us see that ψK 'D 1|K|D : |K|D −→ |K|D . Since hpk coincides with hp−1
onkeach (p − 1)-simplex, {hpk }p≥0 defines the D-homotopy hk : |K|D × [0, 1] −→ |K|D
connecting ψk to 1|K|D (see [43, Lemma 2.5]). Set b
hk = (hk , p[0,1] ) : |K|D ×[0, 1] −→|K|D × [0, 1], where p[0,1] is the projection of |K|D × [0, 1] onto [0, 1], and define
the map b
hK : |K|D × [0, 1] −→ |K|D × [0, 1] by b
hK = bh0 ◦ bh1 ◦ bh2 ◦ · · · . Since
p
b
b
b
b
hK = h0 ◦ h1 ◦ · · · ◦ hp−1 on |K |D × [0, 1] (see the end of Step 3) and |K|D × [0, 1] =lim (|K p |D × [0, 1]) (see Propositions 2.12(1) and 2.2(2)), we see that b
hK is a well−→defined smooth map. Thus, the smooth map hK : |K|D × [0, 1] −→ |K|D , definedas the composite
b
h

proj

K
|K|D × [0, 1] −→|K|D × [0, 1] −→ |K|D ,is a D-homotopy connecting ψK to 1|K|D .
Step 5: Smoothness of ψK : |K|0D −→ |K|D . We show that ψK is smooth as a map
from |K|0D to |K|D . By the construction, we have only to show that ψ p (:= ψ0p ◦p
˚p = ∆˚ p (Lemma 2.17(3)), it suffices· · · ◦ ψp−1
) : ∆psub −→ ∆p is smooth. Since ∆subpp
to see that ψ is smooth on the neighborhood (supp0 ψ0p )◦ ∪ · · · ∪ (suppp−1 ψp−1)◦of the boundary.

64

HIROSHI KIHARAFirst, we show that for k < l, ψlp preserves (suppk ψkp )◦ and makes the following
diagram commute:ψlp(suppk ψkp )◦(suppk ψkp )◦

(8.1)
p
ψk

p
ψk

∆p

By the construction in Step 3, we have(suppk ψkp )◦=a(suppk ψkp )◦{i0 ,...,ik } ,i0 <···<iksupp ψlp⊂a

V{i00 ,...,i0l } ,i00 <···<i0l

where (suppk ψkp )◦{i0 ,...,ik } := (suppk ψkp )◦ ∩ V{i0 ,...,ik } . Thus, we see how ψlp |V{i0 ,...,i0 }0lmaps the points of (suppk ψkp )◦{i0 ,...,ik } . Since V{i0 ,...,ik } ∩V{i00 ,...,i0l } = ∅ if {i0 , . . . , ik } 6⊂{i00 , . . . , i0l }, we may assume that {i0 , . . . , ik } ⊂ {i00 , . . . , i0l }. For simplicity, we assume that {i0 , . . . , ik } = {0, . . . , k} and {i00 , . . . , i0l } = {0, . . . , l}. By the definition,
ψlp |V{0,...,l} sends (x0 , . . . , xp ) to (x00 , . . . , x0p ) such that (x00 , . . . , x0l ) = r(x0 , . . . , xl )for some r ≥ 1 and (x0l+1 , . . . , x0p ) = s(xl+1 , . . . , xp ) for some s ≤ 1. Thus, un0∼∆
˚ k × ∆p−k , ψ p |V
der the identification U{0,...,k} =l {0,...,l} sends (y, z) to (y, z ) with0̂pp0
z1 + · · · + zp−k ≥ z10 + · · · + zp−k
. Hence, ψl preserves (suppk ψk )◦ and makes
˚ k × ∆p−k ,diagram (8.1) commute; note that under the identification U{i0 ,...,ik } ∼=∆0̂ψkp |(suppk ψkp )◦ sends (y, z) to (y, (0)).
From this, we have the commutative diagram in Set
ψp

p
p
ψk+1
◦···◦ψp−1(suppk ψkp )◦{i0 ,...,ik }(suppk ψkp )◦{i0 ,...,ik }
p
ψk
p
ψk

∆p

p
ψ0p ◦···◦ψk−1

∆p

∼
=

(i0 , . . . , ik )˚ k × ∆p−k
∆
0̂

proj

˚k ,
∆

which is a diagram in D if the subsets of ∆p , ∆k , and ∆p−k are endowed with thesubdiffeologies of the standard simplices (Definition 2.10). Thus, using Proposition
8.7, we can see that ψkp : (suppk ψkp )◦{i0 ,...,ik } sub −→ ∆p , and hence ψ p : ∆psub −→∆p is smooth.

Corollary 8.8. Lemma 8.4 remains true even if | |0D is replaced by | |D .
Proof. The result follows from Lemma 8.4 and Theorem 8.6.



9. HOMOTOPY COFIBRANCY THEOREM

65

9. Homotopy cofibrancy theorem
In this section, we prove the homotopy cofibrancy theorem (Theorem 1.10),
which is a diffeological version of a theorem of tom Dieck [67, Theorem 4]. For the
proof, we establish a diffeological version of [62, Proposition 4.1] (Section 9.1) andthen introduce and study the notion of a Hurewicz cofibration in D (Section 9.2).
9.1. Diffeological spaces associated to a covering. In this subsection,
we introduce the diffeological spaces BRU and BXU associated to a covering U ={Uα }α∈A of a diffeological space X and prove that BXU is D-homotopy equivalent
to X under the D-numerability condition on U (Proposition 9.5).
Recall the nerve functor N : Cat −→ S, where Cat denotes the category ofsmall categories ([49, p. 271], [30, p. 5]). Similarly to the topological case, we candefine the classifying space functor B : Cat −→ D by the commutative diagram
Cat

N

| |D

S

D.

B
We first extend these three functors. For a finitely complete category A, Cat(A)denotes the category of category objects in A (see [49, p. 267]). Thus, Cat(D)denotes the category of diffeological categories, which are defined to be category
objects in D. Let sD denote the category of simplicial diffeological spaces (seeSection 3.1 for the notation).
Definition 9.1.
(1) The nerve N C of a diffeological category C can be
regarded as a simplicial diffeological space in an obvious manner. Hence,
the nerve functor N : Cat(D) −→ sD is defined.
(2) We define the|X• |D of a simplicial diffeological space X• to be
R n realization
the coend
Xn × ∆n ([49, Section 6 in Chapter IX]), thereby defining
the realization functor−→ D. Then, |X• |D is just the quotient` | |D : sD
diffeological space
Xn × ∆n / ∼, where the equivalence relation ∼ isndefined by (x, θ∗ (u)) ∼ (θ∗ x, u) for x ∈ Xn , u ∈ ∆m , and θ : [m] −→ [n].
(3) We define the classifying space functor B : Cat(D) −→ D by the commutative diagram
Cat(D)

N

sD

| |D

D.

B
Define the functor δ : Set −→ D to assign to a set A the set A endowed withthe discrete diffeology. Observing that δ has left and right adjoints, we see that δ isa fully faithful functor preserving limits and colimits. Hence, the induced functors
δ : S = sSet −→ sD and δ : Cat = Cat(Set) −→ Cat(D) are also fully faithful andthe following diagram is commutative:Cat

N

Cat(D)

| |D

δ
N

sD

D
=

δ

S

| |D

D.

66

HIROSHI KIHARA

Therefore, Cat and S can be regarded as full subcategories of Cat(D) and sD,
respectively. Conversely, a diffeological category C with ob C and mor C discrete(resp. a simplicial diffeological space X• with each Xn discrete) is often regarded
simply as a small category (resp. a simplicial set).
The following is a basic result on the realization of a simplicial diffeologicalspace.
Lemma 9.2. For a simplicial diffeological space X• , define the diffeological(n)spaces |X• |D by the following conditions:(0)(i) |X• |D = X0 .
(n−1)
(n)
(ii) Suppose that |X• |D
is defined. Then, |X• |D is defined by the pushout
diagram˙n
Xn × ∆∪

˙nsXn−1 ×∆(n−1)

sXn−1 × ∆n

|X• |D(n)

Xn × ∆n|X• |D ,where sXn−1 is the subset ∪ si Xn−1 of Xn endowed with the quotientiP`
diffeology for the surjection si :Xn−1 −→ sXn−1 .
i

i

(n)
Then, lim |X• |D = |X• |D holds.
−→
(n)

(n)

Proof. For a simplicial set K, define the diffeological space |K|D by |K|D =|K |D , where K n is the n-skeleton of K (see the proof of Lemma 3.5). Since theunderlying set functor U : D −→ Set preserves limits and colimits ([43, Proposition
2.1(1)]), we have the equalities in Set
n

(n)

(n)

|X• |D = |U X• |D and |X• |D = |U X• |D ,which imply the equalities in Set
(n)

(n)

|X• |D = |U X• |D = lim |U X• |D = lim |X• |D .
−→−→Noticing that the diffeology of |X• |D is final for the canonical maps Xn × ∆n −→(n)|X• |D (n ≥ 0), we see that lim |X• |D = |X• |D holds in D.
−→We now introduce the diffeological categories RU and XU associated to a covering U = {Uα }α∈A of a diffeological space X by subsets.
First, we define RU to be the category whose objects are the nonempty finite
subsets σ of A with Uσ := ∩ Uα 6= ∅ and whose morphisms are their reverseα∈σinclusions. Regard RU as a diffeological category by endowing ob(RU ) and mor(RU )with discrete diffeologies. Then, N RU is just the barycentric subdivision of thenerve N U of the covering U (see [64, p. 123] and [62, p. 108]).
Next, we define the category XU byob(XU )

= {(x, σ) | σ is a nonempty finite subset of A and x ∈ Uσ },mor(XU )= {(x, σ) → (x, τ ) | σ ⊃ τ }9. HOMOTOPY COFIBRANCY THEOREM

67

∼
and
mor(X
U ) with diffeologies via the obvious bijections ob(XU ) =` endow ob(XU ) and`Uσ and mor(XU ) ∼
Uσ .
=
σ

σ⊃τ

The morphism of diffeological categories
π : XU −→ RU
is defined by π(x, σ) = σ.
Remark 9.3. See [62, Section 4] for simple but somewhat imprecise definitionsof RU and XU . Compare them with the definitions above; the key point is that
distinct elements σ and τ of ΣN U should be distinguished as objects of RU even if
Uσ and Uτ coincide.
To investigate the relation between BXU and X, we construct a smooth injection BXU ,−→ X × BRU . Since BXU = |N XU |D and BRU = |N RU |D , we set
(n)
(n)
(n)
(n)
BXU = |N XU |D and BRU = |N RU |D . Then, we have the pushout diagrams
`
`
(n−1)
(n−1)
˙n
˙n
Uσn × ∆BXU
X ×∆
X × BRUσn )···)σ0`σn )···)σ0(n)Uσn × ∆nBXU ,σn )···)σ0(n)X × ∆n`

X × BRUσn )···)σ0

and the identities
(n)

(n)

BXU = lim BXU , X × BRU = lim X × BRU−→−→(see Lemma 9.2 and Proposition 2.2(2)). From these, we observe that the settheoretic identitiesaa
˚ n and X × BRU =˚n
BXU =Uσ × ∆X ×∆nσn )···)σ0σn )···)σ0hold. We thus see that the canonical inclusions Uσn × ∆n ,−→ X × ∆n define asmooth injection
BXU ,−→ X × BRU .
Then, the map pr : BXU −→ X is defined to be the X-component of this injection;
the BRU -component of this injection is just the map Bπ : BXU −→ BRU .
Let [σp ) · · · ) σ0 ] denote the copy of ∆p indexed by a nondegenerate p-simplex
σp ) · · · ) σ0 of N RU . Then, the canonical mapsaaUσp ×[σp ) · · · ) σ0 ] −→ BXU andX×[σp ) · · · ) σ0 ] −→ X×BRUσp )···)σ0σp )···)σ0

are D-quotient maps.
Remark 9.4. We see that the smooth injection (pr, Bπ) : BXU ,−→ X × BRU
need not be a D-embedding or a topological embedding.
Let X = R and consider the open cover U = {(−∞, 1), (−1, ∞)}. Then, we
have the isomorphism
1
1
X × BRU ∼= R × ([0, ] ∪{ 12 } [ , 1]).
2
2
Under this isomorphism, BXU is identified with the subset(−∞, 1) × (0) ∪ (−1, 1) × [0, 1] ∪ (−1, ∞) × (1)68

HIROSHI KIHARA

set-theoretically. Consider a smooth curve
11
c : R −→ X × BRU ∼= R × ([0, ] ∪{ 12 } [ , 1])
2
2such that• c(0) = (1, 1).
• −1 < c1 (t) < 1, 12 < c2 (t) < 1 for t 6= 0.
(Here, c1 and c2 are the R- and ([0, 12 ] ∪{ 21 } [ 12 , 1])-components of c respectively.)
Noticing that the map
11(−∞, 1) × (0) t (−1, 1) × ([0, ] ∪{ 12 } [ , 1]) t (−1, ∞) × (1) −→ BXU
2
2
is a D-quotient map, we see that the corestriction c : R −→ BXU is not smooth,and hence that BXU ,−→ X × BRU is not a D-embedding.
Next, consider the topological quotient map]U(−∞, 1) × (0) t (−1, 1) × [0, 1] t (−1, ∞) × (1) −→ BX(see [43, Lemma 2.11]). Then, the set
11
V = {(x, y) ∈ (−1, 1) × [0, 1] | (x − )2 + (y − 1)2 < } ∪ (0, 2) × (1)2
4
is an open neighborhood of (1, 1) ∈ BXU . But, V is not an open neighborhood of(1, 1) ∈ BXU with respect to the subspace topology of X × BRU . Hence, BXU ,−→X × BRU is not a topological embedding.
The rest of this subsection is devoted to the proof of the following result, which
is a diffeological version of [62, Proposition 4.1].
Proposition 9.5. Let U = {Uα }α∈A be a covering of a diffeological space X bysubsets. If U is D-numerable, then pr : BXU −→ X is a D-homotopy equivalence.
For the proof, we need a lemma on the barycentric subdivision of a D-polyhedron.
For a simplicial complex K, we have the natural bijection
ηK : |sd K|D −→ |K|D
(see [64, p. 123] for the barycentric subdivision sd K of K); we can easily see that
ηK is smooth and that ηeK is a homeomorphism (Proposition 2.11 and [43, Lemma
2.11]). Thus, |sd K|D is topologically identified with |K|D via ηfK.
Lemma 9.6. Let K be a simplicial complex. Then, the smooth bijection
ηK : |sd K|D −→ |K|D
has a D-homotopy inverse ρK : |K|D −→ |sd K|D such that under the topological
identification of |sd K|D and |K|D , ρK preserves each closed simplex of |sd K|.
Proof. We have only to construct a smooth map ρ0K : |K|D −→ |sd K|0D
which preserves each closed simplex of |sd K|. In fact, Corollary 8.8 shows that the
ρ0

ψsdK

K
composite |K|D −−−→ |sd K|0D −−−−−−→ |sd K|D is the desired D-homotopy inverseof ηK , where ψL : |L|0D −→ |L|D is the smooth map constructed in the proof of
Theorem 8.6.

9. HOMOTOPY COFIBRANCY THEOREM

69

We can reduce the construction of ρ0K to the case of K = ∆(p). More precisely,
to construct such a natural map ρ0K : |K|D −→ |sd K|0D , it suffices to constructsmooth maps
ρ0p : ∆p = |∆(p)|D −→ |sd ∆(p)|0D(p = 0, 1, 2, . . .)satisfying the following conditions:(i) ρ0p is equivariant with respect to the obvious actions of the symmetricgroup Sp+1 .
p−1p−1(ii) ρ0p preserves each (p − 1)-face ∆(i)and coincides with ρ0p−1 on ∆(i)p−1with ∆p−1 .
under an affine identification of ∆(i)

(See [43, Definition 4.1] for the notation.)
Consider the open cover Vp = {V(i0 ,··· ,ik ) | 0 ≤ i0 < · · · < ik ≤ p} of |sd ∆(p)|,where V(i0 ,...,ik ) is the star of the barycenter b(i0 ,...,ik ) of an open simplex (i0 , . . . , ik )of ∆p . Since |sd ∆(p)| is topologically identified with ∆p and the nerve of Vp isjust sd ∆(p) ([64, p. 114]), we have only to construct a smooth partition of unity{ρ0(i0 ,...,ik ) } on ∆p subordinate to the open cover Vp satisfying the two conditions
0
corresponding to conditions (i) and (ii), which are denoted by conditions (i) and00(ii) respectively (see Lemma 8.3); the precise formulation of conditions (i) and0(ii) is left to the reader.
For p = 0, {ρ0(i0 ,...,ik ) }(= {ρ0(0) }) is defined in the obvious manner. Supposethat we have constructed a partition of unity {ρ0(i0 ,...,ik ) } on ∆p−1 subordinate
00
to Vp−1 satisfying conditions (i) and (ii) . Then, the functions ρ0(i0 ,...,ik ) with˙ p by conditions (i)0 and (ii)0 . Extend the(i0 , . . . , ik ) 6= (0, . . . , p) are defined on ∆0functions ρ(i0 ,...,ik ) to smooth functions on ∆p with supp ρ0(i0 ,...,ik ) ⊂ V(i0 ,...,ik ) (see[43, Lemma 9.11]) and average them over Sp+1 . We then obtainPthe desired smooth
partition of unity {ρ0(i0 ,...,ik ) } on ∆p by setting ρ0(0,...,p) = 1 −ρ0(i0 ,...,ik ) .

k<pProof of Proposition 9.5. We prove the result in two steps.
Step 1: Construction of a splitting ψ to pr : BXU −→ X. We choose a smooth
partition of unity {ϕα } subordinate to U , which defines the smooth function ϕ :X −→ |N U |0D (see Lemma 8.3). Noticing that BRU = |sd N U |D , we then define
the smooth map φ to be the composite
ϕ

ψN U

ρN U

'

'

X −−→ |N U |0D −−−→ |N U |D −−−→ BRU(see Theorem 8.6 and Lemma 9.6 for ψN U and ρN U , respectively).
Consider the solid arrow diagram in D
X
ψ

BXU

(pr,Bπ)(1X ,φ)X × BRU .

Then, the dotted arrow ψ exists, making the diagram commute. In fact, for x ∈ X,
consider the open neighborhood Vσ of x defined in the proof of Lemma 8.3. Then,(1X , φ)|Vσ factorizes the D-embedding Uσ × |sd ∆(p)|D ,−→ X × BRU , which isthe product of the inclusion Uσ ,−→ X and the inclusion |sd ∆(p)|D ,−→ BRU ofthe barycentric subdivision of the closed simplex corresponding to σ (see Lemma

70

HIROSHI KIHARA

8.1(1) and [43, the proof of Lemma 4.2]). Since the closed p-simplices
of |sd ∆(p)|D`are of the form [σ ) · · · ], we have the D-quotient map
Uσ × [σ ) · · · ] −→Uσ × |sd ∆(p)|D (see [43, Lemma 2.5] and Proposition 2.2(2)). Since the composite`(1X ,φ)f
O −→ Vσ −−−−→ Uσ × |sd ∆(p)|D lifts locally along
Uσ × [σ ) · · · ] −→ Uσ ×
|sd ∆(p)|D for any plot f of Vσ , we see that (1X , φ) factorizes through a smoothmap ψ : X −→ BXU .
Note that pr ◦ ψ = 1X holds.
Step 2: Construction of a D-homotopy between ψ ◦ pr and 1BXU . We must showthat ψ ◦ pr 'D 1BXU . For this, we construct a D-homotopy h between the twocomposites
ψ◦pr

BXU 1

BXU

BXU

(pr,Bπ)X × BRU

so that it factorizes through the smooth injection BXU ,−→ X × BRU . Since(pr, Bπ) ◦ ψ ◦ pr = (pr, φ ◦ pr), we define the X-component hX of h to be theconstant homotopy at pr.
To construct the BRU -component hBRU , we consider the composites
φ◦pr

BXU

Bπ

ηN U

BRU −−−→ |N U |D'`(see Lemma 9.6 for ηN U ). Recall that the canonical map
Uσp × [σp ) · · · )σ0 ] −→ BXU is a D-quotient map and consider the composites
φ◦prUσp × [σp ) · · · ) σ0 ]

BXU

Bπ

ηN U

BRU −−−→ |N U |D .
'For x ∈ Uσp , consider the open neighborhood Vσ of x defined in the proof of
Lemma 8.3 and set σp0 = σ ∪ σp and Wσp0 = Vσ ∩ Uσp . Then, Wσp0 is an open
neighborhood of x contained in Uσp and the images of Wσp0 × [σp ) · · · ) σ0 ] by
the two composites are contained in the closed simplex of |N U |D corresponding to
σp0 . Thus, ηN U ◦ φ ◦ pr 'D ηN U ◦ Bπ by Corollary 8.8. Hence, by composing themap ρN U : |N U |D −→ BRU , we haveφ ◦ pr 'D ρN U ◦ ηN U ◦ φ ◦ pr 'D ρN U ◦ ηN U ◦ Bπ 'D Bπ(see Lemma 9.6). Construct three D-homotopies corresponding to these three Dhomotopy relations using the linear homotopies and the D-homotopies hK constructed in the proof Theorem 8.6 (see the proofs of Corollary 8.8 and Lemma
9.6). We then define the D-homotopy hBRU to be the composite of the three Dhomotopies.
Last, we must verify that the D-homotopy h : BXU ×I −→ X ×BRU factorizes
through BXU ,−→ X × BRU . Let σp0 and Wσp0 be as above and note that thecompositehWσp0 × [σp ) · · · ) σ0 ] × I −→ BXU × I −−−→ X × BRU
factorizes through the D-embedding Uσp0 × |sd ∆(p0 )|D ,−→ X × BRU , which isthe product of the inclusion Uσp0 ,−→ X and the inclusion |sd ∆(p0 )|D ,−→ BRUof the barycentric subdivision of the closed simplex corresponding to σp0 . Since|sd ∆(p0 )|D is the union of the simplices of the form [σp0 ) · · · ], the D-homotopy

9. HOMOTOPY COFIBRANCY THEOREM

71

h : BXU × I −→ X × BRU factorizes through (pr, Bπ) : BXU ,−→ X × BRU(see the argument showing that (1X , φ) factorizes through (pr, Bπ) : BXU ,−→X × BRU ).

Remark 9.7. The proof of Proposition 9.5 is much more subtle than that of[62, Proposition 4.1] since we need two kinds of notions of a diffeological polyhedronand the natural map |sd K|D −→ |K|D is not an isomorphism unlike the topologicalcase (see the definition of φ : X −→ BRU in the proof of Proposition 9.5).
9.2. Hurewicz cofibrations in D. In this subsection, we introduce the notion of a Hurewicz cofibration in D using a weak version of the homotopy extension
property, and then establish its basic properties.
Definition 9.8. A smooth map i : A −→ X is called a Hurewicz cofibration(or an h-cofibration) if i : A −→ X satisfies the following weak homotopy extension
property (WHEP):
Assume given a solid arrow diagram in D
X × (0)

∪

A×I

f +h

Z

A×(0)

X × I,
where the vertical arrow is the canonical map from the pushout of the diagrami×(0)X × (0) ←−−− A × (0) ,−→ A × I
to X × I. If ht = h0 for sufficiently small t, the dotted arrow exists, making thediagram commute.
The reason why we define an h-cofibration in D not by HEP but by WHEP is
explained in Remark 9.13(2).
Remark 9.9.
(1) Let i : A −→ X be a smooth map and assume given
a solid arrow diagram as in Definition 9.8. If i : A −→ X is an hcofibration, then we have an extension ĥ : X × I −→ Z after deforming
the homotopy h : A × I −→ Z using a nondecreasing smooth function
λ : I −→ I such that λ ≡ 0 on [0, ] and λ ≡ id on [2, 1] for some  > 0.
Further, if we deform the homotopy h : A × I −→ Z so thath0 if t ∈ [0, ]ht =h1 if t ∈ [1 − , 1]
for some  with 0 <  < 12 , we have an extension ĥ : X × I −→ Z suchthatĥ0 if t ∈ [0, 0 ]ĥt =ĥ1 if t ∈ [1 − 0 , 1]
for any fixed 0 with 0 < 0 < . Thus, we always choose such homotopies
and extensions if necessary.

72

HIROSHI KIHARA(2) Let i : A −→ X be an h-cofibration. Since we have a solution of theextension problem
X × (0)∪

1

A×I

X × (0)

A×(0)

∪

A×I

A×(0)

X × I.
after deforming the homotopy A × I −→ X × (0) ∪ A × I (see Part 1),
A×(0)we see that there exist an open diffeological subspace U of X and a retract
diagram
i

A

r

U

A,

1
and hence that i : A −→ X is an D-embedding. Therefore, given an
h-cofibration i : A −→ X, we can regard the source A as a diffeologicalsubspace of X.
We first show the following lemma.
Lemma 9.10.
(1) If i : A ,−→ X is an h-cofibration, then i×1 : A×U ,−→X × U is also an h-cofibration for any U ∈ D.
(2) Assume given a pushout diagram in D
A0
A

i0

i

X0
X.

If i0 is an h-cofibration, then i is also an h-cofibration.
(3) The class of h-cofibrations is closed under composites, coproducts, andretracts.
(4) Assume given a sequence in D
i0

i1

i2

X0 ,−−−→ X1 ,−−−→ X2 ,−−−→ · · ·If the ik are h-cofibrations, then i : X0 ,−→ X := lim Xk is also an h→cofibration.
Proof. (1) The result follows from Proposition 2.2(2).
(2) We must solve the extension problem in Definition 9.8 under the assumptionthat ht = h0 for sufficiently small t. Since i0 : A0 ,−→ X 0 is an h-cofibration, there
exists a morphism k̂ : X 0 × I −→ Z making the diagram
X 0 × (0)∪A0 ×(0)A0 × I

f |X 0 +h|A0 ×Ik̂

X0 × IZ

9. HOMOTOPY COFIBRANCY THEOREM

73

commute. Noticing that X × I = X 0 × I ∪A0 ×I A × I (Proposition 2.2(2)), we seethatk̂ + h : X × I = X 0 × I ∪A0 ×I A × I −→ Z
is the desired solution.
(3) Obvious (see Remark 9.9(1) for the closure property under composites).
(4) Since X × I = lim Xn × I (Proposition 2.2(2)), the extension problem in→question can be solved (see Remark 9.9(1)).

A diffeological pair (X, A) is an N DR-pair (= neighborhood deformation retract pair) if there is a smooth map u : X −→ I such that u−1 (0) = A and a
D-homotopy ϕ : X × I −→ X relative to A such that ϕ0 = 1X and ϕ1 (x) ∈ A foru(x) < 1.
By the following lemma, the notion of an NDR-pair gives a useful criterion fora map to be an h-cofibration.
Lemma 9.11. Let (X, A) be an N DR-pair. Then the inclusion i : A −→ X isan h-cofibration.
Proof. Choose a smooth map u and a D-homotopy ϕ as in the definition of
an NDR-pair. We have to solve the extension problem in Definition 9.8 under theassumption that ht = h0 for sufficiently small t.
Choose  > 0 such that ht = h0 for t ∈ [0, ] and a non-decreasing function
ρ : I −→ I such that ρ−1 (0) = [0, 31 ] and ρ−1 (1) = [ 23 , 1]. Set U1 = {(x, t) ∈X × I | (1 − u(x))t < } and U2 = {(x, t) ∈ X × I | (1 − u(x))t > 32 }, which form
an open cover of X × I. Define the smooth maps ĥi : Ui −→ Z (i = 1, 2) byĥ1 (x, t)=f (ϕ(x, ρ((1 − u(x))t))),ĥ2 (x, t)=h(ϕ(x, 1), (1 − u(x))t).

Since ĥ1 = ĥ2 on U1 ∩ U2 , we have the smooth mapbh=bh1 + bh2 : X × I = U1 ∪ U2 −→ Z,
which is the desired extension of f + h.

We call cofibrations in Definition 2.13 Quillen cofibrations or q-cofibrations.
Then, the following result holds.
Proposition 9.12. Every q-cofibration in D is an h-cofibration in D.
Proof. Let i : A −→ X be a q-cofibration. We show that i is an h-cofibrationin two steps.
˙ p ,−→ ∆p (p ≥ 0). Since the result is
Step 1: The case where i is the inclusion ∆˙ p)obvious for p = 0, we may assume that p > 0. We have only to show that (∆p , ∆is an NDR-pair (see Lemma 9.11).
˙ p ), we construct a smooth map u and a smooth homotopy
For the pair (∆p , ∆ϕ as in the definition of an NDR-pair. Let µ : I −→ I be a nondecreasing smoothfunction such that µ = id on [0, 1 − 2] and µ ≡ 1 on [1 − , 1] for some  ∈ (0, 21 ).
Define the smooth map u : ∆p −→ I by u(t0 , . . . , tp ) = µ((p + 1)p+1 t0 · · · tp )(see Lemma 2.17(3)). Then, we can construct the desired smooth homotopy ϕ :74

HIROSHI KIHARA∆p × I −→ ∆p using [43, Lemma 9.11].
Step 2: The case where i is a general q-cofibration. Consider the cofibration-trivial
fibration factorization of ip∞i−−→ G∞ (I, i) −−−−−−→ X
A −−−∞
constructed in [43, the proof of Theorem 1.3] and note that i∞ is a sequential
relative I-cell complex. By solving the lifting problemi∞A

G∞ (I, i)
p∞

i

X

X,

1X

we can show that i is a retract of i∞ , and hence that i is a h-cofibration (see Step
1 and Lemma 9.10).

From Proposition 9.12 and Remark 9.9(2), we have the implications
q-cofibration ⇒ h-cofibration ⇒ D-embedding.
Remark 9.13.
(1) For an h-cofibration i : A ,−→ X, the canonical map
X × (0) ∪A×(0) A × I ,−→ X × I need not be a D-embedding; consider the˙ 1 ,−→ ∆1 and see the proof of [43, Proposition A.2(1)].
h-cofibration ∆(2) Topological h-cofibrations are defined by the HEP ([52, p. 41]). However,diffeological h-cofibrations, introduced in this subsection, are defined by the
W HEP so that Proposition 9.12 holds true. In fact, since the implications
i : A −→ X satisfies the HEP⇐⇒ X × (0)∪A × I ,−→ X × I has a retraction

A×(0)=⇒ X × (0)∪A × I ,−→ X × I is a D-embeddingA×(0)˙ 1 ,−→ ∆1 does not satisfy the HEP (see Part 1).
hold, even ∆(3) I do not know under what conditions the notions of an h-cofibration and
an N DR-pair are equivalent or whether there exists a model structure on
D whose cofibrations are just h-cofibrations (cf. [52, p. 43 and p. 46],
[53, p. 340]).
We need several more results on h-cofibrations to prove Theorem 1.10.
Lemma 9.14. Every smooth map f : X −→ Y has a functorial factorizationipX ,−→ Y 0 −−→ Ysuch that i is an h-cofibration and p is a D-homotopy equivalence.
Proof. We define the mapping cylinder M f by the pushout diagram in D
X

f

Y

i1

X ×I

M f,

9. HOMOTOPY COFIBRANCY THEOREM

75

where the map i1 is defined by i1 (x) = (x, 1). Consider the factorization
i

X

Mf
p

f

Y,
i0

where the maps i and p are defined to be the composite X ,−−→ X × I −→ M f andf ◦proj+1Y
the sum M f = X × I ∪X Y −−−−−−−→ Y, respectively. Then, we can observe that
(M f, X) is an NDR-pair, and hence that this factorization is the desired one (seeLemma 9.11).
For a diffeological space A, let A/D denote the category of diffeological spaces
under A. Then, the notion of a D-homotopy under A and that of a D-homotopy
equivalence under A are defined in the obvious manner (cf. [52, p. 44]). Since
A/D is a simplicial category (see Section 7.1), these homotopical notions can bealso defined as A/D-homotopical notions (see Section 4.3).
Lemma 9.15. Let i : A −→ X and j : A −→ Y be h-cofibrations and let
f : X −→ Y be a map under A. If f is a D-homotopy equivalence, then f is a
D-homotopy equivalence under A.
Proof. The result is a diffeological version of Proposition in [52, p. 44] andcan be proved by a similar argument; see Remark 9.9(1).
Proposition 9.16. Assume given a commutative diagram in D
i

A

X
f

d

B

j

Y

in which i and j are h-cofibrations and d and f are D-homotopy equivalences. Then,(f, d) : (X, A) −→ (Y, B) is a D-homotopy equivalence of pairs.
Proof. This result is a diffeological version of Proposition in [52, p. 45].
Recall the fact that if a morphism φ of a category C has a left inverse α and a rightinverse β, then α = β and hence φ is an isomorphism. Then, we have only to showthat a given D-homotopy inverse e of d and a given D-homotopy h connecting e ◦ d
to 1A extend to a D-homotopy inverse g of f and a D-homotopy H connecting g ◦ f
to 1X (recall Remark 9.9(1)).
We can choose a D-homotopy inverse g of f as in [52, p. 46]. Choose Dhomotopies γ : X × I × (0) −→ X and Γ : A × I × I −→ X as in [52, p. 46] andconsider the extension problem
X × I × (0)∪A×I ×Iγ+Γ

X

A×I×(0)

X × I × I.
(Note that unlike in [52, p. 46], the domain of γ is X × I × (0).) Since i ×
1 : A × I ,−→ X × I is an h-cofibration (Lemma 9.10(1)), we have a solution
Λ of the above extension problem. Then, the sum Σ of the three D-homotopies

76

HIROSHI KIHARA

Λ|X×(0)×I , Λ|X×I×(1) and Λ|X×(1)×I is a D-homotopy connecting g ◦ f to 1X . We
thus obtain the desired D-homotopy H by reparametrizing Σ (see Remark 9.9(1)).
Lemma 9.17. Assume given a commutative diagram in D
A1

f1

h1

B1

f2

A0
h0

h2

B0

g1

A2
B2

g2

in which f1 and g1 are h-cofibrations. If h0 , h1 , and h2 are D-homotopy equivalences, then so is their pushout
A1 ∪A0 A2 −→ B1 ∪B0 B2 .
Proof. From Proposition 9.16 (and its proof), we see that if f2 and g2 arealso h-cofibrations, then the result holds. Thus, we have only to show that the map
1A1 ∪A0 p : A1 ∪A0 A02 −→ A1 ∪A0 A2 induced by the commutative diagram
A1

f1

1A1

A1

A0

i

A02

1A0
f1

p

A0

A2

f2

i1

is a D-homotopy equivalence, where A02 = M f2 and the factorization A0 ,−−→
p
A02 −−→ A2 of f2 is the one constructed in the proof of Lemma 9.14.
Note that A02 ,−→ A1 ∪A0 A02 is an h-cofibration (Lemma 9.10(2)) and consider
the extension problem in D
(A1 ∪A0 A02 ) × (0) ∪A02 ×(0) A02 × I

1A ∪
1

0 +h
A0 A2

A1 ∪A0 A02 ,k(A1 ∪A0 A02 ) × I
where h is a reparametrization of the canonical deformation of A02 onto A2 suchthat ht = h0 for sufficiently small t (see Remark 9.9(1)). Then, we have a solution
k of the extension problem.
Note that since p : A02 −→ A2 has a canonical splitting, p : A02 −→ A2 , andhence 1A1 ∪A0 p : A1 ∪A0 A02 −→ A1 ∪A0 A2 is a D-quotient map. Thus, we see that
k1 = k(·, 1) is factored as
A1 ∪A0 A02
1A1 ∪A0 p

A1 ∪A0 A2

k1

j

A1 ∪A0 A02

9. HOMOTOPY COFIBRANCY THEOREM

77

in D, and hence that j ◦ (1A1 ∪A0 p) 'D 1A1 ∪A0 A02 . Using [43, Lemma 2.5], we also
see that the D-homotopy (1A1 ∪A0 p) ◦ k is factored ask(A1 ∪A0 A02 ) × I

A1 ∪A0 A02
1A1 ∪A0 p(1A1 ∪A0 p)×1k̄

(A1 ∪A0 A2 ) × I

A1 ∪A0 A2 ,which shows that (1A1 ∪A0 p) ◦ j 'D 1A1 ∪A0 A2 .



Lemma 9.18. Assume given a commutative diagram
i1

X0
f0

X1

i2

f1

Y0

j1

i3

X2

···

f2

Y1

j2

Y2

j3

···

in which the ik and jk are h-cofibrations. If the maps fk are D-homotopy equivalences, then so is their colimit
lim fk : lim Xk −→ lim Yk .
→→→Proof. The result can be proved using Proposition 9.16 and its proof.



9.3. Proof of Theorem 1.10. In this section, we prove Theorem 1.10 using
the results in Sections 9.1 and 9.2.
Let us begin by proving a few lemmas. Regard the subclass WD of D as a fullsubcategory of D and recall that˙ p ,−→ ∆p | p ≥ 0}.
I = {∆
See [53, Definitions 15.1.1 and 15.1.2] for a sequential relative I-cell complex.
Lemma 9.19. Let f : X0 −→ X1 be a morphism of WD and h0 : A0 −→ X0 a
D-homotopy equivalence with A0 cofibrant. Then, there exist a sequential relative
I-cell complex i : A0 ,−→ A1 and a D-homotopy equivalence h1 : A1 −→ X1 ,making the diagram
A0

i

A1

h0

X0

h1
f

X1

commute.
Proof. Define the mapsih1
A0 ,−−−−→ A1 −−−−−→ X1

by applying the cofibration-trivial fibration factorization in [43, the proof of Theorem 1.3] to f ◦ h0 . We can see from Corollary 4.12 that i and h1 are the desiredmaps.
Lemma 9.20.

(1) Assume given a diagram in D
X1

f1

X0

f2

X2

with f1 h-cofibration. If X0 , X1 , and X2 are in WD , then so is X1 ∪X0 X2 .

78

HIROSHI KIHARA(2) Assume given a sequence in D
i0

i1

i2

X0 ,−−−→ X1 ,−−−→ X2 ,−−−→ · · ·such that the in are h-cofibrations. If the Xn are in WD , then so is
X = lim Xn .
→Proof. Using Lemma 9.19, we can derive Parts 1 and 2 from Lemmas 9.17
and 9.18 respectively; recall Proposition 9.12.
Proof of Theorem 1.10. Since pr : BXU −→ X is a D-homotopy equivalence (Proposition 9.5), we prove that BXU is in WD .
Recall from Section 9.1 the pushout diagram
`

(n−1)

˙n
Uσn × ∆BXU

Uσn × ∆nBXUσn )···)σ0

(9.1)
`

(n)

σn )···)σ0

and the identity
(n)

(9.2)

BXU = lim BXU .
→(n)First, we show that BXU(0)

is in WD by induction on n. Since BXU =

`

Uσ ,

σ

the result holds for n = 0. Suppose that the result holds up to n − 1. Then, we see(n)from pushout diagram (9.1) that BXU is in WD , using Proposition 9.12, Lemma
9.10, Corollary 4.13 and Lemma 9.20(1).
(n−1)
(n)
From pushout diagram (9.1), we also see that BXU,−→ BXU is an h(n)cofibration (Lemma 9.10). Thus, we obtain the result from that for BXU , (9.2),
and Lemma 9.20(2).
Remark 9.21. For an ordinal number γ, let ωγ denote the γ th infinite cardinal
number ([13, p. 66]) and define the subclass WD γ of WD by
WD γ = {A ∈ D | A 'D |K|D for some simplicial set K with card K ≤ ωγ },where we set card K = |`Kn |. The class WD 0 is just the subclass of diffeologicaln≥0spaces having the D-homotopy type of the realization of a countable simplicial set(cf. [56, Section 1]).
We give a basic result on WD γ (Part 1), a diffeological version of [56, Proposition 2](Part 2), and a refined version of Theorem 1.10 (Part 3).
(1) Define the fibrant approximation functor ˆ· : S −→ S by assigning to asimplicial set K the Kan complex K̂ such that K ,−→ K̂ −→ ∗ is theusual trivial cofibration-fibration factorization of K −→ ∗ (see, eg, [42]).

10. LOCALLY CONTRACTIBLE DIFFEOLOGICAL SPACES

79

Then, we have the implications
card K ≤ ωγ ⇔ card K̂ ≤ ωγ
⇒ |πi (K̂, k0 )| ≤ ωγ for any i and any k0⇒ K̂ is homotopy equivalent to some Kancomplex L with card L ≤ ωγ .
From these, we obtain the following equivalence:A ∈ WD γ ⇔ A ∈ WD and |πiD (A, a)| ≤ ωγ for any i and any a.
(see Theorem 1.5, and Corollaries 1.6 and 4.12). A similar result holds
for the similarly defined subclass WC 0 γ of WC 0 .
(2) Let us consider the case of γ = 0. We see from [56, Theorem 1] that for
A ∈ C 0 , the following equivalences hold:A ∈ WC 0 0 ⇔ A is homotopy equivalent to a countable
CW -complex.
⇔ A is homotopy equivalent to a conuntable
polyhedron.
Thus, the class WC 0 0 is just the intersection of Milnor’s class W0 and C 0 .
Further, using Corollary 4.11 and [56, Proposition 2], we have thefollowing result:e Lindelöf, then X is
If X is a diffeological space in WD with X
in WD 0 .
Theorem 1.10 along with this result gives a useful criterion for a diffeological space to be in WD 0 , which is used in the proof of Proposition
11.4.
(3) Theorem 1.10 can be refined as follows:
Let X be a diffeological space and U = {Uα }α∈A a D-numerable
covering of X by subsets. If |A| ≤ ωγ and Uσ ∈ WD γ for any
nonempty finite σ ⊂ A, then X is in WD γ .
For the proof, we have only to show that Lemma 9.19 can be refined asfollows: If X0 , X1 ∈ WD γ and A0 = |K0 |D with card K0 ≤ ωγ , we cantake A1 such that A1 = |K1 |D for some K1 with card K1 ≤ ωγ . This canbe done by using Part 1 and Theorem 1.5(1).
The topological analogue, which is a refined version of [67, Theorem4], can be also shown by a similar argument.
10. Locally contractible diffeological spaces
In this section, we introduce the notion of a locally contractible diffeologicalspace and prove that such a diffeological space is in VD (Theorem 1.11).
Recall that a diffeological space X is called D-contractible if X is D-homotopy
equivalent to the terminal object ∗.
Definition 10.1. A diffeological space X is called locally D-contractible ifevery x ∈ X has a neighborhood basis consisting of D-contractible open diffeologicalsubspaces.

80

HIROSHI KIHARA

D-contractible (resp. locally D-contractible) diffeological spaces are simply
called contractible (resp. locally contractible) diffeological spaces if there is noconfusion in context.
Note that if a diffeological space X is contractible (resp. locally contractible),e
X is contractible (resp. locally contractible) as a topological space (see [43, Section2.4]).
We define the iterated barycentric subdivision sdk ∆p by sdk ∆p = |sdk ∆(p)|Dand identify it with |∆(p)|D = ∆p topologically (see [64, p. 124] for the iterated
barycentric subdivision sdk K of a simplicial complex K). For the proof of Theorem
1.11, we introduce the D-polyhedron (∆p × I)(k) connecting sdk ∆p to ∆p .
First, we introduce the D-polyhedra (∆p × I)0 such thatpp × I)0 = ∆p
• (∆^top × Itop , where ∆top and Itop denotes the topologicalstandard p-simplex and the topological unit interval, respectively.
• The subsets ∆p × (0) and ∆p × (1) are D-subpolyhedra which are isomorphic to sd ∆p and ∆p respectively.
For this, we inductively define the simplicial complexes (∆(p) × I)0 whose Dpolyhedra are the desired (∆p ×I)0 . For p = 0, we define (∆(p)×I)0 by (∆(p)×I)0 =∆(0) × I (∼= I = ∆(1)). For p = 1, we define (∆(p) × I)0 to be the cone of thesimplicial complex (0) × I ∪ ∆(1) × (1) ∪ (1) × I, where the cone CK of a simplicial
complex K is defined by``` `CK = ∗ K and ΣCK = ∗ ΣK {∗σ | σ ∈ ΣK }(see the left side of Fig 10.1). Suppose that (∆(p − 1) × I)0 is constructed. Then thesimplicial complex (∆(p) × I)0 is defined to be the cone of the simplicial complex
Lp , where Lp is the union of ∆(p) × (1) and p + 1 copies of (∆(p − 1) × I)0 whose
˙ p × I.
polyhedron is topologically identified with ∆p × (1) ∪∆˙ p ×(1) ∆p0Using the construction of the D-polyhedron (∆ × I) repeatedly, we obtain theD-polyhedron (∆p × I)(k) such that• (∆p^× I)(k) = ∆ptop × Itop .
• The subsets ∆p × (0) and ∆p × (1) are D-subpolyhedra which are isomorphic to sdk ∆p and ∆p respectively.
The simplicial complex corresponding to (∆p × I)(k) is denoted by (∆(p) × I)(k) .
∆1

∆1

I

I

D-polyhedron (∆1 × I)0

D-polyhedron (∆1 × I)(2)
Fig 10.1

10. LOCALLY CONTRACTIBLE DIFFEOLOGICAL SPACES

81

Lemma 10.2. Let X be a diffeological space and σ: ∆p −→ X a continuous
map. Suppose that σ satisfies the following conditions:• σ̇ = σ |∆˙ p is smooth.
• σ is smooth as a map from sdk ∆p to X.
Then there exists a smooth map σ 0 : ∆p −→ X such that σ 0 is C 0 -homotopic to σ
˙ p.
relative to ∆Proof. Consider the D-subpolyhedronp−1
sdk ∆p × (0) ∪ ∪ (∆(i)× I)(k)
i

p

(k)

(k)
is a copy of (∆p−1 × I)(k) for 0 ≤ i ≤ p (see, where (∆p−1(i) × I)of (∆ × I)
[43, Definition 4.1]). Noticing that the composite
proj

σ̇

p−1
˙ p −−−→ X∪ (∆(i)× I)(k) −−−→ ∆iis smooth (Proposition 2.11), we then have the commutative solid arrow diagram
in D
σ+σ̇◦proj(k)sdk ∆p × (0) ∪ ∪ (∆p−1
X(i) × I)ih(∆p × I)(k)∗(see Lemma 8.1). Since the left and right vertical arrows are a trivial cofibration
and a fibration respectively (Lemmas 8.1 and 2.20 and Theorem 2.14), there is adotted arrow h making the diagram commute. Then, σ 0 := h(·, 1) is the desired
smooth map from ∆p to X.

Proof of Theorem 1.11. Let ιX denote the natural inclusion S D X ,−→e
S X. Note that S D X and S X̃ are Kan complexes and consider the following statements:(Ap ) For any locally contractible diffeological space X and any fixed base pointx, πi (ιX ) : πi (S D X, x) −→ πi (S X̃, x) is injective for i = p − 1 and surjective fori = p.
(Bp ) For any locally contractible diffeological space X and any continuous map
τ : ∆p+1 −→ X with τ |∆˙ p+1 smooth, there exists a smooth map τ 0 : ∆p+1 −→ X
˙ p+1 .
such that τ 0 is C 0 -homotopic to τ relative to ∆We arrange the statements in the following order:(A0 ), (B0 ), (A1 ), . . . , (Ap ), (Bp ), (Ap+1 ), . . .
Noticing that (Ap ) obviously holds for p = 0, we prove all statements in two steps,
which implies the result (see Corollary 1.6(3)).
Step 1. Suppose that the implications hold up to (Ap ). Then, we show that (Bp )holds.
Choose a covering U of X consisting of contractible open sets. Take a sufficiently large integer k and choose for each (p + 1)-simplex ∆p+1
of sdk ∆p+1 , anαp+1element Uα of U such that τ (∆α ) ⊂ Uα (see, eg, [64, p. 178]). By Lemma 10.2,
we have only to construct a C 0 -homotopy
h : ∆p+1 × I −→ X

82

HIROSHI KIHARA

such that• h|∆p+1 ×(0) = τ .
˙ p+1 × I.
• h is the constant homotopy of τ |∆˙ p+1 on ∆k p+1
• h|∆p+1 ×(1) is smooth as a map from sd ∆to X.
˙ p+1 × I ∪ skp (sdk ∆p+1 ) × I, whereFirst, let us construct h on ∆p+1 × (0) ∪ ∆
k p+1
k p+1
˙ p+1 ×I ∪skl (sd ∆ ) denotes the l-skeleton of sd ∆ . Define h on ∆p+1 ×(0)∪ ∆k p+1
sk0 (sd ∆ )×I to be the sum of τ and the constant homotopy of τ |∆˙ p+1 ∪ sk0 (sdk ∆p+1 ) .
˙ p+1 × I ∪ skl (sdk ∆p+1 ) × I for l < p
Suppose that h is defined on ∆p+1 × (0) ∪ ∆Tlso that the image h(∆α(l) × I) is contained in
Uα and h|∆l ×(1) is
α(l)

p+1
∆l (l) ⊂∆ααsmooth for any l-simplex ∆lα(l) of sdk ∆p+1 . We have only to construct for each(l + 1)-simplex ∆l+1
of sdk ∆p+1 a solution of the lifting problem in C 0α(l+1)

˙ l+1∆l+1
× (0) ∪ ∆×Iα(l+1)
α(l+1)

h

T

Uα

∆l+1
⊂∆p+1
α
(l+1)
α

(10.1)∆l+1
×Iα(l+1)∗

such that its restriction to ∆l+1
× (1) is smooth.
α(l+1)
We give an outline of the construction; it can be made precise by solving lifting
problems in the model categories C 0 and D. Since the left and right vertical arrowsin (10.1) is a trivial cofibration and a fibration respectively, there exists a Tcontinuous
solution Γ to the lifting problem. Set γ = Γ(·, 1) and Uα(l+1) =
Uα .
⊂∆p+1
∆l+1
α
(l+1)
α

Then, there exists a smooth map γ 0 : ∆l+1−→ Uα(l+1) such that γ 0 |∆˙ l+1
α(l+1)=

α(l+1)γ |∆˙ l+1α(l+1)

by the injectivity of πl (ιUα(l+1) ). Further, we can use the surjectivity of

˙ l+1
πl+1 (ιUα(l+1) ) to choose γ 0 so that h + γ 0 : ∆l+1
× (0) ∪ ∆× I ∪ ∆l+1
×α(l+1)
α(l+1)
α(l+1)
l+1(1) −→ Uα(l+1) extends to ∆α(l+1) ×I; this extension is the desired solution of lifting
problem (10.1).
Last, recall that Uα is contractible and extend h to ∆p+1
× (1) smoothly andαp+1further to ∆p+1
×I
continuously
for
each
(p
+
1)-simplex∆α
α . It completes theconstruction of the desired C 0 -homotopy h : ∆p+1 × I −→ X.
Step 2. Suppose that the implications hold up to (Bp ). Then, we show that (Ap+1 )holds.
Let σ be a smooth map from ∆p to X with σ|∆˙ p ≡ 0, and suppose that
πp (ιX )([σ]) = 0. Then there exists a continuous map θ : ∆p+1 −→ X such thatθ|∆p(0) = σ and θ|Λp+1 = 0 (see [43, Section 4.1] for the notation). Thus, from (Bp ),0

we see that [σ] = 0 in πp (S D X).
Let τ be a continuous map from ∆p+1 to X with τ |∆˙ p+1 = 0. Then, we see
from (Bp ) that there exists a smooth map τ 0 : ∆p+1 −→ X such that τ 0 |∆˙ p+1 = 0
and πp (ιX )([τ 0 ]) = [τ ].
11. APPLICATIONS TO C ∞ -MANIFOLDS

83

11. Applications to C ∞ -manifolds
In Section 11.1, we prove the main theorems on C ∞ -manifolds (Theorems 1.11.3) using Theorems 1.7-1.9. In Sections 11.2 and 11.3, we study (semi)classicality
condition and hereditary C ∞ -paracompactness, respectively. In Section 11.4, weapply the results obtained in Sections 11.2-11.3 to many important C ∞ -manifolds,
showing that most of the C ∞ -manifolds introduced by Kriegl, Michor, and theircoauthors are hereditarily C ∞ -paracompact and semiclassical.
In Sections 11.2-11.4, we use some notions introduced in Appendix B and theresults obtained in Appendix C.
11.1. Proofs of Theorems 1.1-1.3. In this subsection, we give proofs ofTheorems 1.1-1.3.
From Theorems 1.10 and 1.11, we first derive the following theorems, which
form the foundation of applications of smooth homotopy theory of diffeologicalspaces to C ∞ -manifolds. See Section 1.2 for the definitions of hereditary C ∞ paracompactness and semiclassicality.
Theorem 11.1. If a C ∞ -manifold M is hereditarily C ∞ -paracompact andsemiclassical, then M is in WD .
Proof. We prove the result in two steps.
Step 1. Let V be a subset of a convenient vector space of E which is open withrespect to the locally convex topology. Then, V has a convex open cover. Thus,
we see from Theorem 1.10 that if V is C ∞ -paracompact, then V is in WD .
Step 2. We show that M is in WD . Let {(Uα , uα )}α∈A be a semiclassical atlas(i.e., an atlas as in the definition of a semiclassical C ∞ -manifold) of M and σ anonempty finite subset of A. Since uα : Uσ −→ uα (Uσ ) is a diffeomorphism for any
α ∈ σ and uα (Uσ ) is open with respect to the locally convex topology of Eα , Uσ isin WD (Step 1) and hence M is in WD (see Theorem 1.10).

Theorem 11.2. Every C ∞ -manifold is in VD .
Proof. Since every C ∞ -manifold is locally contractible as a diffeological space([46, Lemma 4.17]), the result follows immediately from Theorem 1.11.

We have the following corollary; compare Corollary 11.3 with Examples A.3
and A.6.
Corollary 11.3. Let M be a C ∞ -manifold and x a point of M . Then, thesmooth homotopy groups π∗D (M, x) and the smooth homology groups H∗ (M ) aref, x) and the homology groups
naturally isomorphic to the homotopy groups π∗ (Mf) of the underlying topological space M
f respectively.
H∗ (MProof. By Theorem 11.2 and Corollary 1.6, the natural inclusionf
S D M ,−→ S M
is a weak equivalence in S, which induces isomorphismsf),H∗ (M ) −→
H∗ (M
∼
=

f, x).
π∗D (M, x) −→
π∗ (M
∼
=



84

HIROSHI KIHARA

Theorems 1.1-1.3 are now immediate consequences of Theorems 1.7-1.9. Recall
from Remark 7.1 that C ∞ and C ∞ /M are S-full subcategories of D and D/Mrespectively and note that PC ∞ G/M is isomorphic to PDG/M as S-categories (seeRemark 5.5(3)).
Proof of Theorem 1.1. The result follows from Theorem 1.7 by using Theorems 11.1-11.2.
Proof of Theorem 1.2. The result follows from Theorem 1.8, by using Theorems 11.1-11.2.
Proof of Theorem 1.3. The result follows from Theorem 1.9 and Remark
5.18 by using Theorems 11.1-11.2.

We end this subsection by establishing a variant of Theorem 11.1. Recall thata topological space X is called hereditarily Lindelöf if any open set of X is Lindelöf.
Recall also the subclass WD 0 of WD from Remark 9.21.
Proposition 11.4. For a semiclassical C ∞ -manifold M , consider the following
conditions:(i) M is hereditarily Lindelöf and C ∞ -regular.
(ii) M is Lindelöf and hereditarily C ∞ -paracompact.
(iii) M is in WD 0 .
Then, the implications (i) =⇒ (ii) =⇒ (iii) hold.
Proof. The result follows from [46, Theorem 16.10] (or Proposition C.4),
Theorem 11.1, and Remark 9.21(2).

11.2. Classical atlases. In this subsection, we introduce two forms semiclassical C ∞ -manifolds appear.
A C ∞ -manifold M is called classical if M admits a semiclassical atlas {(Uα , uα )}α∈A
such that the chart changing uαβ : uβ (Uα ∩ Uβ ) −→ uα (Uα ∩ Uβ ) is a homeomorphism with respect to the locally convex topologies of Eα and Eβ for any α, β ∈ A.
If M has a classical atlas {(Uα , uα )}α∈A , then the classical topology on M is defined to be the final topology for the inclusions {uα (Uα ) ∼= Uα ,−→ M }α∈A , whereuα (Uα ) is endowed with the locally convex topology; the set M endowed with theclassical topology is denoted by Mcl . The inclusion uα (Uα ) ∼= Uα ,−→ Mcl is an
f −→ Mcl is continuous.
open topological embedding and id : M
Lemma 11.5. If a C ∞ -manifold M is modeled on Fréchet spaces or Silva spaces,
f = Mcl holds.
then M is classical and M
Proof. The result follows from [46, Theorem 4.11(1)-(2)]; note that the strong
dual of a Fréchet-Schwartz space is just a Silva space ([40, Corollary 12.5.9 and thecomment on it in p. 270]).

From the viewpoint of application of Lemma 11.5, the following lemma is important. Recall the definition of a submanifold from [46, 27.11].
Lemma 11.6. If a C ∞ -manifold M is modeled on Fréchet spaces (resp. Silva
spaces), then every submanifold of M is also modeled on Fréchet spaces (resp. Silva
spaces).

11. APPLICATIONS TO C ∞ -MANIFOLDS

85

Proof. The result follows from the fact that a closed linear subspace F of aFréchet space (resp. Silva space) E is a Fréchet space (resp. Silva space) ([61, p.
49], [73, Proposition 1]).

Next, we see that a C ∞ -manifold in Keller’s Cc∞ -theory (or a Cc∞ -manifold in
short) defines a classical C ∞ -manifold. See Appendix B for the category Cc∞conv
and the faithful functor J : Cc∞conv −→ C ∞ .
Lemma 11.7. If a C ∞ -manifold M comes from a Cc∞ -manifold (i.e., M ∼= JM 0for some M 0 ∈ Cc∞conv ), then M is classical.
Proof. Obvious.



Lemmas 11.5 and 11.7 apply to many important C ∞ -manifolds (see Section
11.4).
Remark 11.8.
(1) Lemma 11.5 can be regarded as a specialization ofLemma 11.7 (see Remark B.4).
(2) Lemma 11.7 remains true even if we replace Keller’s Cc∞ -theory with another classical infinite dimensional calculus satisfying the three conditions
in Remark B.1.
11.3. Hereditary C ∞ -paracompactness. The condition of being hereditarily C ∞ -paracompact and the condition of being hereditarily Lindelöf and C ∞ regular are important from a smooth homotopical viewpoint (see Theorem 11.1 and
Proposition 11.4); recall also that the implication
hereditarily Lindelöf and C ∞ -regular ⇒ herditarily C ∞ -paracompact
holds (see Proposition 11.4).
In this subsection, we study these two conditions, recalling the basics of C ∞ paracompactness from [46, Section 16] and Appendix C. More concretely, for eachof these two conditions, we give a necessary and sufficient condition of the form
(11.1) topological condition on M + condition on the model vector spaces of M
(Propositions 11.13 and 11.15) along with its application (Corollaries 11.14 and11.17).
Remark 11.9. Given a smooth topological condition P on a C ∞ -manifold M , itis of practical importance to give a necessary and sufficient condition of the form asin (11.1); such a characterization enables us to directly apply the results of topology
and functional analysis. Proposition C.3 also gives such a characterization.
Some results are discussed in a rather general context using the notation ofSection 5; C denotes one of the categories C ∞ , D, C 0 and T , and U : C −→ T
denotes the underlying topological space functor for C.
We begin by studying the heredity of the relevant properties on objects of C;
see Remark 5.3 for the heredity of C-regularity.
Lemma 11.10. Hereditary C-paracompactness and hereditary Lindelöf property
pass from an object of C to every C-subspace whose underlying topology is the subspace topology.
Proof. We prove the result for hereditary C-paracompactness; the proof forhereditary Lindelöf property is similar.

86

HIROSHI KIHARA

Let X be a hereditarily C-paracomapct object of C and A a C-subspace of X
such that U A is a topological subspace of U X. Suppose that an open set V of A
and an open cover {Vi } of V are given. Then, we can choose a family {Ui } of open
sets of X such that Vi = Ui ∩ A and use the C-paracompactness of U := ∪ Ui toconstruct a C-partition of unity on V subordinate to {Vi }.

From Lemma 11.10, we obtain the following result.
Corollary 11.11. If a C ∞ -manifold M is hereditarily C ∞ -paracompact (resp.
hereditarily Lindelöf and C ∞ -regular), then every submanifold of M is also hereditarily C ∞ -paracompact (resp. hereditarily Lindelöf and C ∞ -regular).
Proof. Since the underlying topology of a submanifold is the subspace topology of the ambient C ∞ -manifold ([46, 27.11]), the result follows from Lemma 11.10
and Remark 5.3.
Remark 11.12. A hereditary topological property is defined to be a topologicalproperty that carries over from a topological space to its subspaces. However, we saythat a property of objects of C is hereditary if it carries over from an object of C toevery open C-subspace, since the underlying topology of a C-subspace need not be thesubspace topology in the case of C 6= T (consider the C-subspace {0} ∪ { n1 | n ∈ N}
of R for C = D, C 0 ). Hereditary C-paracompactness, hereditary Lindelöf property,
and C-regularity are obviously hereditary.
Many important hereditary properties in C, such as these three properties, carryover from an object of C to its C-subspace whose underlying topology is the subspace
topology (see Lemma 11.10 and its proof ).
Next, we give a necessary and sufficient condition for a C ∞ -manifold to behereditarily Lindelöf and C ∞ -regular.
Proposition 11.13. For a C ∞ -manifold M , the following are equivalent:(i) M is hereditarily Lindelöf and C ∞ -regular.
(ii) M is Lindelöf and regular as a topological space and is modeled on hereditarily Lindelöf, C ∞ -regular convenient vector spaces.
Proof. The result follows from Proposition C.3.



Corollary 11.14. Suppose that a C ∞ -manifold M is Lindelöf and regular as
a topological space and that every model vector space of M is a separable Hilbert
space, a nuclear Fréchet space, a nuclear Silva space, or a strict inductive limit ofa sequence of nuclear Fréchet spaces. Then, every submanifold of M is hereditarily
Lindelöf and C ∞ -regular, and hence hereditarily C ∞ -paracompact.
Proof. Note that for a metric space, separability, second countablity, and
Lindelöf property are equivalent ([69, Theorem 16.11]). Then, we see that any c∞ open set V of a model vector space E of M is Lindelöf (see [46, the proof of Theorem
16.10]), and hence that E is hereditarily Lindelöf. Thus, the result follows from [46,Corollary 16.16 and Theorem 16.10], Proposition 11.13, and Corollary 11.11.

Next, we give a necessary and sufficient condition for a C ∞ -manifold to behereditarily C ∞ -paracompact.
Proposition 11.15. For a C ∞ -manifold M , the following conditions are equivalent:11. APPLICATIONS TO C ∞ -MANIFOLDS

87(i) M is hereditarily C ∞ -paracompact.
(ii) M is paracompact Hausdorff as a topological space and is modeled onhereditarily C ∞ -paracompact convenient vector spaces.
For the proof, we need the following lemma. An object X of C is called locallyhereditarily C-paracompact if any x ∈ X has a hereditarily C-paracompact open
neighborhood.
Lemma 11.16. For X ∈ C with U X T1 -space, the following are equivalent:(i) X is hereditarily C-paracompact.
(ii) U X is paracompact Hausdorff and X is locally hereditarily C-paracompact.
Proof. (i) =⇒ (ii) Obvious (see the end of Section 5.1).
(ii) =⇒ (i) Assume given an open set V of X and an open cover {Vi }i∈I of V .
Then, we must construct a C-partition of unity on V subordinate to {Vi }i∈I .
In this proof, for a subset S of X, S and Sˆ denote the closure of S in U X andthe intersection of S with V , respectively.
First, we choose a locally finite open cover {Uα }α∈A of X such that each Uα
is hereditarily C-paracompact. Noticing that U X is paracompact Hausdorff (andhence normal), we choose for each α ∈ A, open sets Uα0 and Uα00 such that
• Uα0 ⊂ Uα0 ⊂ Uα00 ⊂ Uα00 ⊂ Uα ,• ∪α Uα0 = X.
Next, consider the open cover {Uα ˆ}α∈A of V . Since U X is normal, we canchoose for each x ∈ Uα ˆ, an open neighborhood Wαx of x satisfying the following
conditions:• Wαx ⊂ Uα ∩ Vi for some i.
x ∈ Uα0 =⇒ Wαx ⊂ Uα00 ,•x∈/ Uα0 =⇒ Wαx ⊂ Uα \Uα0
Since Uα ˆ is C-paracompact, we choose a C-partition of unity {ϕxα } on Uα ˆ subordinate to the open cover {Wαx }; note that ϕxα extends to a C-function on V bysetting ϕxα = 0 on V − Wαx . Further, we define the (set-theoretic) map
r : Uα ˆ −→ I
by choosing for each x ∈ Uα ˆ, an element i of I with Wαx ⊂ Uα ∩ Vi .
Let us see that the familyΦ = {supp ϕxα | α ∈ A, x ∈ Uα0 ˆ}of closed sets in V is locally finite. For any z ∈ V , we can choose an open neighborhood U such that]{α ∈ A | U ∩ Uα ˆ 6= ∅} < ∞.
For α ∈ A with U ∩ Uα ˆ 6= ∅, we choose a smaller open neighborhood U (α) of z
satisfying the following conditions:• z∈/ Uα =⇒ U (α) = U \Uα00 .
• z ∈ Uα =⇒ U (α) meets the supports of only finitely many ϕxα .
Then, the intersection∩ U (α) is an open neighborhood of z which meets onlyU ∩Uα ˆ6=∅

finitely many members of Φ.
Thus, we see that
ϕi :=

P
0 ˆ
α∈A, x∈Uα
i

ϕxα

88

HIROSHI KIHARA

is a C-function on V with supp ϕi ⊂ Vi , where Uα0 ˆi is the inverse image of i ∈ I
under the (set-theoretic) mapr

Uα0 ˆ ,−→ Uα ˆ −→ I.
Further, we see from the definition that
P
P
ϕi =
i∈I

ϕxα > 0on V.

0 ˆα∈A, x∈UαTherefore, we obtain the desired C-partition of unity {φi }i∈I by settingP
φi = ϕi / ϕi .

i∈IProof of Proposition 11.15. (i) =⇒ (ii) The implication follows from Remark 5.3 and Proposition C.3.
(ii) =⇒ (i) The implication follows from Lemma 11.16.
Corollary 11.17. Suppose that a C ∞ -manifold M is paracompact Hausdorff
as a topological space and that every model vector space of M is a C ∞ -paracompactFréchet space (e.g., a Hilbert space or a nuclear Fréchet space), a nuclear Silva
space, or a strict inductive limit of a sequence of nuclear Fréchet spaces. Then,every submanifold of M is hereditarily C ∞ -paracompact.
For the proof, we need the following lemma.
Lemma 11.18. Let X be an object of C with U X metrizable. Then, X ishereditarily C-paracompact if and only if X is C-paracompact.
Proof. The result follows from [46, Theorem 16.15] (see the comment after
Remark 5.3).
Note that Lemma 11.18 has little meaning in the case of C = C 0 and T ; in fact,
every object X with U X metrizable is hereditarily C-paracompact for C = C 0 , T .
Proof of Corolary 11.17. Note that a Fréchet space is hereditarily C ∞ paracompact if and only if it is C ∞ -paracompact (see Lemma 11.18 and [46,Theorem 4.11(1)]). Then, we see that a Hilbert space and a nuclear Fréchetspace are hereditarily C ∞ -paracompact (see [46, Corollary 16.16 and Theorem
16.10]). Also recall that model vector spaces as in Corollary 11.14 are hereditarily
C ∞ -paracompact. Then, the result follows from Proposition 11.15 and Corollary
11.11.
We end this subsection with the following remark, in which we precisely state
the relation between Theorem 11.1 and the results of Palais and Heisey and therelation between Proposition 11.4 and the result of Milnor (see Remark 1.12(2)).
Remark 11.19. In this remark, a topological manifold is defined to be a Hausdorff topological space locally homeomorphic to an open set of a locally convex vector
space; this definition is the usual one but not that in [46, 27.1].
(1) We explain that Theorem 11.1 can be regarded as a smooth refinementof a result generalizing those of Palais [59, Theorem 14] and Heisey [33,Theorem II.10], which give sufficient conditions for an infinite dimensional
topological manifold to have the homotopy type of a CW -complex.
We can prove that for a topological manifold M , the following implications hold:11. APPLICATIONS TO C ∞ -MANIFOLDS

89

M is paracompact and is modeled on hereditarily paracompact locally convex vector spaces⇒ M is hereditarily paracompact⇒ M has the homoropy type of a CW -complex.
In fact, the argument in the proof of Theorem 11.1 is applicable to the
topological context by using [67, Theorem 4] instead of Theorem 1.10 (seeLemma 11.16 for the first implication). This result is a topological versionof Theorem 11.1, which generalizes the results of Palais and Heisey (see[33, Proposition III.1]).
(2) We explain that Proposition 11.4, a variant of Theorem 11.1, can be regarded as a smooth refinement of a result generalizing that of Milnor [56,Corollary 1], which states that every separable metrizable topological manifold has the homotopy type of a countable CW -complex.
Recall that a Lindelöf, regular topological space is paracompact ([69,Corollary 20.8]). Then, we see that for a topological manifold M , thefollowing implications hold:
M is hereditarily Lindelöf and regular⇒ M is Lindelöf and hereditarily paracompact⇒ M has the homotopy type of a countable CW -complex(see Part 1 and [56, Proposition 2]). This is a topological analogue of
Proposition 11.4, which generalizes the result of Milnor.
11.4. Hereditarily C ∞ -paracompact, semiclassical C ∞ -manifolds. In
this subsection, we show that many important C ∞ -manifolds are hereditarily C ∞ paracompact and semiclassical, and hence are in WD . In fact, we show that manyof the C ∞ -manifolds studied in [46, Chapter IX and Section 47] are hereditarily
Lindelöf, C ∞ -regular, and classical, and hence are in WD 0 ; recall that for a C ∞ manifold M , the implication
M is classical =⇒ M is semiclassicalholds (Section 11.2) and that for a semiclassical C ∞ -manifold M , the implications
M is hereditarily Lindelöf and C ∞ -regular

M is in WD 0

M is hereditarily C ∞ -paracompact

M is in WD

hold (Theorem 11.1 and Proposition 11.4).
Theorem 11.20. Let M be a C ∞ -manifold.
(1) Suppose that M is paracompact and modeled on Hilbert spaces, nuclear
Fréchet spaces, or nuclear Silva spaces. Then, every submanifold of M ishereditarily C ∞ -paracompact and classical, and hence is in WD .
(2) Suppose that M is Lindelöf and regular as a topological space and ismodeled on separable Hilbert spaces, nuclear Fréchet spaces, or nuclear
Silva spaces. Then, every submanifold of M is hereditarily Lindelöf, C ∞ regular, classical, and hence is in WD 0 .
(3) Suppose that M is Lindelöf and regular as a topological space and is semiclassical and modeled on strict inductive limits of sequences of nuclear

90

HIROSHI KIHARA

Fréchet spaces. Then, M is hereditarily Lindelöf and C ∞ -regular, andhence is in WD 0 .
Proof. (1) Note that a (separated) C ∞ -manifold modeled on Fréchet spaces orSilva spaces is Hausdorff (see Remark B.4). Then, the result follows from Corollary
11.17, Lemmas 11.5-11.6, and Theorem 11.1.
(2) The result follows from Corollary 11.14, Lemmas 11.5-11.6, and Proposition
11.4.
(3) The result follows from Corollary 11.14 and Proposition 11.4.

Example 11.21. Theorem 11.20(1) applies to the Grassmannian Gr(H) ofa polarized Hilbert space H ([60, Section 7.1]) and the Hilbert manifold models
L2r (S, M ) for mapping spaces ([19, p. 781]); note that in the literature, infinite
dimensional manifolds are often assumed to be paracompact ([19, p. 765], [60, p.
24]).
Let us apply Theorem 11.20 to various C ∞ -manifolds of mappings (see [46,Chapter IX]).
Let M and N be second countable finite dimensional C ∞ -manifolds; we assume
that M and N are connected and that dim M ≤ dim N whenever we consider the
manifolds of embeddings. Then, the C ∞ -manifold C∞ (M, N ) of C ∞ -maps from M
to N is defined ([46, Theorem 42.1]) and the Lie group Diff(M ) of diffeomorphismsof M and the C ∞ -manifold Emb(M, N ) of C ∞ -embeddings of M into N are defined
as open submanifolds of C∞ (M, M ) and C∞ (M, N ) respectively ([46, Theorems
43.1 and 44.1]).
Suppose further that M and N are C ω -manifolds with M compact. Then, the∞C -manifold C ω (M, N ) of C ω -maps from M to N is defined ([46, Theorem 42.6]),
and the Lie group Diff ω (M ) of C ω -diffeomorphisms of M and the C ∞ -manifold
Embω (M, N ) of C ω -embeddings of M into N are defined as open submanifolds ofC ω (M, M ) and C ω (M, N ) respectively ([46, Theorems 43.4 and 44.3]).
Corollary 11.22. Let M and N be second countable finite dimensional C ∞ manifolds.
(1) Suppose that M is compact. Then, every submanifold of C∞ (M, N ) (e.g.
Diff(M ) and Emb(M, N )) is hereditarily Lindelöf, C ∞ -regular, and classical, and hence is in WD 0 .
(2) Each connected component of the C ∞ -manifolds C∞ (M, N ), Diff(M ), andEmb(M, N ) is hereditarily Lindelöf, C ∞ -regular, and classical, and hence
is in WD 0 .
(3) Suppose that M and N are C ω -manifolds with M compact. Then, every
submanifold of C ω (M, N ) (e.g. Diff ω (M ) and Embω (M, N )) is hereditarily Lindelöf, C ∞ -regular, and classical, and hence is in WD 0 .
We see from [46, Theorem 42.14] that C∞ (M, N ) coincides with the hom-object
C (M, N ) (= D(M, N )) as a diffeological space if and only if M is compact or N
is discrete. For Corollary 11.22(2), note the obvious fact that a diffeological space
X is in WD if and only if each connected component of X is in WD .
The C ∞ -manifold Embprop (M, N ) of proper (equivalently closed) embeddings
is defined as an open submanifold of Emb(M, N ) ([46, Theorem 44.1]). We can∞11. APPLICATIONS TO C ∞ -MANIFOLDS

91

observe from [46, Theorem 42.14] that Embprop (M, N ) is a disjoint union of connected components of Emb(M, N ). Hence, Corollary 11.22(2) also holds true for
the C ∞ -manifold Embprop (M, N ).
For the proof of Corollary 11.22, we need the following facts on the C ∞ manifolds C∞ (M, N ) and C ω (M, N ); Part 1 is a corrected version of [46, Proposition 42.3] (see Remark 11.24) and its proof complements that of [46, Proposition
42.3] (see Remark C.6).
Lemma 11.23. Let M and N be second countable finite dimensional C ∞ manifolds.
(1) If M is compact, then C∞ (M, N ) is Lindelöf and regular and is modeled on nuclear Fréchet spaces. If M is noncompact, then each connectedcomponent of C∞ (M, N ) is Lindelöf and regular and is modeled on strict
inductive limits of sequences of nuclear Fréchet spaces.
(2) Suppose that M and N are C ω -manifolds with M compact. Then, C ω (M, N )is Lindelöf and regular and is modeled on nuclear Silva spaces.
Proof. (1) Model vector spaces See [46, Theorem 42.1 and the proof of
Lemma 30.4].
Lindelöf property The proof of [46, Proposition 42.3] applies to each connectedcomponent of C∞ (M, N ), showing its Lindelöf property (see Remark 11.24).
If M is compact, then C∞ (M, N ) = C ∞ (M, N ) holds in D, and hence π0 C∞ (M, N )f, Ne ]C 0 (see Theorem 1.1). Since M
f and Ne have the homotopy
is isomorphic to [Mtypes of a finite complex and a countable complex respectively, π0 C∞ (M, N ) is
countable. Thus, C∞ (M, N ) is Lindelöf.
Regularity Throughout this proof, we use the notation of [46, the proof of Theorem
42.1].
Note that C∞ (M, N ) is Hausdorff ([46, Theorem 42.1]) and that its model vector spaces are C ∞ -regular ([46, Theorem 16.10]). Also recall the charts (Uf , uf ) (f ∈C∞ (M, N )) constructed in [46, the proof of Theorem 42.1]. Then, we have only
to show that for any f ∈ C∞ (M, N ), there exist disjoint open sets Wf and Vf of
C∞ (M, N ) such thatf ∈ Wf and Ufc ⊂ Vf(see Lemma C.7 and Remark C.8). Consider the open embedding(πN ,exp)

T N ⊃ U ,−−−−−→ N × N
constructed in [46, the proof of Theorem 42.1]. To simplify the argument below,
we choose a new metric on the tangent bundle T N such that the unit disk bundleof T N is contained in U and redefine U as the open unit disk bundle of T N (withrespect to the new metric); explicitly,U = {v ∈ T N | |v| < 1} ⊂ T N,
V = (πN , exp)(U ) ⊂ N × N.
Recall that
Uf = {g ∈ C∞ (M, N ) | (f (x), g(x)) ∈ V for all x ∈ M, g ∼ f }.

92

HIROSHI KIHARA

We define the subsets Wf and Kf of C∞ (M, N ) by
Wf = {g ∈ C∞ (M, N ) | (f (x), g(x)) ∈ V 31 for all x ∈ M, g ∼ f },Kf = {g ∈ C∞ (M, N ) | (f (x), g(x)) ∈ V 23 for all x ∈ M, g ∼ f },where we set
Ur = {v ∈ T N | |v| < r},
Vr = (πN , exp)(Ur )for 0 < r < 1. We then have the obvious inclusion relations
Wf ⊂ Kf ⊂ Uf .
−1

Note that (f × 1N ) (Vr ) is an open neighborhood of the graph Γf in M × N andthat the equivalence(f (x), g(x)) ∈ Vr for all x ∈ M ⇐⇒ Γg ⊂ (f × 1N )−1 (Vr )holds. Recalling [46, Remark 42.2], we thus see that Wf is open. Similarly, we also
see that Kf is closed. Thus, Wf and Vf := C∞ (M, N )\Kf are disjoint open setssuch thatf ∈ Wf and Ufc ⊂ Vf .
(2) Choose a real analytic exponential map of N . Then, the atlas {(Ufω , uωf )}f ∈C ω (M,N )of C ω (M, N ) is constructed similarly to the atlas {(Uf , uf )}f ∈C ∞ (M,N ) of C∞ (M, N )(see the proofs of [45, Theorem 8.2] and [46, Theorems 42.1 and 42.6]). By
Part 1 and [46, the proof of Theorem 42.8], we can choose a countable atlas{(Ufn , ufn )}n∈N of C∞ (M, N ) such that each fn is in C ω (M, N ). Hence, {(Ufωn , uωfn )}n∈N
is an atlas of C ω (M, N ). Since the model vector spaces C ω (M ← f ∗ T N ) are nuclear Silva spaces ([46, Lemma 30.6 and Theorem 11.4]), their open sets are Lindelöf(see the proof of Corollary 11.14), and hence C ω (M, N ) is Lindelöf. Last, we showthat C ω (M, N ) is regular. Since C ω (M, N ) is Hausdorff ([46, Corollary 42.11])
and its model vector spaces are C ∞ -regular ([46, Theorem 16.10]), we have only
to show that for any f ∈ C ω (M, N ), there exists a coordinate open neighborhood
satisfying condition (b) in Lemma C.7 (see Remark C.8). Recalling that the inclusion C ω (M, N ) ,−→ C∞ (M, N ) is smooth ([46, Corollary 42.11]), we can obtain
the desired coordinate open neighborhood of f in C ω (M, N ) by restricting that in
C∞ (M, N ) constructed in Part 1.

Remark 11.24. In [46, Proposition 42.3], it is stated that C∞ (M, N ) is Lindelöf. However, this is incorrect. In fact, suppose that M is noncompact and that
dim N ≥ 1. Then, C∞ (M, N ) has uncountably many constant maps, which represent different connected components (see [46, Theorem 42.14]). Hence, C∞ (M, N )is not Lindelöf.
Proof of Corollary 11.22. (1) The result follows from Theorem 11.20(2)and Lemma 11.23(1).
(2) By Part 1, we may assume that M is noncompact. By [54, Sections 10, 11,
and 13], the C ∞ -manifolds in question are classical (see Lemma 11.7). Thus, we see
from Theorem 11.20(3) and Lemma 11.23(1) that the result holds for C∞ (M, N ),and hence that each connected component of the two open submanifolds is also
Lindelöf. Thus, the result also holds for the two open submanifolds.
(3) The result follows from Theorem 11.20(2) and Lemma 11.23(2).
11. APPLICATIONS TO C ∞ -MANIFOLDS

93

Let us investigate important submanifolds of C∞ (M, N ) and C ω (M, N ) otherthan those mentioned in Corollary 11.22.
Example 11.25. Let M and N be second countable finite dimensional C ∞ manifolds; M and N are assumed to be C ω -manifolds with M compact, whenever
we consider C ω -maps between them. Let X be one of the following C ∞ -manifolds:• The C ∞ -manifold C∞ (q) of smooth sections of a given smooth map q :N −→ M and the C ∞ -manifold C ω (q) of analytic sections of a givenC ω -map q : N −→ M (see [46, Thorem 42.20]).
• The Lie group Diff(M, σ) of symplectic diffeomorphisms of a symplectic
manifold (M, σ) and the Lie group Diff ω (M, σ) of C ω -symplectic diffeomorphisms of a C ω -symplectic manifold (M, σ) (see [46, Theorem 43.12]).
If M is compact, then every submanifold of X is hereditarily Lindelöf, C ∞ -regular,
and classical, and hence is in WD 0 (see Corollary 11.22(1) and (3)). Even if M
is noncompact, X (= C∞ (q) or Diff(M, σ)) is classical (see [54, Sections 10 and
14] and Lemma 11.7), and thus each connected component of X is hereditarily
Lindelöf, C ∞ -regular, and classical, and hence is in WD 0 (see Corollary 11.22(2)and Proposition 11.4).
This result applies to the following C ∞ -manifolds that can be formulated as∞
C -manifolds of smooth sections:
The C ∞ -manifold B(M ) of nondegenerate bilinear structures on
M , the C ∞ -manifold Mq (M ) of pseudo Riemannian metrics on
M of signature q, the C ∞ -manifold Ω2nd (M ) of nondegenerate 2forms on M , and the C ∞ -manifold H(M ) of almost Hermitian
structures on M (see [46, Section 45] and [26, Section 1]).
Hence, if M is compact, the C ∞ -manifold Symp(M ) of symplectic structures on M
is in WD 0 . Even if M is noncompact, each connected component of Symp(M ) isin WD 0 (see [46, 45.21]).
Next, we apply Theorem 11.20 to the C ∞ -manifold B(M, N ) of submanifolds
of N of type M ([46, p. 474]) and the C ∞ -manifold B ω (M, N ) of real analytic
submanifolds of N of type M ([46, p. 477]).
Corollary 11.26. Let M and N be a second countable, connected C ∞ -manifolds
with dim M ≤ dim N < ∞.
(1) Suppose that M is compact. Then, every submanifold of B(M, N ) is hereditarily Lindelöf, C ∞ -regular, and classical, and hence is in WD 0 .
(2) Suppose that M is noncompact. Then, each connected component ofB(M, N ) is hereditarily Lindelöf, C ∞ -regular, and classical, and hence
is in WD 0 .
(3) Suppose that M and N are C ω -manifolds with M compact. Then, every submanifold of B ω (M, N ) is hereditarily Lindelöf, C ∞ -regular, andclassical, and hence is in WD 0 .
For the proof of Corollary 11.26, we need basic facts on C ∞ -manifolds B(M, N )and B ω (M, N ) (Lemma 11.28). The following lemma is needed to prove Lemma
11.28. See [46, 41.5 and 41.10] for the CO-topology and the W O∞ -topology.
Lemma 11.27. Let M and N be second countable finite dimensional C ∞ manifolds.

94

HIROSHI KIHARA(1) For any compact subset K of M , there exists a compact submanifold K0
with boundary such that K ⊂ K0 .
(2) If f ∈ C∞ (M, N ) is in the connected component of a diffeomorphism, thenf is surjective.
(3) The canonical mapsid

C∞ (M, N ) −−→ C ∞ (M, N )W O∞ and C ∞ (M, N )W O∞ ,−→ C 0 (M, N )COare continuous, where C ∞ (M, N )W O∞ (resp. C 0 (M, N )CO ) is the set ofsmooth (resp. continuous) maps from M to N endowed with the W O∞ (resp. CO-)topology.
Proof. Set m = dim M .
(1) Choose a closed embedding ι : M ,−→ R2m+1 ([6, (7.10)]) and define thesmooth function ϕ : R2m+1 −→ R by ϕ(x1 , · · · , x2m+1 ) = x21 + · · · + x22m+1 . Then,the compositeιϕ
M ,−→ R2m+1 −−→ R
is a proper smooth map. Set K(r) = (ϕ ◦ ι)−1 [0, r] for r ∈ R. Choose r0 > 0such that (ϕ ◦ ι)(K) ⊂ [0, r0 ] and r0 is a regular value of ϕ ◦ ι ([6, (6.1)]). Then,
K0 = K(r0 ) is the desired compact submanifold with boundary.
(2) For simplicity, we assume that M and N are connected. Then, we can choosea diffeomorphism g and a connected compact submanifold K (with boundary) of
M such that f is smoothly homotopic to g relative to M − K̊ (see [46, Lemma 42.5]
and Part 1). Then, we see thatf∗ = g∗ : Hm (M, M − K̊; Z/2) −→ Hm (N, N − g(K̊); Z/2),
and hence that f∗ is an isomorphism. Since∼ Hm (K, ∂K; Z/2) ∼Hm (M, M − K̊; Z/2) =
= Z/2
and

Hm (M − {x}, M − K̊; Z/2) ∼= Hm (K − {x}, ∂K; Z/2) = 0

for x ∈ K̊ (see [15, p. 304 and p. 307]), f : M −→ N is surjective.
(3) See [46, Section 41 and Remark 42.2].

Lemma 11.28. Let M and N be a second countable, connected C ∞ -manifolds
with dim M ≤ dim N < ∞.
(1) If M is compact, then B(M, N ) is Lindelöf and regular and is modeled on
nuclear Fréchet spaces. If M is noncompact, then each connected component of B(M, N ) is Lindelöf and regular and is modeled on strict inductive
limits of sequences of nuclear Fréchet spaces.
(2) Suppose that M and N are C ω -manifolds with M compact. Then, B ω (M, N )is Lindelöf and regular and is modeled on nuclear Silva spaces.
Proof. For the proof, recall the following fact: Since an arc-generated space
X is locally arcwise connected, the connected components of X are just the arcwise
connected components of X and they are open sets.
(1) Lindelöf property and Model vector spaces Consider the principal Diff(M )bundle π : Emb(M, N ) −→ B(M, N ) ([46, Theorem 44.1]). First, suppose that M
is compact. Then, Emb(M, N ), and hence B(M, N ) is Lindelöf (see Corollary
11.22(1)). We see from [46, the proof of Theorem 44.1 and Section 30.1] that themodel vector spaces of B(M, N ) are nuclear Fréchet spaces. Next, suppose that M11. APPLICATIONS TO C ∞ -MANIFOLDS

95

is noncompact. Then, for any f ∈ Emb(M, N ), the restriction π : Emb(M, N )f −→B(M, N )π(f ) is a principal G-bundle for some open Lie subgroup G of Diff(M ),where Emb(M, N )f and B(M, N )π(f ) are the connected components of Emb(M, N )and B(M, N ) containing f and π(f ) respectively. Since Emb(M, N )f is Lindelöf(Corollary 11.22(2)), B(M, N )π(f ) is also Lindelöf. We see from [46, the proof of
Theorem 44.1 and Section 30.4] that the model vector spaces of B(M, N ) are strict
inductive limits of sequences of nuclear Fréchet spaces.
Regularity In the case of dim M = dim N , B(M, N ) is 0-dimensional (see [46,the proof of Theorem 44.1]), and hence discrete. Thus, we assume that dim M <dim N . Since B(M, N ) is Hausdorff ([46, Theorem 44.1]) and its model vector
spaces are C ∞ -regular ([46, Theorem 16.10]), we have only to construct, for any
bi = π(i) ∈ B(M, N ), a coordinate open neighborhood satisfying condition (b) inLemma C.7 (see also Remark C.8).
In this proof, given a map p : E −→ B and a subset S of B, we write E|S for
the inverse image of S under p.
We first recall the construction of charts of B(M, N ) from [54, Section 13].
Let us fix an embedding i ∈ Emb(M, N ). Since L = i(M ) is a submanifold of N ,there exist a tubular neighborhood WL of L; more precisely, we have a submersion
pL : WL −→ L, a vector bundle qL : VL −→ L, and a diffeomorphism over L
φL

VL −−−−→ WL .
∼=(We construct pL , qL , and φL by choosing a Riemannian metric on N .) Using
these, a chart around bi (= π(i))
ci −−→ Cc∞ (M ←− i∗ VL )B(M, N ) ⊃ Q∼=

is constructed.
Next, consider the space J 1 (i∗ VL ) of 1-jets of local sections of i∗ VL and thecanonical epimorphism π01 : J 1 (i∗ VL ) −→ i∗ VL of vector bundles over M (see [46,Section 41] for the basic facts and notation on jets). We fix a metric on the jetbundle J 1 (i∗ VL ). By identifying the orthogonal complement of Ker π01 with i∗ VL
via π01 , we endow i∗ VL with the metric induced by that of J 1 (i∗ VL ). Then, we havethe inequality|σ| ≥ |π01 (σ)|(11.1)for σ ∈ J 1 (i∗ VL ).

Given a finite rank vector bundle E −→ X with metric, Dr (E) (resp. D̊r (E))
denotes the subbundle of E whose fiber is the disk (resp. open disk) of radius r.
ci as follows using the chart described
We thus define the subsets Wi and Ki of Q
above:Wi ∼= {s ∈ Cc∞ (M ←− i∗ VL ) | j 1 s(M ) ⊂ D̊ 13 (J 1 (i∗ VL ))},Ki ∼= {s ∈ Cc∞ (M ←− i∗ VL ) | j 1 s(M ) ⊂ D 32 (J 1 (i∗ VL ))}.
Since the relationscibi = π(i) ∈ Wi ⊂ Ki ⊂ Q
hold and Wi is open in B(M, N ) ([46, 41.13]), we have only to show that Ki is
closed in B(M, N ).
Noticing that π : Emb(M, N ) −→ B(M, N ) is a topological quotient map (seeLemma 5.10), we show that π −1 Ki is closed in Emb(M, N ). Since π −1 Ki is closed

96

HIROSHI KIHARA

ci , it suffices to show that every j0 ∈ (π −1 Qci )c has a neighborhood disjointin π −1 Q−1from π Ki .
ci )c satisfies one of the following conditions:
We can easily see that j0 ∈ (π −1 Q(i) j0 (M ) 6⊂ WL .
(ii) j0 (M ) ⊂ WL and pL ◦ j0 : M −→ L is not a diffeomorphism.
Suppose that condition (i) is satisfied. We first deal with the case where
j0 (M ) 6⊂ D 23 (WL ), where D 23 (WL ) is the image of D 32 (i∗ VL ) under the composite
φLi∗ VL −→VL −−
→ WL ,−→ N.
∼
∼
=

=

In this case, we see from Lemma 11.27(3) and (11.1) that the subset{j ∈ Emb(M, N ) | j(M ) 6⊂ D 32 (WL )}is an open neighborhood of j0 disjoint from π −1 Ki . Next, we deal with thecase where j0 (M ) ⊂ D 23 (WL ). Then, there exists x0 ∈ M such that j0 (x0 ) ∈D 23 (WL )\D 23 (WL ); such a situation must be considered when i is a nonclosed embedding (see Fig 11.1).
D 32 (WL )j0 (M )
L = i(M )Fig 11.1
If jb0 is not in the connected component of bi, then the connected component of
j0 is an open neighborhood of j0 disjoint from π −1 Ki . Thus, we assume that jb0
is in the connected component of bi. Then, there exists h ∈ Diff(M ) such that{x ∈ M | j0 (x) 6= i ◦ h(x)} is relatively compact in M (see [46, Lemma 42.5]).
Set K = {x ∈ M | j0 (x) 6= i ◦ h(x)} and choose compact submanifolds K0 and K1(with boundary) of M such that K ⊂ K0 ⊂ K̊1 (see the proof of Lemma 11.27(1)).
Further, we can choose a tubular neighborhood Wj0 (M ) of j0 (M ) and a submersion
pj0 (M ) : Wj0 (M ) −→ j0 (M ) such that Wj0 (M ) |j0 (M )−j0 (K0 ) ⊂ WL |L−i◦h(K0 ) and the
diagram
Wj0 (M ) |j0 (M )−j0 (K0 )WL |L−i◦h(K0 )pj0 (M )j0 (M ) − j0 (K0 )pL=

L − i ◦ h(K0 )commutes.
Define the open sets V1 , V2 , and V3 of Emb(M, N ) by
V1 = {j ∈ Emb(M, N ) | j(x0 ) ∈ N − D 23 (WL )|i◦h(K1 ) },

11. APPLICATIONS TO C ∞ -MANIFOLDS

97

V2 = {j ∈ Emb(M, N ) | j(K0 ) ⊂ Wj0 (M ) |j0 (K̊1 ) },V3 = {j ∈ Emb(M, N ) | j(M ) ⊂ Wj0 (M ) }(see Lemma 11.27(3)) and define V to be the connected component of j0 of V1 ∩V2 ∩ V3 . Since V is obviously an open neighborhood of j0 , we have only to showthat V is disjoint from π −1 Ki . We thus show that every j ∈ V satisfies one of thefollowing conditions:• j(M ) 6⊂ D 23 (WL ).
• j(M ) ⊂ D 23 (WL ) and pL ◦ j : M −→ L is not injective.
Observe from Lemma 11.27(2) that the image of M − K0 under the compositejpj (M )

0−−→ j0 (M )M ,−−−−→ Wj0 (M ) −−−

contains j0 (M ) − j0 (K̊1 ) = L − i ◦ h(K̊1 ). Now suppose that j(M ) ⊂ D 32 (WL ).
Noticing that x0 ∈ K and that pL ◦ j(x0 ) ∈ L − i ◦ h(K1 ), we then see that thejpL

composite M ,−→ WL −→ L is not injective.
Suppose that condition (ii) is satisfied. We first deal with the case of compact
M . If the differential of pL ◦ j0 : M −→ L is invertible at every point x ∈ M , thenpL ◦ j0 is a covering (see [6, Theorem 8.12]). Since pL ◦ j0 is not a diffeomorphism,
it is an d-fold cover with d > 1. Considering the mapping degree, we thus see that asufficiently small neighborhood of j0 is disjoint from π −1 Ki (see Lemma 11.27(3)).
Hence, we may assume that the differential of pL ◦ j0 is not invertible at somex0 ∈ M . Then, we can choose a nonzero vector v0 ∈ Tx0 M with pL ∗ j0 ∗ (v0 ) = 0.
Noticing that pL ∗ j∗ (v0 ) is small for j near j0 , we see that a sufficiently smallneighborhood of j0 is disjoint from π −1 Ki (see Fig. 11.2). We next deal withthe case of noncompact M . If the connected component of j0 does not containany element of π −1 Ki , then it is an open neighborhood of j0 disjoint from π −1 Ki .
Thus, we assume that the connected component of j0 contains an element of π −1 Ki .
Then, pL ◦ j0 restricts to a diffeomorphism between M \K and L\pL ◦ j0 (K) forsome compact subset K of M (see [46, Lemma 42.5]). If the differential of pL ◦ j0 isinvertible at every point x ∈ M , then pL ◦ j0 is a diffeomorphism (see [6, Theorem
8.12]), which is a contradiction. Thus, the differential of pL ◦ j0 is not invertible atsome x0 ∈ M , which shows that a sufficiently small neighborhood of j0 is disjoint
from π −1 Ki (see the argument in the case of compact M ).

WL
j0 (M )
L = i(M )Fig 11.2∞

ω(2) The C -structure of B (M, N ) is introduced similarly to that of B(M, N ),though the model vector spaces are nuclear Silva spaces unlike the case of B(M, N )98

HIROSHI KIHARA([46, Theorem 44.3]). Since the principal Diff ω (M )-bundle π ω : Embω (M, N ) −→B ω (M, N ) exists ([46, Theorem 44.3]) and Embω (M, N ) is Lindelöf (Corollary
11.22(3)), B ω (M, N ) is Lindelöf.
Last, we show that B ω (M, N ) is regular. Since B ω (M, N ) is Hausdorff ([46,Theorem 44.3]) and its model vector spaces are C ∞ -regular ([46, Theorem 16.10]),
we have only to show that for any bi = π ω (i) ∈ B ω (M, N ), there exists a coordinate
open neighborhood satisfying condition (b) in Lemma C.7 (see also Remark C.8).
Noticing that π ω : Embω (M, N ) −→ B ω (M, N ) is a D-quotient map (Lemma
5.10), we have from [46, Corollary 42.11] the commutative diagram in C ∞Embω (M, N )Emb(M, N )πωπB ω (M, N )B(M, N ).

Thus, we can obtain the desired coordinate open neighborhood of bi in B ω (M, N )by restricting that in B(M, N ) constructed in Part 1 (see the proof of [46, Theorem
44.3]).

Proof of Corolarry 11.26. (1) The result follows from Theorem 11.20(2)and Lemma 11.28(1).
(2) Since B(M, N ) is classical ([54, Section 13] and Lemma 11.7), the result
follows from Theorem 11.20(3) and Lemma 11.28(1).
(3) The result follows from Theorem 11.20(2) and Lemma 11.28(2).
Let Bclosed (M, N ) be the open submanifold of B(M, N ) consisting of closed
submanifolds ([46, p. 474]). We can observe that Bclosed (M, N ) is a disjoint unionof connected components of B(M, N ) (see the comment on Embprop (M, N ) after Corollary 11.22). Hence, Corollary 11.26(2) remains true even if we replace
B(M, N ) with Bclosed (M, N ).
Theorem 11.20 also applies to direct limits of finite dimensional C ∞ -manifolds,
which are extensively studied by Glöckner and coauthors ([27], [28], [29]). Many
infinite dimensional C ∞ -manifolds important in algebraic topology are obtained assuch direct limits.
More precisely, we consider an ascending sequence {Mn } of second countable finite dimensional C ∞ -manifolds whose bonding maps are closed embeddings. Then,the direct limit M exists in the category C ∞ , and if lim dim Mn = ∞, thenn→∞

M is modeled on the Silva space R∞ ([27, Proposition 3.6]). For F = R, C, theinfinite classical groups such as GL(∞, F), SL(∞, F), and U (∞), and the infinite
Grassmannians G(k, ∞; F) are described as such direct limits; these C ∞ -manifolds
are introduced in a different manner in [46, Section 47]. The infinite GrassmannianG(∞, ∞; F), which is not introduced in [46, Section 47], can be defined to be thedirect limit lim G(k, ∞; F), since→lim G(k, ∞; F) = lim lim G(k, l; F) = lim G(k, k; F),
→

→
k

→
l

→

where G(k, l; F) is the finite Grassmannian consisting of k-dimensional linear subspaces of Fk+l .
Corollary 11.29. Let {Mn } be an ascending sequence of second countable
finite dimensional C ∞ -manifolds whose bonding maps are closed embeddings. Then,11. APPLICATIONS TO C ∞ -MANIFOLDS

99

every submanifold of the direct limit M = lim Mn is hereditarily Lindelöf, C ∞ →regular, and classical, and hence is in WD 0 . In particular, the infinite classicalgroups such as GL(∞, F), SL(∞, F), and U (∞), and the infinite Grassmannians
G(k, ∞; F) (0 ≤ k ≤ ∞) are in WD 0 .
Proof. Note that the direct limit M is Lindelöf and regular and is modeled
on R∞ ([27, Proposition 3.6 and its proof]) and that R∞ is a nuclear Silva space([40, Corollary 21.2.3]). Then, the result follows from Theorem 11.20(2).
100

HIROSHI KIHARAAppendix A. Pathological diffeological spaces
The subclasses WD and VD of D were introduced in Section 1.3 as subclassese ∈ WC 0 }, we obtainof D-homotopically good objects. SettingeWC 0 = {A ∈ D | A
from Corollary 1.6 the following Venn diagram:
D
eWC 0

VD
WD
(A.1)

In this appendix, we show that all the inclusions shown in Venn diagram (A.1)
are proper by providing examples of diffeological spaces in the classes D\(VD ∪eWC0 ), eWC 0 \VD , VD \eWC 0 , and (VD ∩ eWC 0 )\WD (Examples A.3, A.6, and A.10).
Our study in this appendix organizes the study of pathological diffeological spaces
presented by Iglesias-Zemmour [38, 8.38] and Christensen-Wu [8, Examples 3.12
and 3.20].
We begin with the following lemma on topological spaces.
Lemma A.1. Let X be a topological space having the homotopy type of a CW complex. Then, the following are equivalent:(i) X is homotopy equivalent to a Lindelöf space.
(ii) X is homotopy equivalent to a countable CW -complex.
(iii) Hi (X) and πi (X, x) are countable for any i ≥ 0 and any x ∈ X.
(iv) πi (X, x) is countable for any i ≥ 0 and x ∈ X.
Proof. For (i) ⇐⇒ (ii), see [56, Proposition 2]. For (ii) ⇐⇒ (iii) ⇐⇒ (iv),see Remark 9.21(1) and (2).
Remark A.2. We can prove the following analogue of Lemma A.1 using theresults in [52, pp. 75-76] and [53, Theorem 4.5.2]:
Let X be a topological space having the homotopy type of a CW complex. If X is homotopy equivalent to a compact space, thenH∗ (X) = ⊕ Hi (X) and π1 (X, x) are finitely generated for any
i≥0x ∈ X. Further if X is 1-connected, then πi (X, x) is finitely
generated for any i ≥ 1 and any x ∈ X.
Example A.3. The n-dimensional Hawaiian earring E n is defined to be the∞diffeological subspace ∪ Ekn of Rn+1 , where
k=1

1 2
1 2
Ekn = {(x0 , . . . , xn ) ∈ Rn+1 | (x0 − √) + x21 + · · · + x2n = ( √
) }.
n
n
k
k
fn does not have the homotopy type of a CW -complex.
(1) E

APPENDIX A. PATHOLOGICAL DIFFEOLOGICAL SPACES

101(2) The natural homomorphismfn )πnD (E n ) −→ πn (Eis not surjective.
(3) E n is in D\(VD ∪ eWC 0 ).
Proof. (1) By the argument of [8, Example 3.12] (see also [46, p. 18]), we
see that the underlying topology of E n is just the sub-topology of Rn+1 . ObserveQfn ) −→ Q
fn
that the canonical epimorphism πn (Ek≥1 πn (Ek ) =k≥1 Z exists (cf. [58,
Example 1 in Section 71]). Then, the result follows from Lemma A.1.
(2) We can prove the result by the argument in the proof of Part 1 and anargument similar to that in [8, Example 3.12].
(3) The result follows from Parts 1-2 and Corollary 1.6(3).
Remark A.4. This remark relates to the n-dimensional Hawaiian earring E n .
n
fn is homeomorphic to the topological subspace Etopof Rn+1
We can easily see that E
defined by
∞

n
Etop
= ∪ {(x0 , . . . , xn ) ∈ Rn+1 | (x0 − k1 )2 + x21 + · · · + x2n = ( k1 )2 },k=1

which is called the n-dimensional topological Hawaiian earring. See [20, 21] forn).
the exact calculations of πn (Etop
Next, we investigate the quotient of a diffeological group by a dense subgroup.
Lemma A.5. Let G be a diffeological group, and H a diffeological subgroup ofe and the inclusion H ,−→ G is not a weak equivalence in D,
G. If H is dense in Gthen G/H is in eWC 0 \VD .
] is an indiscrete space ([43, Lemma 2.11]),
Proof. By the assumption, G/H
and hence, contractible. On the other hand, we see that π∗D (G/H) 6= 0 from thehomotopy exact sequence of the sequence
H ,−→ G −→ G/H([38, 8.15 and 8.21]). Thus, we obtain the result by Corollary 1.6(3).

For an irrational number θ, the irrational torus Tθ2 of slope θ is defined to be
the quotient diffeological group T 2 /Rθ , where T 2 = R2 /Z2 is the usual 2-torus andRθ is the image of the injective homomorphism R −→ T 2 sending x to [x, θx] ([38,8.38]).
Example A.6. The quotient diffeological groups R/Q and the irrational torusTθ are ineWC 0 \VD .
Proof. The result is immediate from Lemma A.5.



Lastly, we show the following lemma and then identify conditions under whicha diffeological space of the form RX is in VD , eWC 0 and WD (see Proposition 2.2(3)and Remark 2.3(2) for the functor R).
Lemma A.7. Let X be an arc-generated space and Z a cofibrant diffeologicalspace. Then, any smooth map f : RX −→ Z is locally constant.

102

HIROSHI KIHARA

Proof. Let γ : ∅ −→ Z be the canonical map from the initial object ∅ to Z.
Consider the cofibration-trivial fibration factorizationi∞∅

G∞ (I, γ)p∞γZ
as in [43, the proof of Theorem 1.3] and recall that i∞ is a sequential relative I-cellcomplex. Then, by solving the lifting problem in Di∞∅

G∞ (I, γ)
p∞
1

Z

Z,

we obtain the retract diagram in D
Z ,−→ G∞ (I, γ) −→ Z.
Thus, we may assume that Z is the colimit of a sequence in Di2

i1

Z0 ,−→ Z1 ,−→ Z2 ,−→ · · ·such that Z0 is a coproduct of copies of ∆0 and each in fits into a pushout diagramof the form
` ˙ pλ
` pλ
∆
∆
λ∈Λn

λ∈Λn

Zn−1

in

Zn .

Since X is locally arcwise connected, we may also assume that X is arcwise
connected. Suppose that there exists a nonconstant smooth map f : RX −→ Z.
Then, we can choose x0 , x1 ∈ X and a continuous curve d : R −→ X satisfying thefollowing conditions:• f (x0 ) 6= f (x1 ).
• d((−∞, 0]) = x0 , d([1, ∞)) = x1 .
We see from Proposition 2.2(3) that the compositefd

R −−−→ RX −−−→ Z
is a nonconstant smooth curve.
Recall [43, Lemma 9.9] and set
m = min {n | (f ◦ d)(R) ⊂ Zn }.
Then, we obtain the nonconstant smooth map
f ◦d

R −−−−−→ Zm .
˚ pλ −→ Zm is a D-embedding for λ ∈ Λm , we regard ∆˚ pλSince the inclusion ∆λλ−1 ˚ pλas a diffeological subspace of Zm . Noticing that (f ◦ d) (∆λ ) is a nonempty
open set of R for some λ ∈ Λm , we may assume that the nonconstant smoothAPPENDIX A. PATHOLOGICAL DIFFEOLOGICAL SPACES103

˚ pλ . Then, we can choose a linear map
map f ◦ d : R −→ Zm corestricts to ∆λ` : Rpλ +1 −→ R such that the compositef ◦d`˚ pλ −−−→ R
R −−−−−→ ∆λis a nonconstant smooth map (see Lemma 2.17(3)). By the definition of the functor
R, the compositec`◦f ◦dR −−−→ R −−−−−−→ R
is smooth for any continuous curve c, which is a contradiction.

Remark A.8. Lemma A.7 remains true even if a cofibrant target Z is replaced
by a C ∞ -manifold M . This is proved as follows. Suppose that there exists a smooth
map f : RX −→ M which is not locally constant. Then, we can choose a continuousuα− Uα ,−→ M , and a continuous linear
curve d : R −→ X, a chart Eα ⊃ uα (Uα ) ←−∼=functional ` : Eα −→ R such that• f ◦ d : R −→ M corestricts to Uα .
• The composite
f ◦d

u

`

α
R −−→ Uα −−→uα (Uα ) ,−→ Eα −−−→ R∼=

is a nonconstant smooth map.
(See [40, p. 127] and [46, Corollary 2.11]). This yields a contradiction (see theproof of Lemma A.7).
We prove the following result; Lemma A.7 is used for the proof of Part 3.
Lemma A.9. Let X be an arc-generated space.
(1) RX is in VD .
(2) RX is ineWC 0 if and only if X is in WC 0 .
(3) RX is in WD if and only if X has the homotopy type of a discrete set.
Proof. Parts 1 and 2 are immediate from the equality ˜· ◦ R = IdC 0 (Remark
2.3(2)).
(3) (⇐) Note that if f, g : X −→ Y are C 0 -homotopic, then Rf, Rg : RX −→RY are D-homotopic (see the argument in the proof of Corollary 1.6(2)). Then,the implication is obvious.
(⇒) Note that`every arc-generated space is locally` arcwise connected. Then, we
see that X = Xi in C 0 , and hence that RX = RXi in D, where {Xi } is the setiiof arcwise connected components of X. Thus, we may assume that X is arcwise
connected.
If RX is in WD , then 1RX : RX −→ RX is D-homotopic to a constant map(Lemma A.7), and hence RX 'D ∗ holds.
nnExample A.10.
(1) REtop
is in VD \ eWC 0 for n ≥ 1, where Etop
is then-dimensional topological Hawaiian earring.
nn(2) RStop
is in (VD ∩ eWC 0 )\WD for n ≥ 1, where Stopis the n-dimensional
topological sphere.
n
Proof. (1) Recall that Etop
is an arc-generated space which is not in WC 0 (seeExample A.3(1) and Remark A.4). Then, the result follows from Lemma A.9(1)-(2).
(2) The result follows from Lemma A.9(1)-(3).
104

HIROSHI KIHARA

By Examples A.3, A.6, and A.10, we have shown that all the inclusions in Venn
diagram (A.1) are proper.
nRemark A.11. Since S n is in WD (Theorem 11.1), [RStop, S n ]D = ∗ byLemma A.7. From Example A.10(2), we thus see that even if A ∈ VD ∩ eWC 0
e X]
e C 0 need not be bijective (cf. Theorem 1.7).
and X ∈ WD , [A, X]D −→ [A,Appendix B. Keller’s Cc∞ -theory and diffeological spaces
Keller’s Cc∞ -theory is a representative example of classical infinite dimensional
calculi (Remark B.1) and is widely used by many authors (cf. [54] and the articles
of Glöckner, Neeb, and their coauthors); see [27, 1.3] and references therein forthe basics of Keller’s Cc∞ -theory. In this appendix, we discuss the relation between
C ∞ -manifolds in Keller’s Cc∞ -theory, C ∞ -manifolds in convenient calculus, anddiffeological spaces. In particular, we show that the C ∞ -manifolds in Keller’s Cc∞ theory cannot be fully faithfully embedded into the category D (cf. Proposition
2.6).
C ∞ -manifolds in Keller’s Cc∞ -theory are defined by gluing open sets of locallyconvex vector spaces via diffeomorphisms; we require that C ∞ -manifolds in Keller’s
Cc∞ -theory are Hausdorff. Let Cc∞ denote the category of C ∞ -manifolds in Keller’s
Cc∞ -theory. We call an object and a morphism of Cc∞ a Cc∞ -manifold and a Cc∞ map to distinguish them from convenient ones.
Let Cc∞conv denote the full subcategory of Cc∞ consisting of Cc∞ -manifolds
modeled on convenient vector spaces. We can then define the faithful functor
J : Cc∞conv −→ C ∞by assigning to M ∈ Cc∞conv the object JM of C ∞ having the same underlying set
and atlas as M .
Remark B.1. In this paper, a classical infinite dimensional calculus is a word
opposed to the word “convenient calculus”. A classical infinite dimensional calculus
is assumed to satisfy the following conditions:• A smooth map is a continuous map between open sets of locally convex
vector spaces (in some specified class).
• The smooth curves into a locally convex vector space are just the usual
ones (see [46, 1.2]).
• Smooth maps are closed under composites.
Keller’s Cc∞ -theory is a classical infinite dimensional calculus in this sense. The
definition of C ∞ -manifolds and that of the faithful functor J apply to every classicalinfinite dimensional calculus.
Proposition B.2. The faithful functor
J : Cc∞conv −→ C ∞is not full.
Proof. Choose a nonbornological convenient vector space E (see [68]) andconsider the bornologification id : Eborn −→ E ([46, Lemma 4.2]). Then, J id :JEborn −→ JE is an isomorphism in C ∞ (see [46, Corollary 2.11]). Noticing that
id : E −→ Eborn is not even continuous, we see that
J : Cc∞conv (E, Eborn ) −→ C ∞ (JE, JEborn )∞
APPENDIX B. KELLER’S CC-THEORY AND DIFFEOLOGICAL SPACES

is not surjective.

105

We define the faithful functor I : Cc∞ −→ D similarly to the faithful functor
I : C ∞ −→ D (see Section 2.2); explicitly, I assigns to M ∈ Cc∞ the set M endowedwith the diffeology
DIM = {Cc∞ -parametrizations of M }.
Corollary B.3. The faithful functor
I : Cc∞ −→ D
is not full
Proof. We prove the result in two steps.
Step 1. We show that the diagram of faithful functors
Cc∞conv

J

Cc∞

I

C∞
I

D

is commutative. For this, we have only to show that for a domain U of Rn and alocally convex vector space E, the following equivalence holds:
f : U −→ E is a Cc∞ -map ⇔ f : U −→ E is a C ∞ -map.
The implication (⇒) is obvious. For (⇐), use [27, 1.3] and [46, Definition 3.17,
Theorem 3.18, the proof of Corollary 5.11, and Theorem 4.11(1)].
Step 2. Consider the commutative diagram in Step 1. Since the two vertical functors
are full (Proposition 2.6), I : Cc∞ −→ D is not full by Proposition B.2.
Remark B.4. Let Cc∞F (resp. Cc∞S ) denote the full subcategory of Cc∞ consisting of Cc∞ -manifolds modeled on Fréchet spaces (resp. Silva spaces). Similarly,
let CF∞ (resp. CS∞ ) denote the full subcategory of C ∞ consisting of C ∞ -manifolds
modeled on Fréchet spaces (resp. Silva spaces). Since the class of Fréchet spaces
and that of Silva spaces are closed under finite products ([73, Proposition 3]), we
see from [46, Theorem 4.11(1)-(2)] that if M is in CF∞ or CS∞ , then M is necessarily Hausdorff. Further, using an argument similar to that in the proof of Corollary
B.3, we can show that the functor J : Cc∞conv −→ C ∞ restricts to the isomorphismsof categories Cc∞F −→ CF∞ and Cc∞S −→ CS∞ (see [44] for details).
∼
∼
=

=

Proposition 2.6 and Corollary B.3 show that convenient calculus also provides
a good categorical setting for the purpose of investigating infinite dimensional C ∞ manifolds and smooth maps via the smooth homotopy theory of diffeological spaces.
However, Keller’s Cc∞ -theory plays an important role in supplying classical C ∞ manifolds in convenient calculus (Lemma 11.7).
We end this appendix with the following remark on convenient categories ofsmooth spaces.
Remark B.5. Kriegl-Michor [46] used the category F of Frölicher spaces as aconvenient category of smooth spaces. Using the adjoint pair F : D  F : I with
I fully faithful ([65]), we can construct a model structure on F for which (F, I)
is a pair of Quillen equivalences (see a forthcoming paper). However, we do not

106

HIROSHI KIHARA

adopt F as a convenient category of smooth spaces since it cannot be ensured that
F contains C ∞ (see [46, Lemma 27.5]).
Further, our approach cannot be directly applied to Frölicher spaces because of
the following fact:
The underlying topological space functor ˜· : F −→ C 0 , whichassigns to a Frölicher space A the set A endowed with the final
topology for the smooth curves, does not have a right adjoint.
(Cf. Theorem 1.5(2) and the definition of the class VD .) This fact is shown as
follows. Suppose that ˜· : F −→ C 0 has a right adjoint R : C 0 −→ F. Then, thenatural bijection F(∗, RX) ∼= C 0 (˜∗, X) implies that R preserves underlying sets.
Similarly, the natural bijection F(R, RX) ∼= C 0 (R̃, X) implies that the set CRX ofsmooth curves of RX is just the set of continuous curves of X. By an argument
similar to that in the proof of Lemma A.7, we thus see that if X is arcwise connected,
g is indiscrete, we have athen every smooth function on RX is constant. Since RX
g −→ X.
contradiction by considering the counit id : RX
Appendix C. Smooth regularity and smooth paracompactness
In this appendix, we discuss smooth regularity and smooth paracompactness,clarifying points often misunderstood and overlooked in the literature (Remarks
C.2 and C.6). The results in this appendix are used in Sections 11.3 and 11.4 toestablish the smooth paracompactness of all submanifolds of C∞ (M, N ), C ω (M, N ),B(M, N ), and B ω (M, N ) (see Section 11.4 for these C ∞ -manifolds).
Some results are discussed in a rather general context using the notation ofSection 5; C denotes one of the categories C ∞ , D, C 0 and T , and U : C −→ T
denotes the underlying topological space functor for C.
First, we prove the following result on C-regularity and apply it to topologicalmanifolds and C ∞ -manifolds. An object X of C is called locally C-regular if any
x ∈ X has a C-regular open neighborhood.
Lemma C.1. For X ∈ C with U X T1 -space, the following are equivalent:(i) X is C-regular.
(ii) U X is regular and X is locally C-regular.
Proof. (i) =⇒ (ii) See Remark 5.3.
(ii) =⇒ (i) Assume given a point x ∈ X and an open neighborhood U of x. We
construct a C-function φ : X −→ R with φ(x) = 1, carr φ ⊂ U , and φ(X) ⊂ [0, 1].
First, we choose a C-regular open neighborhood U 0 of x such that U 0 ⊂ U (seeRemark 5.3). Then, we can choose disjoint open sets W 0 and V 0 of X such thatx ∈ W 0 and U 0c ⊂ V 0
and a C-function ϕ : U 0 −→ R such that ϕ(x) = 1, carr ϕ ⊂ W 0 , and ϕ(U 0 ) ⊂ [0, 1].
Since the function ϕ : U 0 −→ R and the zero function 0 : V 0 −→ R coincide on0
U ∩ V 0 , φ := ϕ + 0 : X = U 0 ∪ V 0 −→ R is a C-function on X, which is the desired
bump function.
Remark C.2. In this remark, we specialize Lemma C.1 to topological manifolds(see Remark 11.19 for the definition of a topological manifold and recall from Remark 5.3 that T -regularity is just complete regularity). Since locally convex vector

APPENDIX C. SMOOTH REGULARITY AND SMOOTH PARACOMPACTNESS

107

spaces are completely regular ([40, Sections 2.1 and 2.9]), every topological manifold
is locally T -regular. Hence, Lemma C.1 specializes to the following statement:
A topological manifold is completely regular if and only if it isregular.
This is a corrected version of Eells’ statement that a topological manifold is completely regular [19, p. 765]. In fact, a separation condition strictly stronger than
Hausdorff property is necessary to derive complete regularity of a topological manifold (see [46, the example in Section 27.6]).
Now we establish the relation between C ∞ -regularity of a C ∞ -manifold andC ∞ -regularity of its model vector spaces.
Proposition C.3. For a C ∞ -manifold M , the following are equivalent:(i) M is C ∞ -regular.
(ii) M is regular as a topological space and is modeled on C ∞ -regular convenient vector spaces.
(iii) M is regular as a topological space and admits an atlas consisting of charts
onto C ∞ -regular convenient vector spaces.
Proof. (iii) =⇒ (ii) Obvious.
(ii) =⇒ (i) The implication follows from Lemma C.1.
(i) =⇒ (iii) Since M is obviously regular as a topological space, we have only toconstruct a chart onto a C ∞ -regular convenient vector space around any x ∈ M .
Take a chart of M around xuM ⊃ U −−∼−→ V ⊂ E=

with u(x) = 0, where E is a convenient vector space and V is a c∞ -open set of E.
We may assume that V is radial by replacing V with its star (see [46, Theorem
16.21]). Set V 12 = 21 V and U 12 = u−1 (V 21 ). By the C ∞ -regularity of M , we canchoose a smooth function ϕ : M −→ R such that ϕ(x) = 1, supp ϕ ⊂ U 21 andϕ(M ) ⊂ [0, 1]. Define the smooth function ψ : V −→ R to be the composite
u−1

ϕ|U

V −−−∼−−→ U −−−−−→ R=

and set V 0 = carr ψ. Then, the c∞ -open set V 0 of E and the smooth function
ψ|V 0 satisfy the assumption of [46, Theorem 16.21], and hence V 00 := star V 0 is a
C ∞ -regular c∞ -open set of E which is diffeomorphic to E. Thus, the compositeuM ⊃ U 00 −−∼−→ V 00 −−∼−→ E
=
00

=

−1

is the desired chart around x, where U = u(V 00 ).



Next, we discuss C-paracompactness of a Lindelöf C-space, especially C ∞ paracompactness of a Lindolöf C ∞ -manifold. Recall the following result from [46].
Proposition C.4. Let X be an object of C with U X Lindelöf T1 -space. Then,
X is C-paracompact if and only if X is C-regular.
Proof. See Remark 5.3, the comment after it, and [46, Theorem 16.10].
The following is a refinement of the first statement of [46, Corollary 27.4].



108

HIROSHI KIHARA

Theorem C.5. Let M be a Lindolöf C ∞ -manifold. Then, the following conditions are equivalent:(i) M is C ∞ -paracompact.
(ii) M is C ∞ -regular.
(iii) M is regular as a topological space and is modeled on C ∞ -regular convenient vector spaces.
Proof. The equivalences (i) ⇐⇒ (ii) and (ii) ⇐⇒ (iii) follow from Propositions C.4 and C.3 respectively.
Remark C.6. We can observe from [46, Section 27.6] that in Theorem C.5,
the regularity condition on M in condition (iii) is necessary for the implication(iii) =⇒ (i) to hold.
At first glance, the first statement of [46, Corollary 27.4] does not require theregularity condition. However, Kriegl replied my email that they use the definitionof Lindelöf (e.g., [23, 3.8]) where regularity is assumed. Thus, the implication(iii) =⇒ (i) in Theorem C.5 is a variant of the first statement of [46, Corollary27.4]. I wrote down the details of the proof since they are not so trivial and containpoints often misunderstood.
As mentioned above, the regularity of a C ∞ -manifold X must be verified to derive the C ∞ -paracompactness of X from its Lindelöf property and the C ∞ -regularityof its model vector spaces. However, Kriegl informed me that they forgot to mention
how to obtain the regularity of C∞ (M, N ) in the proof of C ∞ -paracompactness ofC∞ (M, N ) ([46, Proposition 42.3]). We give a regularity criterion for a manifold(Lemma C.7) and apply it to establish the (hereditary) C ∞ -paracompactness not
only of C∞ (M, N ) but also of C ω (M, N ), B(M, N ), and B ω (M, N ) (see Section
11.4).
We end this appendix with the following useful criterion for a manifold to be
regular as a topological space (see the proofs of Lemmas 11.23 and 11.28).
Lemma C.7. For a T1 -space X, the following are equivalent.
(i) X is regular.
(ii) For any x ∈ X, there exists an open neighborhood Ux of x satisfying
following conditions:(a) Ux is regular.
(b) There exist disjoint open sets Wx and Vx such thatx ∈ Wx and Uxc ⊂ Vx .
Proof. (i) ⇒ (ii) Obvious.
(ii) ⇒ (i) Assume given a point x ∈ X and an open neighborhood U of x. We
choose an open neighborhood Ux of x satisfying conditions (a) and (b), and thenchoose disjoint open sets Wx and Vx such thatx ∈ Wx and Uxc ⊂ Vx .
Next, we use condition (a) to choose disjoint open sets Wx0 and Vx0 of Ux such thatx ∈ Wx0 and Ux \U ∩ Ux ⊂ Vx0 .
Then, we havex ∈ Wx ∩ Wx0 , U c ⊂ Vx ∪ Vx0 ,(Wx ∩ Wx0 ) ∩ (Vx ∪ Vx0 ) = ∅.

APPENDIX C. SMOOTH REGULARITY AND SMOOTH PARACOMPACTNESS

109

Remark C.8. As mentioned above, we apply Lemma C.7 mainly to manifolds,
in which case Ux is a specified coordinate open neighborhood of x. If X is a topological manifold in the sense of Remark 11.19, then condition (a) is automaticallysatisfied (see [40, Section 2.1]). If X is a C ∞ -manifold, then the underlying topoe of its model vector space E is a vector space in C 0 (see Proposition
logical space E
2.6 and Remark 2.4) but need not be a topological vector space ([46, 4.16-4.26]).
e is always regular. However, we are mainly conThus, we do not know whether E
e is regular. We also
cerned with the case where E is C ∞ -regular, in which case Ee
see that if E is a Fréchet space or a Silva space, then E is regular (see [46, Theorem
4.11(1)-(2)]).

Bibliography[1] M. Aguilar and C. Prieto, Fiber bundles, unpublished note, available at:https://computo.matem.unam.mx/cprieto/phocadownloadpap/Libros/fiber%20bundles.pdf.
[2] J. Baez and A. Hoffnung, Convenient categories of smooth spaces, Transactions of theAmerican Mathematical Society 363 (2011), no. 11, 5789-5825.
[3] J. Bergner, A model category structure on the category of simplicial categories, Transactions of the American Mathematical Society 359.5 (2007) 2043-2058.
[4] F. Borceux, Handbook of Categorical Algebra: Volume 2, Categories and Structures.
Vol. 50. Cambridge University Press, (1994).
[5] G. E. Bredon, Topology and geometry, Vol. 139, Springer Science & Business Media,(1993)[6] T. Bröcker and K. Jänich, Introduction to differential topology, Cambridge University
Press, (1982).
[7] J. D. Christensen, G. Sinnamon, and E. Wu, The D-topology for diffeological spaces,
Pacific Journal of Mathematics 272 (2014), no. 1, 87-110.
[8] J. D. Christensen and E. Wu, The homotopy theory of diffeological spaces, New York J.
Math 20 (2014) 1269-1303.
[9] J.D. Christensen and E. Wu, Tangent spaces and tangent bundles for diffeological spaces,
Cahiers de Topologie et Geom trie Diff rentielle Cat gorigues 57(1) (2016), 3-50.
[10] J.D. Christensen and E. Wu, Tangent spaces of bundles and of filtered diffeological spaces,
Proceedings of the American Mathematical Society 145.5 (2017): 2255-2270.
[11] J. D. Christensen and E. Wu, Smooth classifying spaces, arXiv preprint arXiv:1709.10517(2017).
[12] J.D. Christensen and E. Wu, Diffeological vector spaces, Pacific Journal of Mathematics
303.1 (2019): 73-92.
[13] K. Ciesielski, Set theory for the working mathematician, Vol. 39. Cambridge University
Press, 1997.
[14] A. Dold, Partitions of unity in the theory of fibrations, Annals of Mathematics (1963)
223-255.
[15] A. Dold, Lectures on Algebraic Topology, Springer-Verlag, 1972.
[16] E. J. Dubuc, Concrete quasitopoi, Applications of sheaves, Springer, Berlin, Heidelberg,(1979) 239-254.
[17] W. G. Dwyer and D. M. Kan, Homotopy theory and simplicial groupoids, Indagationes
Mathematicae (Proceedings). Vol. 87. No. 4. North-Holland, 1984.
[18] W. G. Dwyer and J. Spalinski, Homotopy theories and model categories, Handbook ofalgebraic topology (1995) 73-126.
[19] J. Eells Jr, A setting for global analysis, Bulletin of the American Mathematical Society
72.5 (1966): 751-807.
[20] K. Eda and K. Kawamura, The singular homology of the Hawaiian earring, Journal of
the London Mathematical Society 62.1 (2000) 305-310.
[21] K. Eda and K. Kawamura, Homotopy and homology groups of the n-dimensional Hawaiian earring, Fundamenta Mathematicae 165.1 (2000) 17-28.
[22] S. Eilenberg and S. MacLane, Acyclic models, Amer. J. math. 75.1 (1953) 189-199.
[23] R. Engelking, General topology. (1989).
[24] A. Frölicher and A. Kriegl, Linear spaces and differentiation theory, Vol. 13, John Wileyand Sons Inc, (1988).
[25] P. Gabriel and M. Zisman, Calculus of fractions and homotopy theory, Vol. 35, Springer
Science & Business Media, (2012)
110

111

[26] O. Gil-Medrano and P. W. Michor, Geodesics on spaces of almost hermitian structures,
Israel Journal of Mathematics 88.1 (1994): 319-332.
[27] H. Glöckner, Fundamentals of direct limit Lie theory, Compositio Mathematica 141.6(2005): 1551-1577.
[28] H. Glöckner, Direct limits of infinite-dimensional Lie groups compared to direct limits inrelated categories, Journal of Functional Analysis 245.1 (2007): 19-61.
[29] H. Glöckner, Direct limits of infinite-dimensional Lie groups, Developments and trends
in infinite-dimensional Lie theory. Birkhäuser Boston, 2011. 243-280.
[30] P. G. Goerss and J. F. Jardine, Simplicial Homotopy theory, Birkhäuser, Verlag, Basel(1999).
[31] M. J. Greenberg, Lectures on algebraic topology, Vol. 9, New York, Amsterdam: WA
Benjamin, (1967).
[32] G. Hector, Géométrie et topologie des espaces difféologiques, Analysis and geometry infoliated manifolds (Santiago de Compostela, 1994) (1995) 55-80.
[33] R. E. Heisey, Manifolds modelled on R∞ or bounded weak-∗ topologies, Transactions of
the American Mathematical Society 206 (1975): 295-312.
[34] P. S. Hirschhorn, Model categories and their localizations, No. 99, American Mathematical Soc, (2009).
[35] M. Hovey, Model categories, No. 63. American Mathematical Soc., 2007.
[36] D. Husemoller, Fibre bundles, volume 20 of Graduate Texts in Mathematics, (1994).
[37] P. Iglesias-Zemmour, The moment maps in diffeology, American Mathematical Soc.,
2010.
[38] P. Iglesias-Zemmour, Diffeology, Vol. 185, American Mathematical Soc, (2013).
[39] P. Iglesias, Y. Karshon, and M. Zadka, Orbifolds as diffeologies, Transactions of theAmerican Mathematical Society 362.6 (2010) 2811-2831.
[40] H. Jarchow, Locally convex spaces, Springer Science & Business Media, 2012.
[41] P. T. Johnstone, Topos theory, Courier Corporation, 2014.
[42] H. Kihara, Minimal fibrations and the organizing theorem of simplicial homotopy theory,
Ricerche di Matematica 63 (2014), no. 1, 79-91.
[43] H. Kihara, Model category of diffeological spaces, Journal of Homotopy and Related
Structures, 14.1 (2019): 51-90.
[44] H. Kihara, Convenient categories for global analysis, in preparation.
[45] A. Kriegl and P. W. Michor, The convenient setting for real analytic mappings, Acta
Mathematica 165.1 (1990): 105-159.
[46] A. Kriegl and P. W. Michor, The convenient setting of global analysis, Vol. 53, American
Mathematical Society (1997).
[47] A. Kriegl and P. W. Michor, Smooth and continuous homotopies into convenient manifolds agree, unpublished preprint, 2002.
[48] M. V. Losik, Fréchet manifolds as diffeological spaces, Izvestiya Vysshikh Uchebnykh
Zavedenii. Matematika 5 (1992): 36-42.
[49] S. Mac Lane, Categories for the Working Mathematician, Vol. 5. Springer Science &Business Media, (1998).
[50] J. P. Magnot and J. Watts, The diffeology of Milnor’s classifying space, Topology andits Applications 232 (2017): 189-213.
[51] J. P. May, Simplicial objects in algebraic topology, University of Chicago Press, 1993[52] J. P. May, A concise course in algebraic topology, University of Chicago press, 1999
[53] J. P. May and K. Ponto, More concise algebraic topology: localization, completion, andmodel categories, University of Chicago Press, (2011).
[54] P. W. Michor, Manifolds of differentiable mappings, Shiva Math, Series, Shiva Publ.,
Kent, UK (1980).
[55] J. Milnor, Construction of universal bundles, II, Annals of Mathematics (1956): 430-436.
[56] J. Milnor, On spaces having the homotopy type of a CW-complex, Transactions of theAmerican Mathematical Society (1959) 272-280[57] C. Müller and C. Wockel, Equivalences of smooth and continuous principal bundles with
infinite-dimensional structure group, Advances in Geometry 9.4 (2009) 605-626.
[58] J. R. Munkres, Topology, Prentice Hall, 2000.
[59] R. S. Palais, Homotopy theory of infinite dimensional manifolds, Topology 5.1 (1966):
1-16.

112

HIROSHI KIHARA[60] A. Pressley and G. B. Segal, Loop groups, Clarendon Press, 1986.
[61] H. H. Schaefer and M. P. Wolff, Topological vector spaces, 2nd ed., Graduate Texts in
Mathematics, vol. 3, Springer-Verlag, New York, 1999.
[62] G. Segal, Classifying spaces and spectral sequences, Publications Mathématiques del’IHÉS 34 (1968): 105-112.
[63] P. Selick, Introduction to homotopy theory, Vol. 9, American Mathematical Soc, 2008.
[64] E. H. Spanier, Algebraic topology, MacGraw-Hill, New York (1966).
[65] A. Stacey, Comparative smootheology, Theory and Applications of Categories 25 (2011),
no. 4, 64-117.
[66] N. Steenrod, The topology of fibre bundles, Vol. 14. Princeton university press, 1999.
[67] T. T. Dieck, Partitions of unity in homotopy theory, Composito Math 23 (1971): 159-167.
[68] M. Valdivia, On the completion of a bornological space, Archiv der Mathematik 29.1
(1977): 608-613.
[69] S. Willard, General Topology. Courier Corporation, 2012.
[70] C. Wockel, Lie group structures on symmetry groups of principal bundles, Journal of
Functional Analysis 251.1 (2007): 254-288.
[71] C. Wockel, A generalization of Steenrod’s approximation theorem, Arch. Math.(Brno)45.2 (2009) 95-104.
[72] E. Wu, Homological algebra for diffeological vector spaces, Homology, Homotopy andApplications 17.1 (2015): 339-376.
[73] K. Yoshinaga, On a locally convex space introduced by JSE Silva, Journal of Scienceof the Hiroshima University. Series A (Mathematics, Physics, Chemistry) 21.2 (1957):
89-98.