Connected Lie Groupoids are Internally Connected and Integral Complete in Synthetic Differential Geometry
We extend some fundamental definitions and constructions in the established generalisation of Lie theory involving Lie groupoids by reformulating them in terms of groupoids internal to a well-adapted model of synthetic differential geometry. In particular we define internal counterparts of the defin...
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irk-123456789-1485572019-02-19T01:25:56Z Connected Lie Groupoids are Internally Connected and Integral Complete in Synthetic Differential Geometry Burke, M. We extend some fundamental definitions and constructions in the established generalisation of Lie theory involving Lie groupoids by reformulating them in terms of groupoids internal to a well-adapted model of synthetic differential geometry. In particular we define internal counterparts of the definitions of source path and source simply connected groupoid and the integration of A-paths. The main results of this paper show that if a classical Hausdorff Lie groupoid satisfies one of the classical connectedness conditions it also satisfies its internal counterpart. 2017 Article Connected Lie Groupoids are Internally Connected and Integral Complete in Synthetic Differential Geometry / M. Burke // Symmetry, Integrability and Geometry: Methods and Applications. — 2017. — Т. 13. — Бібліогр.: 27 назв. — англ. 1815-0659 2010 Mathematics Subject Classification: 22E60; 22E65; 03F55; 18B25; 18B40 DOI:10.3842/SIGMA.2017.007 http://dspace.nbuv.gov.ua/handle/123456789/148557 en Symmetry, Integrability and Geometry: Methods and Applications Інститут математики НАН України |
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We extend some fundamental definitions and constructions in the established generalisation of Lie theory involving Lie groupoids by reformulating them in terms of groupoids internal to a well-adapted model of synthetic differential geometry. In particular we define internal counterparts of the definitions of source path and source simply connected groupoid and the integration of A-paths. The main results of this paper show that if a classical Hausdorff Lie groupoid satisfies one of the classical connectedness conditions it also satisfies its internal counterpart. |
| format |
Article |
| author |
Burke, M. |
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Burke, M. Connected Lie Groupoids are Internally Connected and Integral Complete in Synthetic Differential Geometry Symmetry, Integrability and Geometry: Methods and Applications |
| author_facet |
Burke, M. |
| author_sort |
Burke, M. |
| title |
Connected Lie Groupoids are Internally Connected and Integral Complete in Synthetic Differential Geometry |
| title_short |
Connected Lie Groupoids are Internally Connected and Integral Complete in Synthetic Differential Geometry |
| title_full |
Connected Lie Groupoids are Internally Connected and Integral Complete in Synthetic Differential Geometry |
| title_fullStr |
Connected Lie Groupoids are Internally Connected and Integral Complete in Synthetic Differential Geometry |
| title_full_unstemmed |
Connected Lie Groupoids are Internally Connected and Integral Complete in Synthetic Differential Geometry |
| title_sort |
connected lie groupoids are internally connected and integral complete in synthetic differential geometry |
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Інститут математики НАН України |
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2017 |
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http://dspace.nbuv.gov.ua/handle/123456789/148557 |
| citation_txt |
Connected Lie Groupoids are Internally Connected and Integral Complete in Synthetic Differential Geometry / M. Burke // Symmetry, Integrability and Geometry: Methods and Applications. — 2017. — Т. 13. — Бібліогр.: 27 назв. — англ. |
| series |
Symmetry, Integrability and Geometry: Methods and Applications |
| work_keys_str_mv |
AT burkem connectedliegroupoidsareinternallyconnectedandintegralcompleteinsyntheticdifferentialgeometry |
| first_indexed |
2025-07-12T18:55:22Z |
| last_indexed |
2025-07-12T18:55:22Z |
| _version_ |
1837468515594928128 |
| fulltext |
Symmetry, Integrability and Geometry: Methods and Applications SIGMA 13 (2017), 007, 25 pages
Connected Lie Groupoids are Internally Connected
and Integral Complete in Synthetic Differential
Geometry
Matthew BURKE
4 River Court, Ferry Lane, Cambridge CB4 1NU, UK
E-mail: matthew.burke@cantab.net
URL: http://www.mwpb.uk
Received June 29, 2016, in final form January 13, 2017; Published online January 24, 2017
https://doi.org/10.3842/SIGMA.2017.007
Abstract. We extend some fundamental definitions and constructions in the established
generalisation of Lie theory involving Lie groupoids by reformulating them in terms of
groupoids internal to a well-adapted model of synthetic differential geometry. In partic-
ular we define internal counterparts of the definitions of source path and source simply
connected groupoid and the integration of A-paths. The main results of this paper show
that if a classical Hausdorff Lie groupoid satisfies one of the classical connectedness condi-
tions it also satisfies its internal counterpart.
Key words: Lie theory; Lie groupoid; Lie algebroid; category theory; synthetic differential
geometry; intuitionistic logic
2010 Mathematics Subject Classification: 22E60; 22E65; 03F55; 18B25; 18B40
1 Introduction
In classical Lie theory we use a formal group law to represent the analytic approximation of
a Lie group. Recall that an n-dimensional formal group law F is an n-tuple of power series in
the variables X1, . . . , Xn; Y1, . . . , Yn with coefficients in R such that the equalities
F
(
~X,~0
)
= ~X, F
(
~0, ~Y
)
= ~Y and F
(
F
(
~X, ~Y
)
, ~Z
)
= F
(
~X,F
(
~Y , ~Z
))
hold. In fact there is an equivalence of categories
FGLaw LieGrpsc
(−)int
⊥
(−)∞
(1.1)
between the category FGLaw of formal group laws and LieGrpsc of simply connected Lie groups.
The functor (−)∞ is obtained by considering the Taylor expansion of the multiplication of the
Lie group close to the identity element. For more details see Theorem 3 of Section V.6 and
Theorem 2 of Section V.8 of Part 2 in [26]. The functor (−)int therefore extends local data to
global data and involves finding solutions to smooth time-dependent left-invariant vector fields.
In [5] we generalise the underlying adjunction of (1.1) to an adjunction
Cat∞(E) Cat(E) Catint(E)⊥
i
(−)int
(−)∞
⊥
j
(1.2)
mailto:matthew.burke@cantab.net
http://www.mwpb.uk
https://doi.org/10.3842/SIGMA.2017.007
2 M. Burke
between two subcategories of the category of internal categories in a topos E . When considered
together, this paper and [5] not only provide a more concise exposition of the thesis [4] but also
contain several stronger results. The main improvement over [4] contained in this paper is the
extension of the results about internal connectedness conditions from groups to groupoids.
In [5] we prove the result analogous to Lie’s second theorem in this context: when we apply
the appropriate connectedness conditions (described in Section 1.3) the functor (−)∞j is full
and faithful. In this paper we justify the work in [5] by describing the relationship between (1.1)
and (1.2) in the case that E is a well-adapted model of synthetic differential geometry (see
Section 1.1). This is carried out in Section 4.2 where we show that when we restrict Cat(E)
to the full subcategory on the classical Lie groups, the functor (−)∞ coincides with the formal
group law construction described in the Introduction of [14].
In addition we relate the adjunction (1.2) to the established generalisation of Lie theory
involving Lie algebroids and Lie groupoids. (See for instance [21].) A Lie groupoid is a groupoid
in the category of smooth paracompact manifolds such that the source and target maps are
submersions. A Lie algebroid is a vector bundle A→M together with a bundle homomorphism
ρ : A→ TM such that the space of sections Γ(A) is a Lie algebra satisfying the following Leibniz
law: for all X,Y ∈ Γ(A) and f ∈ C∞(M) the equality
[X, fY ] = ρ(X)(f) · Y + f [X,Y ]
holds. In the theory of Lie groupoids and Lie algebroids we have have a functor
LieAlgd LieGpd,
(−)∞
which is full and faithful but not essentially surjective. Any Lie algebroid integrates to a topo-
logical groupoid, its Weinstein groupoid [6], but there can be obstructions to putting a smooth
Hausdorff structure on it. For instance see [8] for a Lie algebroid whose Weinstein groupoid is
a smooth but non-Hausdorff Lie groupoid and [1] for a Lie algebroid whose Weinstein groupoid
is non-smooth. Therefore when dealing with integrability (for instance in [6]) the category of
smooth manifolds is enlarged to include non-Hausdorff manifolds. Furthermore in [27] Tseng
and Zhu show that the category of differentiable stacks contains all Weinstein groupoids whilst
still retaining the concept of tangent vectors. Another approach, pursued in [5], is to use the
theory of synthetic differential geometry where the Weinstein groupoid construction is always
possible.
In the process of reformulating the theory of Lie groupoids and Lie algebroids in [5] it is
necessary to use internal versions of certain conditions describing connectedness and solutions
to a specific type of vector field. In this paper we will justify these assumptions by showing
that all classical Hausdorff Lie groupoids satisfy these stronger conditions. Since the Weinstein
groupoid construction is always possible in E the assumption that our groupoids are Hausdorff
does not affect the part of the theory dealing with integrability, only the extent to which the
conditions involving completeness and solutions to vector fields generalise the classical ones. So
unless otherwise stated all Lie groupoids in this paper will have Hausdorff arrow space.
1.1 Synthetic differential geometry
In synthetic differential geometry we replace the category Man of smooth paracompact Hausdorff
manifolds with a certain kind of Grothendieck topos E called a well-adapted model of synthetic
differential geometry. In this section we sketch the axioms of a well-adapted model of synthetic
differential geometry and recall a few key properties.
Firstly there is a full and faithful embedding ι : Man � E and therefore a ring R = ιR in E .
In addition we have the objects
Dk =
{
x ∈ R : xk+1 = 0
}
,
Connected Lie Groupoids are Internally Connected 3
which are not terminal. In fact the fundamental Kock–Lawvere axiom holds: the arrow α : Rk+1
→ RDk defined by
(a0, a1, . . . , ak) 7→
(
d 7→ a0 + a1d+ · · ·+ akd
k
)
is an isomorphism. A set of non-classical objects that will be useful in the sequel are the Weil
spectra which are of the following form:
Spec(Weil) =
{
(x1, . . . , xn) :
n∧
i=1
(
xkii = 0
)
∧
m∧
j=1
(pj = 0)
}
,
where n,m ∈ N≥0, ki ∈ N>0 and the pj are polynomials in the xi. We write D∞ =
⋃
iDi and
D = D1.
The following is Definition 3.1 in Part III of [17].
Definition 1.1. A pair of maps fi : Mi → N (i = 1, 2) in Man with common codomain are said
to be transversal to each other iff for each pair of points x1 ∈M1, x2 ∈M2 with f1(x1) = f2(x2)
(= y say), the images of (dfi)xi (i = 1, 2) jointly span TyN as a vector space.
Definition 1.2. A topos E together with a full and faithful embedding ι : Man→ E is a well-
adapted model of synthetic differential geometry iff
• the functor ι preserves transversal pullbacks,
• the functor ι preserves the terminal object,
• the functor ι sends arbitrary open covers in Man to jointly epimorphic families in E ,
• the internal ring ι(R) satisfies the Kock–Lawvere axiom,
• for all Weil spectra DW the functor (−)DW : E → E preserves all colimits.
Remark 1.3. Since ι : Man→ E preserves transversal pullbacks it determines an embedding of
LieGpdH into Grpd(E). Here we have written LieGpdH for the subcategory of LieGpd consisting
of the groupoids that have Hausdorff arrow space.
Remark 1.4. If M is a smooth manifold then we will often abuse notation by writing M to
denote the object ι(M) in the well-adapted model.
Using the Kock–Lawvere axiom we can show that ι(TM) ∼= MD as vector bundles over M
and that the Lie bracket corresponds to an infinitesimal commutator. For more detail see [17].
Furthermore in Section 4.2 we show that formal group laws correspond to groups of the form
(Dn
∞, µ).
1.2 Smooth affine schemes and the Dubuc topos
In Section 2.7 we will need a more detailed description of the coverage that generates the topos E .
Hence in that section we will work in a well-adapted model of synthetic differential geometry
called the Dubuc topos. In this section we briefly sketch the essential features of the Dubuc
topos and refer to [10] for more details. Note that this means that the results of Section 2.7 hold
for all the well-adapted models generated by a site contained in the Dubuc site. For instance by
referring to Appendix 2 of [23] we see that our results hold for the Cahiers topos (see [9]) and
the classifying topos of local Archimedean C∞-rings (see Appendix 2 of [23]).
In addition in Section 4.5 it will be convenient to know that every representable object is
a subobject of Rn for some n ∈ N. Therefore in that section we will work in any well-adapted
model E that is generated by a subcanonical site whose underlying category is a full subcategory
of the category of affine C∞-schemes as defined below. In particular this means that the results
of Section 4.5 hold for the Dubuc topos.
4 M. Burke
Definition 1.5. The category C of affine C∞-schemes has as objects pairs [n, I] where n ∈ N
and I is a finitely generated ideal of C∞(Rn,R). The arrows
[n, I]
f−→ [m,J ]
are equivalence classes of smooth functions f ∈ C∞(Rn,Rm) such that
• we identify f ∼ g iff f ≡ g (mod I),
• for all j ∈ J we have jf ∼ 0.
Now we define a slight generalisation of the notion of open set. Using these open sets we
define the Dubuc coverage by using inverse images of smooth functions.
Definition 1.6. The open subobject U of [n, I] defined by χU : Rn → R is the subobject
[n+ 1, (I, χU ·Xn+1 − 1)]
proj−−→ [n, I],
which intuitively corresponds to the subset χ−1
U (R−{0})∩ [n, I]. The Dubuc coverage J consists
of the families of open subobjects
(Ui � [n, I])i∈I
that are jointly surjective.
The site that we use to generate the Dubuc topos is the full subcategory of the category of
affine C∞-schemes on the germ-determined schemes which are defined as follows:
Definition 1.7. For a smooth function f : Rn → R we write gx(f) for the equivalence class of
functions that is the germ of f at x ∈ Rn and Gx for the ring of germs of smooth functions at x.
For an ideal of smooth functions I / C∞(Rn,R) we write Z(I) for the zero-set of I and
gx(I) =
{
Σk
j=1rjgx(φj) : (rj ∈ Gx) ∧ (φj ∈ I)
}
for the ideal generated by germs of elements of I. Then a scheme [n, I] is germ-determined iff
∀ g ∈ C∞
(
Rn,R
)
,
(
∀x ∈ Z(I), gx(g) ∈ gx(I)
)
=⇒ g ∈ I.
We denote by Cgerm ⊂ C the full subcategory on the objects that are germ-determined. The
Dubuc topos is the Grothendieck topos generated by taking sheaves on the site (Cgerm,J )
where J is the Dubuc coverage.
1.3 Internal connectedness
In classical Lie theory we study how much of the data in a Lie groupoid can be recovered from
the subset of this data that is infinitely close to the identity arrows of the Lie groupoid. Since
global features such as connectedness cannot be captured by the infinitesimal arrows we need
to restrict our attention to Lie groupoids that are source path and source simply connected.
We say that a Lie groupoid G with arrow space G and object space M is source path/source
simply connected iff all of its source fibres are path/simply connected. Let I be the pair groupoid
on the unit interval I that has precisely one invertible arrow between each pair of elements of I.
Then it is easy to see that the global sections of the object GI = Grpd(E)(I,G) in E are equivalent
to arrows I → G in E that are source constant and start at an identity element of G. Therefore G
is source path connected iff
Γ
(
GI) Γ(GιI )−−−−→ Γ
(
G∂I)
Connected Lie Groupoids are Internally Connected 5
is an epimorphism in Set. We have written Γ for the global sections functor and ιI : ∂I→ I for
the full subcategory that is the pair groupoid on the boundary of I. In this case ιI is simply the
inclusion of the long arrow (0, 1) : 2→ I. Similarly G is source simply connected iff it is source
path connected and
Γ
(
GI2) Γ
(
GιI2
)
−−−−−→ Γ
(
G∂I2)
is an epimorphism in Set. We have written ιI2 : ∂I2 → I2 for the full subcategory that is the pair
groupoid on the boundary of I2.
When we work with arbitrary groupoids in a well-adapted model E of synthetic differential
geometry it is necessary to work with epimorphisms between objects of E than between their
sets of global sections. Hence we make the following definitions:
Definition 1.8. A groupoid G in E is E-path connected iff
GI GιI−−→ G∂I
is an epimorphism in E . A groupoid G in E is E-simply connected iff it is E-path connected and
GI2 GιI2−−−→ G∂I2
is an epimorphism in E .
This means that for an arbitrary groupoid in E being E-connected is a stronger condition to
impose than being source connected. In Section 2 we show that a Hausdorff Lie groupoid is
source path/simply connected iff it is E-path/E-simply connected.
1.4 The jet part
The linear approximation of a Lie groupoid has the structure of a Lie algebroid (see for instance
Section 3.5 of [21]). By contrast in [5] we define an analytic approximation of an arbitrary
groupoid in E . This new structure approximates a Lie groupoid in an analogous way to how
a formal group law approximates a Lie group. In this section we briefly sketch the main features
of this analytic approximation.
Using the infinitesimal objects of synthetic differential geometry we can define an infinites-
imal neighbour relation ∼. Intuitively speaking a ∼ b expresses that b is contained in an
infinitesimal jet based at a. For more details see Section 3.1. Using this neighbour relation we
can define the jet part G∞ of a groupoid G with object space G and arrow space M that consists
of all the arrows that are infinitely close to an identity arrow. In [5] we show that this jet part
is closed under composition and so defines a subcategory
G∞
ι∞G−−→ G,
which is however not in general a groupoid.
1.4.1 Symmetry of the neighbour relation
It turns out that the neighbour relation ∼ is not symmetric for all objects of E . In fact it is not
symmetric on the object D of all nilsquares in the real line. In [5] we show that this implies that
the jet part ∇D∞ of the pair groupoid ∇D on D is not a groupoid (although it is a category).
Fortunately in [5] we also show that the symmetry of ∼ in the arrow space of a groupoid G is
not only a necessary condition but also a sufficient condition to ensure that the jet part G∞
of G is a groupoid. We justify this assumption in Section 3.3 by showing that the neighbour
relation is symmetric for all classical Hausdorff Lie groupoids.
6 M. Burke
1.4.2 Path connectedness of the jet part
When we prove Lie’s second theorem in [5] there is an additional condition that is required
which is not visible in the classical theory. In addition to requiring that a groupoid satisfies
certain connectedness and completeness conditions we need to assume that its jet part is E-path
connected. We justify this assumption in Section 3.6 by showing that the jet part of every
classical Hausdorff Lie groupoid is E-path connected.
1.5 Integral completeness
To construct global data from local data in classical Lie theory we use the fact that all smooth
vector fields admit a unique local solution when we fix an initial vector. Unfortunately when
we replace the category Man with a well-adapted model E of synthetic differential geometry we
can no longer use this result. In this section we identify a class of groupoids for which we can
construct global data from local data. It turns out that we do not need to assert the existence
of all solutions to smooth vector fields but instead a weaker condition suffices. In [6] we see
that the crucial lifting property required to prove Lie’s second theorem involves the integration
of a certain type of path in a Lie algebroid (called A-paths) to a certain type of path in a Lie
groupoid (called G-paths).
Let I be the pair groupoid on the unit interval I and G be a Lie groupoid with arrow space G
and object space M . In Section 4.3 we show that A-paths correspond to global sections of the
object GI∞ in E and G-paths correspond to global sections of the object GI. Hence we restrict
attention to groupoids that are integral complete in the following sense:
Definition 1.9. A groupoid G in E is integral complete iff
GI Gι
∞
I−−−→ GI∞
is an isomorphism in Gpd(E).
This assumption is a crucial one in the proof of Lie’s second theorem presented in [5] and
so in Section 4 we justify it by proving that all classical Hausdorff Lie groupoids are integral
complete.
2 Path and simply connectedness
In this section we show that for all Hausdorff Lie groupoids G with arrow space G and object
space M the classical source path and source simply connectedness conditions coincide with
their internal counterparts. (Please see Section 1.3 for the relevant definitions.) In other words,
we show that if G is source path connected then G is E-path connected and if further G is
source simply connected then G is E-simply connected. To do this we will need an explicit
description of the coverage that generates the well-adapted model E . Hence for this section
we will assume that E is the Dubuc topos which is generated by the Dubuc site as defined in
Section 1.2. Note that this means that our results hold for all the well-adapted models generated
by a site contained in the Dubuc site. For instance by referring to Appendix 2 of [23] we see that
our results hold for the Cahiers topos (see [9]) and the classifying topos of local Archimedean
C∞-rings (see Appendix 2 of [23]).
We deduce both the path connected and simply connected results from the following stronger
result.
Notation 2.1. Let B be a compact and contractible subset of a Euclidean space that is a zero
set of an ideal of smooth functions I:
B = [n, I] =
{
~x ∈ Rn : ∀φ ∈ I, φ(~x) = 0
}
,
Connected Lie Groupoids are Internally Connected 7
which means that we can view B as a representable object in the Dubuc topos as well as a subset
of Euclidean space.
Let ∂B denote the boundary of B and ∇B and ∇∂B be the pair groupoids on B and ∂B
respectively. (Recall that the pair groupoid has precisely one invertible arrow between any pair
of objects.) There is a natural inclusion ιB : ∇∂B → ∇B.
Notation 2.2. We write r ∈X R to denote that r is an arrow X → R in E and say that r is
a generalised element of R at stage of definition X.
We prove that if every global element f ∈1 G∇∂B has a filler F ∈1 G∇B (i.e., GιBF = f)
then the arrow
GιB : G∇B → G∇∂B
is an epimorphism in E . Note that being E-path connected is the case when B is the unit
interval I and being E-simply connected is the conjunction of the cases B = I and B = I2.
Our general strategy will be to split the tangent bundle using the submersion s and then show
that various constructions involving Riemannian exponential maps can be forced to respect this
splitting. Once this is done we can work in just one source fibre where the result is substantially
easier.
However first we need to consider the interrelationships between various kinds of open subset
and subobject possible in the context of a well-adapted model of synthetic differential geometry.
2.1 Open subobjects of function spaces
Our aim is to show that a certain arrow between function spaces is an epimorphism. As is the
case for all Grothendieck toposes, the epimorphisms in the Dubuc topos are characterised in
terms of the coverage that is used to generate the topos. However the natural and convenient
notion of open subset for a space of smooth functions is the smooth compact-open topology.
In order to mediate between these two notions of open subobject/subset we introduce another
type of open subobject due to Penon. First we recall that every Penon open subobject of
a representable object is a Dubuc open subobject. Then we recall that any smooth compact-
open subset of the set of global sections of a function space induces a Penon open subobject of
the function space.
2.1.1 Penon open subobjects
In this section we will briefly sketch some of the theory of topological structures in synthetic
differential geometry and refer to [3] and [25] for more comprehensive accounts. Following Penon
in [24] we say that an element r of the line object R in the Dubuc topos is infinitesimal iff
¬¬(r = 0)
holds in the internal logic of the Dubuc topos. Since the line object contains nilpotent elements
it is not a field. However it is easy to see that all of these nilpotents are infinitesimal as defined
above and in fact Theorem 10.1 in [17] tells us that the line object R is a field of fractions, which
is to say that for all elements r ∈ R the proposition
¬(r = 0) ⇐⇒ (r is invertible) (2.1)
holds. We note that in the context of classical logic being a field of fractions implies that
every element is either zero or invertible but this implication does not hold for intuitionistic
8 M. Burke
logic. Using the correspondence in (2.1) we can deduce that the infinitesimals and the invertible
elements of the line object are separated in the following sense: for all r, s ∈ R the proposition
¬(r = 0) ∧ ¬¬(s = 0) =⇒ ¬(r = s)
holds. The following definition is Definition 1.5 in [11].
Definition 2.3. A subobject U ⊂ X in E is Penon open iff the proposition
∀u ∈ U, ∀x ∈ X, (x ∈ U) ∨ (¬(u = x))
holds in the internal logic of E .
Remark 2.4. If U is Penon open then for all u ∈ U then there is an inclusion
{x : ¬¬(x = u)} ⊂ U.
Example 2.5. The subobject
{r ∈ R : ¬(r = 0)}
of R is a Penon open subset.
In fact we can give a characterisation of the Penon open subsets of representable objects
in terms of classical open subsets. Recall from [10] and Lemma 1.3 in III.1 of [23] that the
site generating the Dubuc topos is subcanonical and that by construction the image of any open
subset under the full and faithful embedding of the category of smooth manifolds into the Dubuc
topos is a Dubuc open subobject. The following is Corollary 8 in [12].
Proposition 2.6. A subobject U of a representable object yX in the Dubuc topos is Penon open
iff it is of the form U = ιV ∩ yX for some open subset V ⊂ Rn.
Corollary 2.7. If X is a Penon open subobject of a representable then it is Dubuc open.
Finally we record the result that arbitrary Penon open subobjects are stable under pullback.
Corollary 2.8. Let f : W → X be an arrow and U be a Penon open subobject of X. Then
f∗U = {w ∈W : fw ∈ U}
is a Penon open subobject of W .
Proof. The hypothesis that U is a Penon open subobject of X implies that for all w ∈W and
v ∈ f∗U the proposition
(fw ∈ U) ∨ (¬(fw = fv))
holds. But by definition fw ∈ U iff w ∈ f∗U and it is immediate that ¬(fw = fv) implies that
¬(w = v). �
Connected Lie Groupoids are Internally Connected 9
2.1.2 Smooth compact-open subsets
Recall that for any topos E the global sections functor Γ restricts to a functor from the poset of
subobjects of some function space Y X to the poset of subsets of Γ(Y X). In fact when E is the
Dubuc topos Γ has a right adjoint
SubE
(
Y X
)
Sub
(
Γ
(
Y X
))Γ
⊥
E
for which the unit is an isomorphism (see Lemma 1.5 in [2]). Therefore it is natural to ask
whether we can characterise the subsets U of Γ(Y X) for which EU is a Penon open subobject.
Let M , N be smooth manifolds and f : M → N a smooth function. Let (φ, V ) and (ψ,W ) be
charts for M and N respectively and let K be a compact subset of V such that f(K) ⊂W . Then
following Section 2.1 in [15] we define the weak subbasic neighbourhood Uφ,ψf,ε,K of C∞(M,N) to
be the set of all smooth functions g : M → N such that g(K) ⊂ V and such that for all integers k
the inequality∣∣Dk
(
ψfφ−1
)
(m)−Dk
(
ψgφ−1
)
(m)
∣∣ < ε
holds for all m ∈ φ(K). We call the topology generated by the weak subbasic neighbourhoods
the smooth compact-open topology. The following is Proposition 1.6 of [2].
Proposition 2.9. If U is a smooth compact-open subset of Γ(Y X) then EU is a Penon open
subobject.
Remark 2.10. If f ∈ U then f : 1→ Y X factors through EU � Y X .
2.2 Splitting the tangent bundle
In this section we use the source submersion s : G → M of a Lie groupoid G ⇒ M to split the
tangent bundle TG→ G into horizontal and vertical components. We then confirm that when
we pullback the tangent bundle to a contractible base space this splitting is maintained in the
trivial bundle that results.
Definition 2.11. If pH : H → G and pV : V → G are vector bundles then the direct sum H⊕V
is defined as the pullback vector bundle
H ⊕ V H × V
G G×G,
pG
∆G
pH×pV
∆
where ∆ is the diagonal.
Notation 2.12. Let G be a Lie groupoid with arrows space G, object space M , source map s
and a Riemannian metric σG on G. Let B be a contractible subset of a Euclidean space Rn that is
the zero set of an ideal of smooth functions. Let pV : V = ker(sD)→ G and pH : H = (ker(sD))⊥
→ G be vector bundles where the orthogonal complement is defined using the metric σ.
First we note that the squares
B ×Rm H ⊕ V
B G,
τG
π0 pG
F
B ×Rk H
B G
τH
π0 pH
F
and
B ×Rn V
B G
τV
π0 pV
F
10 M. Burke
are pullbacks for some natural numbers m, k and n because B is contractible. By construction
TG ∼= H ⊕ V as vector bundles.
Lemma 2.13. The square
B ×Rk ×Rn H ⊕ V
B ×Rk H
τG
π0,1 π0◦∆G
τH
commutes.
Proof. There is a unique ψ : B ×Rk ×Rn → H ⊕ V making
B ×Rk ×Rn (B ×Rk)× (B ×Rn)
H ⊕ V H × V
G G×G
B B ×B
π0
ψ
(b,v,b,u)
π0×π0
τH×τV
p
∆G
pH×pV
∆
F
∆
F×F
into a commutative cube because the centre square is a pullback. Furthermore the right and
outer squares are easily seen to be pullbacks. This means that the left square is a pullback and
ψ = τG. Now the result follows from the fact that
B ×Rk ×Rn (B ×Rk)× (B ×Rn) B ×Rk
H ⊕ V H × V H
(b,v,b,u)
τG τH×τV
π0
τH
∆G π0
commutes. �
Corollary 2.14. There is an isomorphism α making
B ×Rk ×Rn GD
B ×Rk MD.
τG
π0,1 sD
α◦τH
Proof. Since sD is an epimorphism there is an isomorphism α making
H ⊕ V MD
H
sD
π0◦∆G
α
commute. Then result follows immediately from Lemma 2.13. �
Connected Lie Groupoids are Internally Connected 11
2.3 Riemannian submersions
In order to transfer fillers between neighbouring source fibres we need to know how to transport
them in parallel to the source fibres. To do this we will use the exponential map on the arrow
space G induced by a Riemannian metric on G. However it is not in general true that for
arbitrary Riemannian metrics ηG and ηM on G and M respectively that s maps geodesics with
respect to ηG to geodesics with respect to ηM .
We now recall a little of the theory of Riemannian submersions which will allow us to construct
Riemannian metrics σG on G and σM on M such that s maps geodesics with respect to σG to
geodesics with respect to σM .
Notation 2.15. Let G and M be smooth manifolds with Riemannian metrics ηG and ηM
respectively. Let s : G → M be a submersion. We write ker(s) = V for the sub-bundle of the
tangent bundle GD that is parallel to the s-fibres and H = (ker(s))⊥ for the bundle orthogonal
to V with respect to the Riemannian metric ηG. Let UG and UM denote the domains of the
exponential maps associated to ηG and ηM respectively.
The next definition is part of the Definition 26.9 in [22].
Definition 2.16. The submersion s is a Riemannian submersion iff(
sD|H
)
g
: Hg →
(
MD
)
s(g)
is an isometric isomorphism.
The next result is Lemma 2.1.1 in [7].
Lemma 2.17. If s : G → M is a submersion then we can choose Riemannian metrics σG
and σM on G and M respectively that make s a Riemannian submersion.
Proof. To begin with choose arbitrary Riemannian metrics ηG and ηM onG andM respectively.
Use ηG to decompose GD = (ker(s))⊥ ⊕ ker(s) = H ⊕ V . Now we can define an alternative
positive definite inner product σH on H as the pullback of ηM along the isometry (sD|H). Also
we can restrict the Riemannian metric ηG to a positive definite inner products σV on V . Then
we define a new Riemannian metric σG on G by declaring all vectors in H to be orthogonal to
all vectors in V . By construction s is a Riemannian submersion with respect to σG and ηM . �
Lemma 2.18. If s : G→ M is a Riemannian submersion and c : [0, 1]→ G is a geodesic in G
such that c′(0) ∈ H then s ◦ c is a geodesic in M . Furthermore if a ∈ [0, 1] then c′(a) ∈ H.
Proof. This is Corollary 26.12 in [22]. �
Corollary 2.19. If s : G → M is a Riemannian submersion then there exist open subsets
WM ⊂ UM and WG ⊂ UG such that
WG G
WM M
expG
sD|
WG s
expM
commutes.
Proof. Immediate from Lemma 2.18. See also Proposition 5.9 in [7]. �
12 M. Burke
2.4 Constructing a tubular extension
In this section we construct a tubular extension B × Ck × Cn → G for every smooth map
F : B → G where G is the arrow space of a Hausdorff Lie groupoid. In the next two sections
we work within this tubular extension to construct the Penon open subobject that we need. We
also show that this extension commutes in the appropriate way with the source map.
Notation 2.20. Let Cn denote the open unit hypercube in Rn. Let B be a contractible and
compact subset of Euclidean space that is the zero set of an ideal of smooth functions.
Lemma 2.21. If W
G
= WG ∩ (sD)−1(WM ) then there are open inclusions νG : Ck → Rk and
νM : Cn → Rn such that νG(~0) = ~0 and νM (~0) = ~0 and maps ιG : B × Ck × Cn → W
G
and
ιM : B × Ck →WM such that
B × Ck × Cn W
G
B ×Rk ×Rn GD
ιG
B×νM×νG
τG
and
B × Ck WM
B ×Rk MD
ιM
B×νM
α◦τH
commute.
Proof. By construction for each b ∈ B the arrow τG(b,−,−) : Rk×Rn → GDb is an isomorphism.
Therefore Xb = τG(b,−,−)−1(W
G
b ) specifies a collection of open sets containing ~0 in Rk+n that
vary smoothly with B. Since B is compact we can find an open ball around ~0 contained in each
of the Xb. Now the existence of ιG and νG follows easily. The existence of ιM and νM follows
similarly. �
Lemma 2.22. If F : B → G is a smooth map that is s-constant and starts at an identity arrow
then there exist smooth maps ξG : B × Ck × Cn → G and ξM : B × Ck →M such that
• both ξG(b, 0, 0) = F (b) and ξN (b, 0, 0) = F (b),
• for all b ∈ B both ξG(b,−,−) and ξM (b,−,−) are open inclusions,
• the diagram
B × Ck × Cn G
B × Ck M
π0,1
ξG
s
ξM
commutes.
Proof. If W
G
= WG ∩ (sD)−1(WM ) then in the cube
B × Cn × Ck W
G
B ×Rk ×Rn GD
B ×Rk MD
B × Ck WM
ιG
π0,1 sD|
W
Gπ0,1
τG
sD
α◦τH
ιM
Connected Lie Groupoids are Internally Connected 13
the centre square commutes by Corollary 2.14, the upper and lower squares commute by
Lemma 2.21 and the left and right squares commute by construction. Therefore the outer square
commutes because WM � MD is a monomorphism. The result now follows from pasting the
square shown to commute in Corollary 2.19 onto the right of the above square; the maps we
require are ξG = expG ◦ιG and ξM = expM ◦ιM . �
2.5 A subobject of the tubular extension admitting fillers
In the previous section we constructed a tubular extension ξGF : B × Ck × Cn → G for every
F ∈ G∇B. In this section we construct a subobject of G∇∂B from this tubular extension such
that every element of this subobject admits a filler. In the next section we find a Penon open
subobject contained in this subobject.
Notation 2.23. Let B be a subset of Euclidean space that is the zero set of an ideal of smooth
functions. Let f ∈ G∇∂B have a filler F ∈ G∇B. We write ξGF for the tubular extension
constructed in Section 2.4.
Remark 2.24. For all ~x0 ∈ Ck the map ∂B → G defined by b 7→ ξGF (b, ~x0,~0) has filler B → G
defined by b 7→ ξGF (b, ~x0,~0).
Definition 2.25. The subobject Tf � G∇∂B consists of all χ ∈ G∇∂B such that
∀ b ∈ B, χ(b) ∈ ξGF
(
b, Ck, Cn
)
or equivalently Tf is the subobject of G∇∂B such that
∀ b ∈ B, ∃ ~x0 ∈ Ck, ∃hχ ∈
(
Cn
)∂B
, χ(b) = ξGF
(
b, ~x0, hχ(b)
)
,
because χ is source constant.
Remark 2.26. By construction F (b) = ξGF (b,~0,~0). Restricting to ∂B gives that f(b) =
ξGF (b,~0,~0) and hence f ∈ Tf .
Lemma 2.27. If χ ∈ Tf then χ has a filler X ∈ G∇B.
Proof. If χ(b) = ξGF (b, ~x0, hχ(b)) then there is an homotopy from χ to (b 7→ ξGF (b, ~x0,~0)) defined
by
I × ∂B → G,
(a, b) 7→ ξGf (b, ~x0, (1− a)hχ(b)) ,
and composing this homotopy with the filler (b 7→ ξGF (b, ~x0,~0)) is a filler for χ. �
2.6 A compact-open set inside a tubular extension
In this section we identify a compact-open set that is contained in space of global sections
of G∇∂B that is contained in the subobject Tf constructed in Section 2.5. Once we have done
this we can deduce using Proposition 2.9 the existence of a Penon open subobject Vf of G∇∂B
such that all maps in Vf have fillers.
Notation 2.28. Let B be a subset of Euclidean space that is the zero set of an ideal of smooth
functions. Let f ∈ G∇∂B have a filler F ∈ G∇B. We write ξGF for the tubular extension
constructed in Section 2.4. Let Dn � Cn be the inclusion of the ball of radius 1
2 centred at the
origin. Let En � Dn be the inclusion of the ball of radius 1
4 centred at the origin,
14 M. Burke
Definition 2.29. The compact-open subset Wf of Γ(G∇∂B) is defined as follows. Let Ub =
f−1ξGF (b, Ek, En). Now (Ub)b∈∂B covers ∂B because b ∈ Ub. Since ∂B is compact we can choose
b1, . . . , bn ∈ ∂B such that (Ubi)
n
i=1 covers ∂B. The compact-open set Wf that we require is
defined by the family (Ubi , ξ
G
F (bi, D
k, Dn))ni=1.
Remark 2.30. Note that f ∈Wf because f(Ub) = ξGF (b, Ek, En) and so f(Ub) ⊂ ξGF (b,Dk, Dn).
Lemma 2.31. The compact-open set Wf of Γ(G∇∂B) is contained in Γ(Tf ).
Proof. Let χ ∈Wf . For each b ∈ ∂B there exists at least one i ∈ {1, . . . , n} such that b ∈ Ubi .
For all such i the elements f(b) and χ(b) are in the open set ξGF (bi, D
k, Dn) of G. Now for each
b ∈ ∂B the map ιG(b,−,−) preserves distances. Furthermore since ξGF = expG ◦ιG the map
ξGF (b,−,−) preserves distances from the origin. Finally recall that f(bi) = ξGF (bi,~0,~0). Hence
d(χ(b), f(b)) ≤ d(χ(b), f(bi)) + d(f(bi), f(b)) < 1
2 + 1
2 = 1,
which tells us that in fact χ ∈ Γ(Tf ). So Wf ⊂ Γ(Tf ). �
Corollary 2.32. If f ∈ G∇∂B has filler F ∈ G∇B then there exists a Penon open subobject
Φ: Vf � G∇∂B and a lift Ψ: Vf → G∇B making
G∇B
Vf G∇∂B
GιB
Φf
Ψf
commute.
Proof. Let Vf = E(Wf ) where E is the left adjoint to the global sections functor as in Sec-
tion 2.1.2. Note that by construction f ∈ Γ(Vf ). �
2.7 Ordinary connectedness implies internal connectedness
Now we are in a position to deduce the main result of this paper. Let G be a (Hausdorff) Lie
groupoid with arrow space G and object space M .
Theorem 2.33. If B is a compact and contractible subset of Euclidean space that is the zero
set of an ideal of smooth functions then the arrow GιB : G∇B → G∇∂B is an epimorphism.
Proof. We perform a sequence of reductions to show that it in fact suffices to prove Corol-
lary 2.32.
Firstly, to show that GιB is an epimorphism, it will suffice to show that for all representable
objects X in E and arrows φ : X → G∇∂B in E there exists a Dubuc open cover (ιi : Xi → X)i∈I
such that for all i ∈ I there exists a lift ψi making
G∇B
Xi G∇∂B
GιB
φιi
ψi
commute.
Connected Lie Groupoids are Internally Connected 15
In fact it will suffice to find for each f ∈1 G∇∂B a Penon open subobject Uf of G∇∂B
containing f and a lift
G∇B
Uf G∇∂B.
GιB
φf
ψf (2.2)
Indeed (Uf )f∈G∇∂B covers G∇∂B as Penon open subobjects and so the pullback cover
(φ−1(Uf ))f∈G∇∂B covers X as Penon open subobject. But now we use Corollary 2.7 and the fact
that X is representable to see that (Uf )f∈G∇∂B covers G∇∂B as Dubuc open subobjects also.
But the existence of ψf and a Penon open φf making (2.2) commute is the conclusion of
Corollary 2.32. �
Corollary 2.34. If G is an s-path connected Lie groupoid then the arrow GιI : G∇I → G2 is an
epimorphism and so, by definition, the groupoid G is internally path connected.
Corollary 2.35. If G is an s-simply connected Lie groupoid then the arrow Gι(I×I) : G∇(I×I) →
G∇∂(I×I) is an epimorphism and so, by definition, the groupoid G is internally simply connected.
3 Properties of the jet part
3.1 The infinitesimal neighbour relation
In this section we introduce the infinitesimal neighbour relation which is used to define the jet
part of a category in [5]. If C is a category in any well-adapted model E of synthetic differential
geometry and M is the space of objects of C then we define the infinitesimal neighbour relation
on objects of the slice topos E/M . In [5] we justify this choice by showing that the jet part
defined using this neighbour relation is closed under composition in C.
Let a, b : X → B where X and B are objects of the topos E/M . Then a ∼ b iff there exists
a cover (ιi : Xi → X)i∈I in E/M such that for each i there exists an object DWi ∈ Spec(Weil),
an arrow φi : Xi ×DWi → B and an arrow di : Xi → DWi such that
Xi Xi ×DWi
B B
ai
(1Xi ,0)
φi
1B
and
Xi Xi ×DWi
B B
bi
(1Xi ,di)
φi
1B
commute, where we have written ai and bi for the restrictions of a and b to Xi.
Remark 3.1. The relation ∼ is not always symmetric. In fact in [5] we see that ∼ is not
symmetric in the case B = D and M = 1.
The relation ≈ is the transitive closure of ∼ in the internal logic of E/M . This means that
for a, b : X → B we have a ≈ b iff there exists a cover (ιi : Xi → X)i∈I and for each i there
exists a natural number ni and elements xi0 , xi1 , . . . , xini ∈Xi B such that
ai = xi0 ∼ xi1 ∼ · · · ∼ xini = bi.
16 M. Burke
3.2 The jet factorisation system and the jet part
In this section we recall the definitions of the jet factorisation system and the jet part of
a groupoid.
An arrow f : A→ B in E/M is jet-dense iff for all b : X → B there exists a cover (ιi : Xi →
X)i∈I and elements ai : Xi → A such that f(ai) ≈ bi. We have written bi for the restriction
of b to Xi. An arrow g : A → B in E/M is jet-closed iff it is a monomorphism and for all
a : X → A and b : X → B such that ga ≈ b there exists a cover (ιi : Xi → X)i∈I and elements
ci : Xi → A such that ai ≈ ci and gci = bi. We have written ai and bi for the restrictions of a
and b respectively to Xi.
In the case M = 1 the right class of the jet factorisation system has been studied before. For
instance it is the class of formal-etale maps in I.17 of [17]. In fact in Section 1.2 of [16] is it called
the class of formally-open morphisms. The sense in which these maps are open is reflected in
the following corollary that follows immediately from the definition of jet closed.
Corollary 3.2. The inclusion of an open subset U into a manifold M is jet closed.
Now we recall from [5] the results about the jet factorisation system that we need in the rest
of this paper.
Lemma 3.3. Let h : A → E be an arrow in E/M . Then there exists a jet closed arrow g and
a jet dense arrow f such that h = gf . The mediating object in the factorisation has the following
description:
B = {x ∈ E : ∃ a ∈ A, ha ≈ x} g−→ E.
Proof. See Lemma 3.23 in [5]. �
Theorem 3.4. Let G be a groupoid in E. Then the subobject
(G∞, s∞) = {g ∈ (G, s) : esg ≈ g}
is closed under composition and hence defines a subgroupoid G∞ � G called the jet part of G.
Proof. See Corollary 4.4 and Proposition 4.18 in [5]. �
Proposition 3.5. Let L∞ be the class of jet dense arrows and R∞ the class of jet closed arrows.
Then the pair (L∞, R∞) defines a (E/M)-factorisation system.
Proof. See Section 3.2 in [5]. �
Proposition 3.6. Let g be jet dense and k be jet closed in E/M . Suppose that the relation ≈
is symmetric on the object E and that the square
A B
C E
f
h
g
k
is a pullback. Then f is also jet dense.
Proof. See Proposition 3.27 in [5]. �
Connected Lie Groupoids are Internally Connected 17
3.3 Neighbour relation is symmetric for Lie groupoids
One of the assumptions that is required to prove Lie’s second theorem in [5] involves the sym-
metry of the neighbour relation ∼ defined in Section 3.1. More precisely, if G is a groupoid
in E with arrow space G and source map s then we need to assume that ∼ is symmetric on the
object (G, s) in E/M . In this section we justify this assumption by proving that if G is a Lie
groupoid then the relation ∼ is symmetric on the object (G, s) in E/M .
So suppose that a, b ∈X (G, s) in E/M and a ∼ b. By definition we have a cover (Xi → X)i
such that for all i there exist Wi ∈ Spec(Weil), φi ∈Xi (G, s)DWi and di ∈Xi DWi making
Xi
Xi ×DWi (G, s)
Xi
(1Xi ,di)
bi
φi
(1Xi ,0)
ai
commute where ai and bi are the restrictions of a and b to Xi. We need to show that b ∼ a.
Definition 3.7. Let s : G→M be an arrow in Man and x ∈ G. Then a pair of open embeddings
(αx, βx) is an s-trivialisation centred at x iff
Ck+n G
Ck M
π
αx
s
βx
commutes and αx(0) = x.
Lemma 3.8. There exists a cover of ιx : (Xi,x → Xi) such that ιxai factors through an s-
trivialisation Cn+k � G around ai(x).
Proof. Let Xi = (Bi, ξi). Since s is a submersion we can choose for each x ∈ Bi an s-
trivialisation νx : Cn+k � G centred at ai(x). Write Ux for the image of νx. Then the family
(ιx : a−1
i (Ux) → Bi)x∈Bi covers Bi in E and for each x ∈ Bi the arrow ιxai factors through Ux.
This means that (ιx : (a−1
i (Ux), ξi)→ Xi))x∈Bi is a covering family in E/M such that ιxai factors
through Ux. So we choose Xi,x = ((a−1
i (Ux), ξi). �
Now using the cover (Xi,x → X)i,x we show that b ∼ a.
Lemma 3.9. If di,x, ai,x, bi,x and φi,x are the restrictions of d, a, b and φ respectively to Xi,x
then the arrows ψi,x : Xi,x ×DWi → (Ux, s) defined by
ψi,x(u, d) = ai,x(u) +s φi,x(u, di,x(u))−s φ(u, d)
exhibit b ∼ a where +s and −s denote the fibrewise addition and subtraction. (I.e., addition in
the last n coordinates of the s-trivialisation.) Hence the infinitesimal neighbourhood relation is
symmetric for all Lie groupoids.
Proof. By construction the diagrams
Xi,x
Xi,x ×DWi (Ux, s)
Xi,x
(1Xi,x ,di,x)
bi,x
φi,x
(1Xi,x ,0)
ai,x
18 M. Burke
commute for all x ∈ Bi. First we check that ψi,x factors through Ux. This follows from the
equality ψ(u, 0) = bi,x(u) and the fact that the inclusion of Ux into G is jet closed. Second we
check that ψi,x defines an arrow in the slice category. But this follows from the fact that the three
terms ai,x(u), φi,x(u, di,x(u)) and φ(u, d) have the same source and the addition defining ψi,x is
carried out in the last n coordinates of the s-trivialisation. Finally since
ψ(u, 0) = ai,x(u) + φi,x(u, di,x(u))− φi,x(u, 0) = bi,x(u)
and
ψ(u, di(u)) = ai,x(u) + φi,x(u, di,x(u))− φi,x(u, di,x(u)) = ai,x(u)
we conclude that b ∼ a. �
3.4 A trivialisation cover of the identity elements
In this section we construct a cover (φem : Cn+k → G)m∈M of e(M) in G with the property that
each φem has a lift ψem making
GI
Cn+k G
Gl
ψem
φem
commute and furthermore when we restrict ψem to e(M) the fillers we obtain are the constant
fillers. First we choose an s-trivialisation at em such that the identity inclusion induces a section
of the projection onto the first k coordinates in the trivialisation.
Lemma 3.10. If m ∈ M then there is an s-trivialisation (αem, βem) at em such that eβem
factors through αem.
Proof. Let (α, β) be any s-trivialisation at em. Then if ν and ξ are defined in the pullback
P Ck+n
Ck G
ν
ξ
α
eβ
then βπξ = sαξ = seβν = βν and so πξ = ν because β is a monomorphism. Now P is an open
set of Ck and 0 ∈ P because eβ(0) = α(0). Since the derivative of ν has full rank at 0 we can
find an open embedding ι : Ck � P such that νι(0) = 0. Now let µ be defined by the pullback
Ck+n Ck+n
Ck Ck
π
µ
π
νι
ρ
and ρ be induced by the pair (1P , ξι). Then eβνι = αξι = αµρ and the s-trivialisation that we
require is (αem, βem) = (αµ, βνι). �
This means that for each ~x ∈ Ck the arrow ψ(ρ(~x), ~y) is an identity arrow. The φem that we
require will be the αem obtained in Lemma 3.10. Now we can construct a lift ψem : Ck+n → GI
for φem as follows. For each (~x, ~y) ∈ Ck+n we have a source constant path
a 7→ (~x, a~y + (1− a)ρ(~x)), (3.1)
which starts at an identity. Since (3.1) is smooth in ~x and ~y it induces an arrow ψem : Ck+n → GI.
Moreover by construction the restriction of ψem to e(M) are the constant paths at identity
arrows.
Connected Lie Groupoids are Internally Connected 19
3.5 A cover of the jet part
In Section 3.4 we constructed a cover (φem : Cn+k → G)m∈M of e(M) in G satisfying certain
properties on restriction to e(M). In this section we show that the φem also induce a cover of
the object (G∞, s∞) in E/M .
Lemma 3.11. There is an inclusion j : (G∞, s∞) �
⋃
m(Um, sφem) such that
⋃
m φem ◦j = ι∞G .
Proof. By hypothesis we have an inclusion (M, 1M ) �
⋃
m(Um, sφem) such that ι ◦ m = e.
Since the inclusion ι is jet closed in E/M the square
(M, 1) (
⋃
m Um, sφem)
(G∞, s∞) (G, s)
e∞
m
⋃
m φem
∃! j
ι∞G
has a unique (monic) filler. �
Corollary 3.12. Let the objects (Vm, s∞φem) of E/M be defined by the pullbacks
(Vm, s∞φem) (Um, sφem)
(G∞, s∞) (G, s)
χm
um
φem
ι∞G
then because colimits are stable under pullback the bottom right square in
(G∞, s∞)
(
⋃
m Vm, s∞φem) (
⋃
m Um, sφem)
(G∞, s∞) (G, s)
j
1G∞
η ⋃
m um
⋃
m χm
⋃
m φem
ι∞G
is a pullback. But then the arrow η induced by the pair (1G∞ , j) is an isomorphism and hence⋃
m∈M χm is a cover of (G∞, s∞).
3.6 Jet part of a Lie groupoid is internal path connected
Now we combine Section 3.4 and Section 3.5 to show that the jet part of a Lie groupoid is E-path
connected. It will suffice to show that when we restrict the fillers ψem : Cn+k → GI defined in
Section 3.4 along um we get an arrow that factors through (GI
∞, s∞). Then ψemum is a filler
for χm.
So let Vm and Wm be defined by the iterated pullback:
(Wm, ι) (Vm, s∞χem) (Um, sφem)
(M, 1) (G∞, s∞) (G, s)
vm
χem
um
φem
e∞ ι∞G
20 M. Burke
and note that the χem are Penon open because the φem are. Then by Proposition 3.6 and
Lemma 3.9 we deduce that vm is jet dense. Since we have chosen ψem such that ψemvmum are
the constant functions cm the square
(Wm, ι)
(
GI
∞, s∞
)
(Vm, s)
(
GI, s
)
cm
vm (ι∞G )Iζm
ψemum
commutes and has a unique filler. This means that the φem form a Penon open cover of G∞
whose fillers factor through GI
∞. By pulling back this cover along generalised elements X → G∞
we deduce that the jet part G∞ is E-path connected.
4 Integral completeness
One of the main assumptions that we require to prove Lie’s second theorem in [5] is that of
integral completeness. Recall from Definition 1.9 that an arbitrary groupoid G in a well-adapted
model E of synthetic differential geometry is integral complete iff
GI Gι
∞
I−−−→ GI∞
is an isomorphism in Gpd(E) where I is the pair groupoid on the unit interval I. In Section 4.3 we
show that the classical A-paths (see for instance [13]) correspond to global sections of GI∞ in E
and the classical G-paths (see also [13]) correspond to global sections of GI in E . In Section 4.5
we show that all classical Lie groupoids are integral complete. But first we give a more explicit
description of the arrow space of I∞.
4.1 Representing object for infinitesimal paths is trivial
In this section we show that the arrow space I2∞ of I∞ is isomorphic to I ×D∞.
Recall from Lemma 3.3 that the arrow space of I∞ is characterised as follows. A generalised
element (a, b) ∈ (I2, π1) is in (I2∞, π1) iff there exists m ∈ (I, 1I) such that (m,m) ≈ (a, b). By
definition of ≈ if b− a ∈ D∞ then a ≈ b. This means that it will suffice to prove the following
result:
Lemma 4.1. If (a, b) : X → I2 and a ≈ b in E then b− a ∈ D∞.
Proof. First suppose that a ∼ b. This means that there exist W ∈ Spec(Weil), φ ∈ IDW and
d ∈ DW such that φ(0) = a and φ(d) = b. Then by the Kock–Lawvere axiom b = a + N for
some nilpotent N .
Suppose now that a ≈ b. This means that there exist a0, . . . , an such that a = a0 ∼ a1 ∼ · · · ∼
an = b. Now we know that for all i ∈ {1, . . . , n} there exists ki ∈ N such that (ai − ai−1)ki = 0.
But then (b− a)Σiki = 0 as required. �
Corollary 4.2. The groupoid I∞ has underlying reflexive graph isomorphic to
I ×D∞ I,
+
e
π1
where e = (1I , 0) and composition I ×D∞ ×D∞ → I ×D∞ defined by
(a, d, d′) 7→ (a, d+ d′).
Proof. For all a ∈ I we have a ≈ a + d and we can define an arrow I × D∞ → I2∞ by
(a, d) 7→ (a, a+ d). The inverse (a, b) 7→ (a, b− a) factors through I ×D∞ by Lemma 4.1. �
Connected Lie Groupoids are Internally Connected 21
4.2 Formal group laws
When we form the infinitesimal part of a category in [5] our construction corresponds to the part
of a Lie group represented by its formal group law. Following [14] we define an n-dimensional
formal group law F to be an n-tuple of power series in the variables X1, . . . , Xn; Y1, . . . , Yn with
coefficients in R such that the equalities
F
(
~X,~0
)
= ~X, F
(
~0, ~Y
)
= ~Y and F
(
F
(
~X, ~Y
)
, ~Z
)
= F
(
~X,F
(
~Y , ~Z
))
(4.1)
hold. We refer to the Introduction of [14] for the construction of a formal group law from a Lie
group. In fact the category of Lie algebras and formal group laws are shown to be equivalent in
Theorem 3 of Section V.6 of Part 2 in [26].
In the following example we show how to reformulate the construction of a formal group law
from a Lie group in terms of the infinitesimal elements of the Lie group.
Example 4.3. Let (G,µ) be a Lie group whose underlying smooth manifold is n-dimensional.
Since G is locally isomorphic to Rn we see that its jet part is a group of the form (Dn
∞, µ) by
a straightforward extension of Lemma 4.1. Now to give a multiplication
µ : Dn
∞ ×Dn
∞ → Dn
∞
is to give arrows
f1, . . . , fn : (D∞)2n → R
taking values in nilpotent elements. Now we have that
(D∞)2n =
⋃
k
(Dk)
2n
and so, since E(−, R) sends colimits to limits the hom-set E(D2n
∞ , R) is given by the limit
· · · → E
(
D2n
k+1, R
)
→ E
(
D2n
k , R
)
→ · · · ,
which by the Kock–Lawvere axiom is equivalently the limit of the polynomial algebras
· · · → R[X1, . . . , X2n]/Ik+1 → R[X1, . . . , X2n]/Ik → · · · ,
where Ik is the ideal generated by (Xk
1 , X
k
2 , . . . , X
k
2n). This means that E(D2n
∞ , R) can be iden-
tified with the ring R[[X1, . . . , X2n]] of formal power series. Now the condition that the fi take
values in the nilpotent elements implies that the constant term of the power series pi corre-
sponding to fi is zero. Under this correspondence, the group axioms for G correspond to the
axioms making p1, . . . , pn into a formal group law.
4.3 Paths of infinitesimals
The correct notion of a path of infinitesimal arrows in a Lie groupoid G is that of an A-path
(see for instance [6]). In the topos E the object of A-paths A(G) associated to G is the subobject
of all φ ∈ GI×D such that for all a ∈ I and all d ∈ D the arrows φ(a, 0) are identity arrows, the
φ(a,−) are source constant and tφ(a, d) = tφ(a + d, 0). Note that since GD ∼= TG the global
sections of A(G) are precisely the A-paths defined in Section 1 of [6].
In this section we show that A(G) ∼= GI∞ in E where I∞ is the jet part of the pair groupoid I
on the unit interval I. Using Corollary 4.2 we see that GI∞ is the subobject of all φ ∈ GI×D∞
22 M. Burke
such that for all a ∈ I and all d ∈ D∞ the arrows φ(a, 0) are identity arrows, the φ(a,−) are
source constant and not only does tφ(a, d) = tφ(a+ d, 0) hold but indeed
φ(a, d+ d′) = φ(a+ d, d′)φ(a, d)
holds for all d, d′ ∈ D∞. This means that there is a natural restriction arrow GI∞ → A(G). In
this section we describe its inverse.
To do this we define an arrow v : GI×D → GI×D∞ which satisfies v(φ)(a, d+ d′) = v(φ)(a+
d, d′)v(φ)(a, d) for all d, d′ ∈ D∞. Recall that D∞ =
⋃
iDi and so it will suffice to find for all
i ∈ N an arrow vi : G
I×D → GI×Di such that vi+j(φ)(a, d + d′) = vj(φ)(a + d, d′)vi(φ)(a, d) for
all d ∈ Di and d′ ∈ Dj .
Now we recall the following slight generalisation of the Bunge axiom that is Proposition 4 in
Section 2.3.2 in [19]:
Lemma 4.4. Let i ∈ N and consider the arrows f1, . . . , fi : D
i−1 → Di defined by
fm(d1, . . . , di−1) = (d1, . . . , dm−1, 0, dm, . . . , di−1).
Then for any microlinear space G the arrow
GDi
G+
−−→ GD
i
is the joint equaliser of Gf1 , . . . , Gfi.
Using Lemma 4.4 we see that it will now suffice to find for all i ∈ N an arrow vi : G
I×D →
GI×D
i
such that for all m, l ∈ {1, . . . , i} the equalities GI×fmvi(φ) = GI×flvi(φ) and
vi+j(φ)(a, (d1, . . . , di+j)) = vj(φ)
(
a+
i∑
m=1
dm, (di+1, . . . , di+j)
)
vi(φ)(a, (d1, . . . , di))
hold for the fi defined in Lemma 4.4.
Lemma 4.5. The restriction GI∞ → A(G) has an inverse.
Proof. The arrows vi : G
I×D → GI×D
i
defined by
vi(φ)(a, (d1, . . . , di)) = φ(a+ Σi−1
m=1dm, di) · · ·φ(a+ d1, d2)φ(a, d1)
satisfy GI×fmvi(φ) = GI×flvi(φ) because φ(a, 0) are identity arrows for all a ∈ I and satisfies
vi+j(φ)(a, (d1, . . . , di+j)) = vj(φ)
(
a+
i∑
m=1
dm, (di+1, . . . , di+j)
)
vi(φ)(a, (d1, . . . , di))
by construction. It is easy to see that the vi define an inverse to the restriction. �
4.4 Integration of paths of infinitesimals is groupoid enriched
Recall that in Definition 1.9 we defined the notion of integral complete groupoid using an
isomorphism in the category Gpd(E). The following result show that we only need to check this
condition on the space of objects which is an object of E .
Proposition 4.6. If Gι∞ : GI → GI∞ is an isomorphism in a well-adapted model E then it is
an isomorphism of groupoids also.
Connected Lie Groupoids are Internally Connected 23
Proof. We need to show that natural transformations extend uniquely, i.e.,
I∞ × 222 G
I× 222.
∀Φ
ι ∃! Ψ
Let ψ0, ψ1 be the unique lifts of φ precomposed with the two inclusions of 1 into 222. If for all
x→ y in I the diagram
Φ(x, 1) Φ(y, 1)
Φ(x, 0) Φ(y, 0)
ψ1(x→y)
Φ(x,0→1) Φ(y,0→1)
ψ0(x→y)
(4.2)
commutes then we can define Ψ(x → y, 0 → 1) to be this common value. To this end define
θ : I→ G to take x→ y to
Φ(x, 1) Φ(y, 1)
Φ(x, 0) Φ(y, 0),
Φ(x,1→0) Φ(y,0→1)
ψ0(x→y)
when we restrict to I∞ (i.e., take y = x+ d) we see that
Φ(x, 1) Φ(x+ d, 1)
Φ(x, 0) Φ(x+ d, 0)
Φ(x,1→0) Φ(x+d,0→1)
Φ(x→x+d,0)
=
Φ(x, 1) Φ(x+ d, 1)
Φ(x→x+d,1)
and so by the uniqueness of lifts θ = ψ1 and (4.2) commutes. �
4.5 Lie groupoids are integral complete
We show that Gι∞I : GI → GI∞ is an isomorphism in Gpd(E). By Proposition 4.6 it will suffice
to show that Gι∞I is an isomorphism in E . More concretely, we show that for all representable
objects X and arrows φ : X → GI∞ there exists a (unique) ψ : X → GI such that Gι∞I ψ = φ.
By Corollary 4.2 arrows φ : X → GI∞ correspond to arrows φ : X × I ×D∞ → G such that
φ(x, a, 0) are identity arrows and φ(x, a,−) are source constant. It is easy to see that arrows
ψ : X → GI correspond to arrows ψ : X×I → G such that ψ(x, 0) are identity arrows and ψ(x,−)
are source constant. At this point it is convenient to assume that the topos E is generated by
a subcanonical site whose underlying category is a full subcategory of the category of affine
C∞-schemes as defined in Definition 1.5. In particular this means that every representable
object is a closed subset of Rn for some n ∈ N. Recall from Lemma 2.26 in [20] that if we are
given a smooth function that has as domain any closed subset of Rn we can lift it to a smooth
function on the whole of Rn. Therefore since every representable X is a closed subset of Rn for
some n ∈ N it will suffice to prove the result in the case X = Rn.
Theorem 4.7. For all φ : Rn× I ×D∞ → G such that φ(x, a, 0) are identity arrows, φ(x, a,−)
is source constant and φ(a, d+ d′) = φ(a+ d, d′)φ(a, d) for x ∈ Rn, a ∈ I and d, d′ ∈ D∞ there
exists a unique ψ : Rn× I → G such that ψ(x, 0) are identity arrows, ψ(x,−) is source constant
and ψ(x, a+ d) = φ(x, a, d)ψ(x, a) for all d ∈ D∞.
24 M. Burke
Proof. To do this we make rigorous the intuitive idea of composing together infinitely many
infinitesimal arrows to get a macroscopic arrow. First let φ0 = sφ(−,−, 0) = tφ(−,−, 0). Then
the pullback
(
Rn × I
)
×φ0 t G G,
Rn × I M
π0,1 t
φ0
is a manifold because t is a submersion. Since φ(x, a,−) is source constant the infinitesimal
action
Rn ×D ×
((
Rn × I
)
×φ0 t G
)
→
((
Rn × I
)
×φ0 t G
)
,
(y, d, x, a, g) 7→ (x, a+ d, φ(x, a, d) ◦ g)
defines a collection of smoothly parameterised vector fields on (Rn×I) ×φ0 tG. Since the solution
curves of smooth vector fields have a smooth dependence on parameters we obtain using the
initial conditions ψ(y, 0) = (y, 0, φ(y, 0, 0)) a parameterised solution ψ : Rn×I → (Rn×I) ×φ0 tG
which satisfies:
• ψ(y, 0) = (ψ1(y, 0), ψ2(y, 0), ψ3(y, 0)) = (y, 0, φ(y, 0, 0)),
• ψ1(y, a+ d) = ψ1(y, a),
• ψ2(y, a+ d) = ψ2(y, a) + d,
• ψ3(y, a+ d) = φ(ψ1(y, a), ψ2(y, a), d) ◦ ψ3(y, a).
Therefore
• ψ1(y, a) = ψ1(y, 0) = y,
• ψ2(y, a) = ψ2(y, 0) + a = a,
• ψ3(y, a+ d) = φ(ψ1(y, a), ψ2(y, a), d)ψ3(y, a) = φ(y, a, d) ◦ ψ3(y, a)),
and ψ3 is the map we require. Now we check that ψ3 does indeed define a G-path. The map ψ3
is source constant in the second variable because
sψ3(y, a+ d) = s (φ(y, a, d) ◦ ψ(x, a)) = sψ3(y, a)
for all d ∈ D. Finally we appeal to Proposition 2.7 in [18] to conclude that ψ3(y, a + d) =
φ(y, a, d)ψ3(y, a) holds for all d ∈ D∞. �
Acknowledgements
The author is very grateful for the constructive comments offered by and the important correc-
tions indicated by the editor and referees. The author would like to acknowledge the assistance
of Richard Garner, my Ph.D. supervisor at Macquarie University Sydney, who provided valuable
comments and insightful discussions in the genesis of this work. In addition the author is grateful
for the support of an International Macquarie University Research Excellence Scholarship.
Connected Lie Groupoids are Internally Connected 25
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http://hdl.handle.net/1959.14/1068209
http://hdl.handle.net/1959.14/1068209
http://arxiv.org/abs/1605.06378
https://doi.org/10.4007/annals.2003.157.575
http://arxiv.org/abs/math.DG/0210152
https://doi.org/10.1515/crelle-2015-0018
http://arxiv.org/abs/1404.5989
https://doi.org/10.1007/BF01389725
https://doi.org/10.2307/2374046
https://doi.org/10.1016/0022-4049(86)90091-5
https://doi.org/10.1016/0022-4049(90)90152-8
https://doi.org/10.1007/978-3-642-56936-4
https://doi.org/10.1007/978-1-4684-9449-5
https://doi.org/10.1017/CBO9780511550812
https://doi.org/10.1007/s11533-005-0005-2
https://doi.org/10.1007/978-1-4757-4588-7
https://doi.org/10.1007/978-1-4757-4588-7
https://doi.org/10.1007/978-1-4419-9982-5
https://doi.org/10.1017/CBO9781107325883
https://doi.org/10.1017/CBO9781107325883
https://doi.org/10.1090/gsm/093
https://doi.org/10.1007/978-1-4757-4143-8
https://doi.org/10.1007/978-3-540-70634-2
https://doi.org/10.1112/S0010437X05001752
http://arxiv.org/abs/math.DG/0405003
1 Introduction
1.1 Synthetic differential geometry
1.2 Smooth affine schemes and the Dubuc topos
1.3 Internal connectedness
1.4 The jet part
1.4.1 Symmetry of the neighbour relation
1.4.2 Path connectedness of the jet part
1.5 Integral completeness
2 Path and simply connectedness
2.1 Open subobjects of function spaces
2.1.1 Penon open subobjects
2.1.2 Smooth compact-open subsets
2.2 Splitting the tangent bundle
2.3 Riemannian submersions
2.4 Constructing a tubular extension
2.5 A subobject of the tubular extension admitting fillers
2.6 A compact-open set inside a tubular extension
2.7 Ordinary connectedness implies internal connectedness
3 Properties of the jet part
3.1 The infinitesimal neighbour relation
3.2 The jet factorisation system and the jet part
3.3 Neighbour relation is symmetric for Lie groupoids
3.4 A trivialisation cover of the identity elements
3.5 A cover of the jet part
3.6 Jet part of a Lie groupoid is internal path connected
4 Integral completeness
4.1 Representing object for infinitesimal paths is trivial
4.2 Formal group laws
4.3 Paths of infinitesimals
4.4 Integration of paths of infinitesimals is groupoid enriched
4.5 Lie groupoids are integral complete
References
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