Curvature in Synthetic Differential Geometry of
Groupoids
Hirokazu Nishimura
Institute of Mathematics, University of Tsukuba
Tsukuba, Ibaraki, 305-8571, Japan
October 28, 2018
Abstract
We study the fundamental properties of curvature in groupoids within
the framework of synthetic differential geometry. As is usual in synthetic
differential geometry, its combinatorial nature is emphasized. In particu-
lar, the classical Bianchi identity is deduced from its combinatorial one.
1 Introduction
The notion of curvature, which is one of the most fundamental concepts in dif-
ferential geometry, retrieves its combinatorial or geometric meaning in synthetic
differential geomety. It was Kock [4] who studied it up to the second Bianchi
identity synthetically for the first time. In particular, he has revealed the com-
binatorial nature of the second Bianchi identity by deducing it from an abstract
In [4] Kock trotted out first neighborhood relations, which are indeed to
be seen in formal manifolds, but which are no longer expected to be seen in
microlinear spaces in general. Since we believe that microlinear spaces should
play the same role in synthetic differential geometry as smooth manifolds have
done in classical differential geometry, we have elevated his ideas to a microlinear
context in [8]. However we were not so happy, because our proof of the second
Bianchi identity there appeared unnecessarily involved, making us feel that we
were somewhat off the point, though the proof was completely correct.
Recently we got accustomed to groupoids, which encouraged us to attack
the same problem once again. Within the framework of groupoids, we find it
pleasant to think multiplicatively rather than additively (cf. Nishimura [11]),
which helps grasp the nature of the second Bianchi identity firmly. Now we
are to the point. What we have to do in order to deduce the classical second
Bianchi identity from the combinatorial one is only to note some commutativity
on the infinitesimal level, though groupoids are, by and large, highly noncom-
mutative. Our present experience is merely an example of the familiar wisdom
in mathematics that a good generalization reveals the nature.
http://arxiv.org/abs/0704.1446v1
2 Preliminaries
2.1 Synthetic Differential Geometry
Our standard reference on synthetic differential geometry is Chapters 1-5 of
Lavendhomme [5]. We will work internally within a good topos, in which the
intended set R of real numbers is endowed with a cornucopia of nilpotent in-
finitesimals pursuant to the general Kock-Lawvere axiom. To see how to build
such a good topos, the reader is referred to Kock [1] or Moerdijk and Reyes [7].
Any space mentioned in this paper will be assumed to be microlinear, unless
stated to the contrary. We denote by D the set {d ∈ R | d2 = 0}, as is usual in
synthetic differential geometry.
Given a group G, we denote by AG the tangent space of G at its identity,
i.e., the totality of mappings t : D → G such that t0 is the identity of G. We
will often write td rather than t(d) for any d ∈ D. As we will see shortly, AG is
more than an R-module.
Proposition 1 For any t ∈ AG and any (d1, d2) ∈ D(2), we have
td1+d2 = td1td2 = td2td1
so that td1 and td2 commute.
Proof. By the same token as in Proposition 3 of §3.2 of Lavendhomme [5].
As an easy corollary of this proposition, we can see that
t−d = (td)
for we have (d,−d) ∈ D(2).
Proposition 2 For any t1, t2 ∈ AG, we have
(t1 + t2)d = (t2)d(t1)d = (t1)d(t2)d
for any d ∈ D, so that (t1)d and (t2)d commute.
Proof. By the same token as in Proposition 6 of §3.2 of Lavendhomme [5].
As an easy corollary of this proposition, we can see, by way of example, that
(t1)d1d2 and (t2)d1d3 commute for any d1, d2, d3 ∈ D, for we have
(t1)d1d2(t2)d1d3 = (d2t1)d1(d3t2)d1 = (d3t2)d1(d2t1)d1 = (t2)d1d3(t1)d1d2
Proposition 3 For any t1, t2 ∈ AG, there exists a unique s ∈ AG such that
sd1d2 = (t2)−d2(t1)−d1(t2)d2(t1)d1
for any d1, d2 ∈ D.
Proof. By the same token as in pp.71-72 of Lavendhomme [5].
We will write [t1, t2] for the above s.
Theorem 4 The R-module AG endowed with the above Lie bracket [·, ·] is a
Lie algebra over R.
Proof. By the same token as in our previous paper [10].
2.2 Groupoids
Groupoids are, roughly speaking, categories whose morphisms are always invert-
ible. Our standard reference on groupoids is MacKenzie [6]. Given a groupoid
G over a base M with its object inclusion map id : M → G and its source and
target projections α, β : G → M , we denote by B(G) the totality of bisections
of G, i.e., the totality of mappings σ : M → G such that α ◦ σ is the identity
mapping on M and β ◦ σ is a bijection of M onto M . It is well known that
B(G) is a group with respect to the operation ∗, where for any σ, ρ ∈ B(G),
σ ∗ ρ ∈ B(G) is defined to be
(σ ∗ ρ)(x) = σ((β ◦ ρ)(x))ρ(x)
for any x ∈ M . It can easily be shown that the space B(G) is microlinear,
provided that both M and G are microlinear, for which the reader is referred
to Proposition 6 of Nishimura [10].
Given x ∈ M , we denote by AnxG the totality of mappings γ : D
n → G
with γ(0, ..., 0) = idx and (α ◦ γ)(d1, ..., dn) = x for any (d1, ..., dn) ∈ D
n. We
denote by AnG the set-theoretic union of AnxG’s for all x ∈ M . In particular,
we usually write AxG and AG in place of AxG and AG respectively. It is easy
to see that AG is naturally a vector bundle over M . A morphism ϕ : H → G
of groupoids over M naturally gives rise to a morphism ϕ∗ : AH → AG of
vector bundles over M . As in §3.2.1 of Lavendhomme [5], where three distinct
but equivalent viewpoints of vector fields are presented, the totality Γ(AG) of
sections of the vector bundle AG can canonically be identified with the totality
of tangent vectors to B(G) at id, for which the reader is referred to Nishimura
[10]. We will enjoy this identification freely, and we dare to write Γ(AG) for
the totality of tangent vectors to B(G) at id. Given X,Y ∈ Γ(AG), we define
a microsquare Y ∗X to B(G) at id to be
(Y ∗X)(d1, d2) = Yd2 ∗Xd1
for any (d1, d2) ∈ D
Given γ ∈ An+1G and e ∈ D, we define γie ∈ A
nG (1 ≤ i ≤ n+ 1) to be
γie(d1, ..., dn) = γ(d1, ..., di−1, e, di, ..., dn)γ(0, ..., 0, e
, 0, ..., 0)−1
for any (d1, ..., dn) ∈ D
n. For our later use in the last section of this paper,
we introduce a variant of this notation. Given γ ∈ An+2G and e1, e2 ∈ D, we
define γi,je1,e2 ∈ A
nG (1 ≤ i < j ≤ n+ 2) to be
e1,e2
(d1, ..., dn) = γ(d1, ..., di−1, e1, di, ..., dj−2, e2, dj−1, ..., dn)γ(0, ..., 0, e1
, ..., e2
, 0, ..., 0)−1
Given γ ∈ A2G, we define τ1γ ∈ A
2G to be
τ1γ (d1, d2) = γ(d1, 0)
for any (d1, d2) ∈ D
2. Similarly, given γ ∈ A2G, we define τ2γ ∈ A
2G to be
γ (d1, d2) = γ(0, d2)
for any (d1, d2) ∈ D
2. Given γ ∈ A2G, we define Σγ ∈ A2G to be
(Σγ)(d1, d2) = γ(d2, d1)
for any (d1, d2) ∈ D
Any γ ∈ A2G can canonically be identified with the mapping e ∈ D 7→ γ1e ∈
AG, so that we can identify A2G and (AG)D. As is expected, this identification
enables us to define γ2 −
γ1 ∈ A
2G for γ1, γ2 ∈ A
2G, provided that γ1(0, ·) =
γ2(0, ·). Similarly, we can define γ2 −
γ1 ∈ A
2G for γ1, γ2 ∈ A
2G, provided that
γ1(·, 0) = γ2(·, 0). Given γ1, γ2 ∈ A
2G, their strong difference γ2
− γ1 ∈ AG
is defined, provided that γ1 |D(2)= γ2 |D(2). Lavendhomme’s [5] treatment of
strong difference
− in §3.4 carries over mutatis mutandis to our present context.
We note in passing the following simple proposition on strong difference
−, which
is not to be seen in our standard reference [5] on synthetic differential geometry.
Proposition 5 For any γ1, γ2, γ3 ∈ A
2G with γ1 |D(2)= γ2 |D(2)= γ3 |D(2), we
− γ1) + (γ3
− γ2) + (γ1
− γ3) = 0
2.3 Differential Forms
Given a groupoid G and a vector bundle E over the same space M , the space
Cn(G,E) of differential n-forms with values in E consists of all mappings ω
from AnG to E whose restriction to AnxG for each x ∈ M takes values in Ex
satisfying the following n-homogeneous and alternating properties:
1. We have
ω(a ·
γ) = aω(γ) (1 ≤ i ≤ n)
for any a ∈ R and any γ ∈ AnxG, where a ·
γ ∈ AnxG is defined to be
γ)(d1, ..., dn) = γ(d1, ..., di−1, adi, di+1, ...dn)
for any (d1, ..., dn) ∈ D
2. We have
ω(γ ◦Dθ) = sign(θ)ω(γ)
for any permutation θ of {1, ..., n}, where Dθ : Dn → Dn permutes the n
coordinates by θ.
3 Connection
Let π : H → G be a morphism of groupoids over M . Let L be the kernel of
π with its canonical injection ι : L → H . It is clear that L is a group bundle
over M . These entities shall be fixed throughout the rest of the paper. Thus
we have an exact sequence of groupoids as follows:
0 → L
A connection ∇ with respect to π is a morphism ∇ : AG → AH of vector
bundles over M such that the composition π∗ ◦ ∇ is the identity mapping of
AG. A connection ∇ with respect to π shall be fixed throughout the rest of the
paper. If G happens to be M ×M (the pair groupoid of M) with π being the
projection h ∈ H 7→ (α(h), β(h)) ∈ M ×M , our present notion of connection
degenerates into the classical one of infinitesimal connection.
Given γ ∈ An+1G, we define γi ∈ AG (1 ≤ i ≤ n+ 1) to be
γi(d) = γ(0, ..., 0, d
, 0, ..., 0)
for any d ∈ D. As in our previous paper [11], we have
Theorem 6 Given ω ∈ Cn(G,AL), there exists a unique d∇ω ∈ C
n+1(G,AL)
such that
((d∇ω)(γ))d1...dn+1
{(ω(γi0))d1...bdi...dn+1((∇γi)di)
−1(ω(γidi))−d1... bdi...dn+1(∇γi)di}
(−1)i
for any γ ∈ An+1G and any (d1, ..., dn+1) ∈ D
4 A Lift of the Connection ∇ to microsquares
Let us define a mapping A2G → A2H , which shall be denoted by the same
symbol ∇ hopefully without any possible confusion, to be
∇γ(d1, d2) = (∇γ
)d2(∇γ
0 )d1
for any γ ∈ A2G.
It is easy to see that
Proposition 7 For any γ ∈ A2G and any a ∈ R, we have
∇(a ·
γ) = a ·
∇(a ·
γ) = a ·
Corollary 8 For any γ1, γ2 ∈ A
2G, we have
∇(γ2 −
γ1) = ∇γ2 −
∇γ1 provided that γ1(0, ·) = γ2(0, ·);
∇(γ2 −
γ1) = ∇γ2 −
∇γ1 provided that γ1(·, 0) = γ2(·, 0).
Proof. This follows from the above proposition by Proposition 10 of §1.2 of
Lavendhomme [5].
Proposition 9 For any t ∈ A1G, we define εt ∈ A
2G to be
εt(d1, d2) = t(d1d2)
Then we have
(∇εt)(d1, d2) = (∇t)(d1d2)
for any d1, d2 ∈ D.
Proof. It suffices to note that
(∇εt)(d1, d2) = (∇(d1t))(d2) = (d1∇t)(d2) = (∇t)(d1d2)
Theorem 10 For any γ1, γ2 ∈ A
2G with γ1 |D(2)= γ2 |D(2), we have
− γ1) = ∇γ2
Proof. Let d1, d2 ∈ D. We have
(∇(γ2
− γ1))(d1d2)
= (∇ε
)(d1, d2)
[By Proposition 9]
= (∇((γ2 −
))(d1, d2)
[By Proposition 7 of §3.4 of Lavendhomme [5]]
= ((∇γ2 −
∇γ1)−
∇τ2γ1)(d1, d2)
[By Corollary 8]
= ((∇γ2 −
∇γ1)−
)(d1, d2)
(d1, d2)
[By Proposition 7 of §3.4 of Lavendhomme [5]]
= (∇γ2
−∇γ1)(d1d2)
[By Proposition 9]
Since d1, d2 ∈ D were arbitrary, the desired conclusion follows at once.
5 The Curvature Form
Proposition 11 For any γ ∈ A2G, there exists a unique t ∈ A1L such that
ι(td1d2) = ((∇γ
0)d1)
−1((∇γ1d1)d2)
−1(∇γ2d2)d1(∇γ
for any d1, d2 ∈ D.
Proof. Let η ∈ A2H to be
η(d1, d2) = ((∇γ
0 )d1)
−1((∇γ1d1)d2)
−1(∇γ2d2)d1(∇γ
for any d1, d2 ∈ D. Then it is easy to see that
η(d, 0) = η(0, d) = idα(η(0,0))
Therefore there exists unique t′ ∈ A1H such that
t′d1d2 = η(d1, d2)
Furthermore we have
π(η(d1, d2))
= π(((∇γ20 )d1)
−1)π(((∇γ1d1)d2)
−1)π((∇γ2d2 )d1)π((∇γ
0 )d2)
= ((γ20)d1)
−1((γ1d1)d2)
−1(γ2d2)d1(γ
= γ(d1, 0)
−1(γ(d1, d2)γ(d1, 0)
−1)−1γ(d1, d2)γ(0, d2)
−1γ(0, d2)
= idα(η(0,0))
Therefore there exists a unique t ∈ A1L with ι(t) = t′. This completes the
proof.
We write Ω(γ) for the above t. Now we have
Proposition 12 The mapping Ω : A2G → A1L consists in C2(G,AL).
Proof. We have to show that
Ω(a ·
γ) = aΩ(γ) (1)
Ω(a ·
γ) = aΩ(γ) (2)
Ω(Σγ) = −Ω(γ) (3)
for any γ ∈ A2G and any a ∈ R. Now we deal with (1), leaving a similar
treatment of (2) to the reader. Let d1, d2 ∈ D. We have
ι(Ω(a ·
γ))d1d2
= ((∇(a ·
γ)20)d1)
−1((∇(a ·
γ)1d1)d2)
−1(∇(a ·
γ)2d2)d1(∇(a ·1
γ)10)d2
= ((∇γ20 )ad1)
−1((∇γ1ad1)d2)
−1(∇γ2d2)ad1(∇γ
0 )d2
= ι(Ω(γ))ad1d2
= ι(aΩ(γ))d1d2
Now we deal with (3). We have
ι(Ω(Σγ))ι(Ω(γ))d1d2
= {((∇γ10 )d2)
−1((∇γ2d2)d1)
−1(∇γ1d1)d2(∇γ
0)d1}{((∇γ
0)d1)
−1((∇γ1d1)d2)
−1(∇γ2d2)d1(∇γ
0)d2}
= idα(γ(0,0))
Thie completes the proof.
We call Ω the curvature form of ∇.
Proposition 13 For any γ ∈ A2G, we have
Ω(γ) = Σ∇Σγ
Proof. As in the proof of Proposition 8 of §3.4 of Lavendhomme [5], let us
consider a function l : D2 ∨D → H given by
l(d1, d2, e) = (∇γ
)d2(∇γ
0)d1Ω(γ)e
for any (d1, d2, e) ∈ D
2∨D. Then it is easy to see that l(d1, d2, 0) = (∇γ)(d1, d2)
and l(d1, d2, d1d2) = (Σ∇Σγ)(d1, d2). Therefore we have
(Σ∇Σγ
−∇γ)e
= l(0, 0, e)
= Ω(γ)e
This completes the proof.
Now we deal with tensorial aspects of Ω. It is easy to see that
Proposition 14 Let X,Y ∈ Γ(AG). Then we have
∇(Y ∗X) = ∇Y ∗ ∇X
Now we have the following familiar form for Ω.
Theorem 15 Let X,Y ∈ Γ(AG). Then we have
Ω(Y ∗X) = ∇[X,Y ]− [∇X,∇Y ]
Proof. It suffices to note that
Ω(Y ∗X)
= Σ∇Σ(Y ∗X)
−∇(Y ∗X)
[By Proposition 13]
= ∇Σ(Y ∗X)
− Σ∇(Y ∗X)
[By Proposition 6 of §3.4 of Lavendhomme [5]]
= ∇(Σ(Y ∗X)
−X ∗ Y )− (Σ∇(Y ∗X)
−∇(X ∗ Y ))
[By Proposition 5]
= ∇(Y ∗X
− Σ(X ∗ Y ))− (∇(Y ∗X)
− Σ∇(X ∗ Y ))
[By Proposition 6 of §3.4 of Lavendhomme [5]]
= ∇(Y ∗X
− Σ(X ∗ Y ))− (∇Y ∗ ∇X
− Σ(∇X ∗ ∇Y ))
[By Proposition 14]
= ∇[X,Y ]− [∇X,∇Y ]
[By Proposition 8 of §3.4 of Lavendhomme [5]]
6 The Bianchi Identity
Let us begin with the following abstract Bianchi identity, which traces back to
Kock [4], though our version is cubical, while Kock’s one is simplicial.
Theorem 16 Let γ ∈ A2G. Let d1, d2, d3 ∈ D. We denote points β(γ(0, 0, 0)),
β(γ(d1, 0, 0)), β(γ(0, d2, 0)), β(γ(0, 0, d3)), β(γ(d1, d2, 0)), β(γ(d1, 0, d3)), β(γ(0, d2, d3))
and β(γ(d1, d2, d3)) by O, A, B, C, D, E, F and G respectively. These eight
points are depicted figuratively as the eight vertices of a cube:
ւ ↓ ւ ↓
↓ ւ ↓ ւ
Theorem 17 For each pair (X,Y ) of adjacent vertices X,Y of the cube, PXY
denotes the following arrow in H, while PY X denotes the inverse of PXY :
POA = (∇γ
0,0)d1
POB = (∇γ
0,0)d2
POC = (∇γ
0,0)d3
PAD = (∇γ
PAE = (∇γ
PBD = (∇γ
PBF = (∇γ
PCE = (∇γ
PCF = (∇γ
PDG = (∇γ
d1,d2
PEG = (∇γ
d2,d3
PFG = (∇γ
d1,d3
For any four vertices X,Y, Z,W of the cube rounding one of the six facial squares
of the cube, RXY ZW denotes PWXPZWPY ZPXY . Then we have
PAOPDAPGDRGFBDRGECFRGDAEPDGPADPOAROCEAROBFCROADB
= idO
Proof. Write over the desired identity exclusively in terms of PXY ’s, and
write off all consective PXY PY X ’s.
Now we are ready to establish the second Bianchi identity in familiar form.
Theorem 18 We have
d∇Ω = 0
Proof. Let γ, d1, d2, d3, O,A,B,C,D,E, F,G be the same as in the previous
theorem. By the very definition of Ω, we have
ROADB = Ω(γ
0)−d1d2
ROBFC = Ω(γ
0)−d2d3
ROCEA = Ω(γ
0)d1d3
Now we have the following three calculations:
PAOPDAPGDRGDAEPDGPADPOA
= PAORAEGDPOA
= ((∇γ
0,0 )d1)
−1Ω(γ1d1)d2d3(∇γ
0,0 )d1
PAOPDAPGDRGECFPDGPADPOA
= PAOPDAPGDPEGPCERCFGEPECPGEPDGPADPOA
= PAORAEGDPEAPCERCFGEPECPAERADGEPOA
= ROCEAPCOPECPAERAEGDPEAPCERCFGEPECPAERADGEPEAPCEPOCROAEC
= Ω(γ20 )d1d3((∇γ
0,0)d3)
−1{((∇γ
−1(∇γ
)d3Ω(γ
)d2d3((∇γ
−1(∇γ
Ω(γ3d3)d1d2{((∇γ
−1(∇γ
)d3Ω(γ
)−d2d3((∇γ
−1(∇γ
0,0)d3Ω(γ
0)−d1d3
= Ω(γ20 )d1d3{((∇γ
0,0)d3)
−1Ω(γ3d3)d1d2(∇γ
0,0)d3}Ω(γ
0)−d1d3
[By Proposition 2, cf. Figure (4)]
= ((∇γ
0,0 )d3)
−1Ω(γ3d3)d1d2(∇γ
0,0)d3
[By Proposition 2, cf. Figure (5)]
C ((∇γ
−1(∇γ
)d3Ω(γ
)−d2d3((∇γ
−1(∇γ
−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−→
Ω(γ3d3)d1d2 ↓ 	 ↓ Ω(γ
)d1d2
−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−→
−1(∇γ
)d3Ω(γ
)−d2d3((∇γ
−1(∇γ
)d1 C
O ((∇γ
0,0 )d3)
−1Ω(γ3d3)d1d2(∇γ
0,0)d3
−−−−−−−−−−−−−−−−−−−−−−−−−→
Ω(γ20)−d1d3 ↓ 	 ↓ Ω(γ
0)−d1d3
−−−−−−−−−−−−−−−−−−−−−−−−−→
0,0 )d3)
−1Ω(γ3d3)d1d2(∇γ
0,0)d3 O
PAOPDAPGDRGFBDPDGPADPOA
= PAOPDARDGFBPADPOA
= ROBDAPBORBDGFPOBROADB
= Ω(γ30)d1d2{((∇γ
0,0)d2)
−1Ω(γ2d2)−d1d3(∇γ
0,0 )d2}Ω(γ
0)−d1d2
= ((∇γ
0,0)d2)
−1Ω(γ2d2)−d1d3(∇γ
0,0 )d2
[By Proposition 2, cf. Figure (6)]
O ((∇γ
0,0 )d2)
−1Ω(γ2d2)−d1d3(∇γ
0,0)d2
−−−−−−−−−−−−−−−−−−−−−−−−−−→
Ω(γ30)−d1d2 ↓ 	 ↓ Ω(γ
0)−d1d2
−−−−−−−−−−−−−−−−−−−−−−−−−−→
0,0 )d2)
−1Ω(γ2d2)−d1d3(∇γ
0,0)d2 O
Therefore we have
(d∇Ω(γ))−d1d2d3
= Ω(γ10)−d2d3{((∇γ
0,0)d1)
−1Ω(γ1d1)d2d3(∇γ
0,0)d1}
Ω(γ20)d1d3{((∇γ
0,0)d2)
−1Ω(γ2d2)−d1d3(∇γ
0,0)d2}
Ω(γ30)−d1d2{((∇γ
0,0)d3)
−1Ω(γ3d3)d1d2(∇γ
0,0)d3}
= {((∇γ
0,0)d1)
−1Ω(γ1d1)d2d3(∇γ
0,0)d1}{((∇γ
0,0)d2)
−1Ω(γ2d2)−d1d3(∇γ
0,0)d2}
{((∇γ
0,0)d3)
−1Ω(γ3d3)d1d2(∇γ
0,0)d3}Ω(γ
0)−d2d3Ω(γ
0)d1d3Ω(γ
0)−d1d2
[By Proposition 2, cf. Figures (7)-(12)]
= {((∇γ
0,0)d2)
−1Ω(γ2d2)−d1d3(∇γ
0,0)d2}{((∇γ
0,0)d3)
−1Ω(γ3d3)d1d2(∇γ
0,0)d3}
{((∇γ
0,0)d1)
−1Ω(γ1d1)d2d3(∇γ
0,0)d1}Ω(γ
0)d1d3Ω(γ
0)−d2d3Ω(γ
0)−d1d2
[By Proposition 2, cf. Figures (13)-(15)]
= {PAOPDAPGDRGFBDPDGPADPOA}{PAOPDAPGDRGECFPDGPADPOA}
{PAOPDAPGDRGDAEPDGPADPOA}ROCEAROBFCROADB
= idO
O ((∇γ
0,0 )d3)
−1Ω(γ3d3)d1d2(∇γ
0,0)d3
−−−−−−−−−−−−−−−−−−−−−−−−−→
Ω(γ10)−d2d3 ↓ 	 ↓ Ω(γ
0)−d2d3
−−−−−−−−−−−−−−−−−−−−−−−−−→
0,0 )d3)
−1Ω(γ3d3)d1d2(∇γ
0,0)d3 O
O (∇γ
0,0)d2)
−1Ω(γ2d2)−d1d3(∇γ
0,0)d2
−−−−−−−−−−−−−−−−−−−−−−−−−→
Ω(γ10)−d2d3 ↓ 	 ↓ Ω(γ
0)−d2d3
−−−−−−−−−−−−−−−−−−−−−−−−−→
0,0)d2)
−1Ω(γ2d2)−d1d3(∇γ
0,0)d2 O
O ((∇γ
0,0 )d3)
−1Ω(γ3d3)d1d2(∇γ
0,0)d3
−−−−−−−−−−−−−−−−−−−−−−−−−→
Ω(γ10)−d2d3 ↓ 	 ↓ Ω(γ
0)−d2d3
−−−−−−−−−−−−−−−−−−−−−−−−−→
0,0 )d3)
−1Ω(γ3d3)d1d2(∇γ
0,0)d3 O
O ((∇γ
0,0 )d2)
−1Ω(γ2d2)−d1d3(∇γ
0,0)d2
−−−−−−−−−−−−−−−−−−−−−−−−−−→
Ω(γ20)d1d3 ↓ 	 ↓ Ω(γ
0)d1d3
−−−−−−−−−−−−−−−−−−−−−−−−−−→
0,0 )d2)
−1Ω(γ2d2)−d1d3(∇γ
0,0)d2 O
O ((∇γ
0,0 )d3)
−1Ω(γ3d3)d1d2(∇γ
0,0)d3
−−−−−−−−−−−−−−−−−−−−−−−−−→
Ω(γ20)d1d3 ↓ 	 ↓ Ω(γ
0)d1d3
−−−−−−−−−−−−−−−−−−−−−−−−−→
0,0 )d3)
−1Ω(γ3d3)d1d2(∇γ
0,0)d3 O
O ((∇γ
0,0 )d3)
−1Ω(γ3d3)d1d2(∇γ
0,0)d3
−−−−−−−−−−−−−−−−−−−−−−−−−→
Ω(γ30)−d1d2 ↓ 	 ↓ Ω(γ
0)−d1d2
−−−−−−−−−−−−−−−−−−−−−−−−−→
0,0 )d3)
−1Ω(γ3d3)d1d2(∇γ
0,0)d3 O
O ((∇γ
0,0)d1)
0,0)d1
−−−−−−−−−−−−−−−−−−−−−−−−−→
0,0)d2)
−d1d3
0,0)d2 ↓ 	 ↓ ((∇γ
0,0)d2)
−d1d3
0,0)d2
−−−−−−−−−−−−−−−−−−−−−−−−−→
0,0)d1)
0,0)d1 O
O ((∇γ
0,0)d1)
0,0)d1
−−−−−−−−−−−−−−−−−−−−−−−−−→
0,0)d3)
)d1d2(∇γ
0,0)d3 ↓ 	 ↓ ((∇γ
0,0)d3)
)d1d2(∇γ
0,0)d3
−−−−−−−−−−−−−−−−−−−−−−−−−→
0,0)d1)
0,0)d1 O
O Ω(γ10)−d2d3
−−−−−−−→
Ω(γ20)d1d3 ↓ 	 ↓ Ω(γ
0)d1d3
−−−−−−−→
Ω(γ10)−d2d3 O
This completes the proof.
References
[1] Kock, A.: Synthetic Differential Geometry, London Mathematical Society
Lecture Note Series, 51, Cambridge University Press, Cambridge, 1981.
[2] Kock, A.:Differential forms with values in groups (preliminary report),
Cahiers Top. Geom. Diff., 22 (1981), 141-148.
[3] Kock, A.:Differential forms with values in groups, Bull. Austral. Math.
Soc., 25 (1982), 357-386.
[4] Kock, A.:Combinatorics of curvature, and the Bianchi identity, Theory and
Applications of Categories, 2 (1996), 69-89.
[5] Lavendhomme, R.: Basic Concepts of Synthetic Differential Geometry,
Kluwer, Dordrecht, 1996.
[6] Mackenzie, K. C. H.:General Theory of Lie Groupoids and Lie Algebroids,
London Mathematical Society Lecture Note Series, 213, Cambridge Uni-
versity Press, Cambridge, 2005.
[7] Moerdijk, I. and Reyes, G. E.: Models for Smooth Infinitesimal Analysis,
Springer-Verlag, New York, 1991.
[8] Nishimura, H.:Another curvature in synthetic differential geometry, Bull.
Belg. Math. Soc. Simon Stevin, 7 (2000), 161-171.
[9] Nishimura, H.:The first Bianchi identity in synthetic differential geometry,
Journal of Pure and Applied Algebra, 160 (2001), 263-274.
[10] Nishimura, H.:The Lie algebra of the group of bisections, Far East Journal
of Mathematical Sciences, 24 (2007), 329-342.
[11] Nishimura, H.:Another coboundary operator for differential forms with val-
ues in the Lie algebra bundle of a group bundle, Mathematics ArXiv,
math.DG/0612067
http://arxiv.org/abs/math/0612067
	Introduction
	Preliminaries
	Synthetic Differential Geometry
	Groupoids
	Differential Forms
	Connection
	A Lift of the Connection  to microsquares
	The Curvature Form
	The Bianchi Identity

  We study the fundamental properties of curvature in groupoids within the
framework of synthetic differential geometry. As is usual in synthetic
differential geometry, its combinatorial nature is emphasized. In particular,
the classical Bianchi identity is deduced from its combinatorial one.

Introduction
The notion of curvature, which is one of the most fundamental concepts in dif-
ferential geometry, retrieves its combinatorial or geometric meaning in synthetic
differential geomety. It was Kock [4] who studied it up to the second Bianchi
identity synthetically for the first time. In particular, he has revealed the com-
binatorial nature of the second Bianchi identity by deducing it from an abstract
In [4] Kock trotted out first neighborhood relations, which are indeed to
be seen in formal manifolds, but which are no longer expected to be seen in
microlinear spaces in general. Since we believe that microlinear spaces should
play the same role in synthetic differential geometry as smooth manifolds have
done in classical differential geometry, we have elevated his ideas to a microlinear
context in [8]. However we were not so happy, because our proof of the second
Bianchi identity there appeared unnecessarily involved, making us feel that we
were somewhat off the point, though the proof was completely correct.
Recently we got accustomed to groupoids, which encouraged us to attack
the same problem once again. Within the framework of groupoids, we find it
pleasant to think multiplicatively rather than additively (cf. Nishimura [11]),
which helps grasp the nature of the second Bianchi identity firmly. Now we
are to the point. What we have to do in order to deduce the classical second
Bianchi identity from the combinatorial one is only to note some commutativity
on the infinitesimal level, though groupoids are, by and large, highly noncom-
mutative. Our present experience is merely an example of the familiar wisdom
in mathematics that a good generalization reveals the nature.
http://arxiv.org/abs/0704.1446v1
2 Preliminaries
2.1 Synthetic Differential Geometry
Our standard reference on synthetic differential geometry is Chapters 1-5 of
Lavendhomme [5]. We will work internally within a good topos, in which the
intended set R of real numbers is endowed with a cornucopia of nilpotent in-
finitesimals pursuant to the general Kock-Lawvere axiom. To see how to build
such a good topos, the reader is referred to Kock [1] or Moerdijk and Reyes [7].
Any space mentioned in this paper will be assumed to be microlinear, unless
stated to the contrary. We denote by D the set {d ∈ R | d2 = 0}, as is usual in
synthetic differential geometry.
Given a group G, we denote by AG the tangent space of G at its identity,
i.e., the totality of mappings t : D → G such that t0 is the identity of G. We
will often write td rather than t(d) for any d ∈ D. As we will see shortly, AG is
more than an R-module.
Proposition 1 For any t ∈ AG and any (d1, d2) ∈ D(2), we have
td1+d2 = td1td2 = td2td1
so that td1 and td2 commute.
Proof. By the same token as in Proposition 3 of §3.2 of Lavendhomme [5].
As an easy corollary of this proposition, we can see that
t−d = (td)
for we have (d,−d) ∈ D(2).
Proposition 2 For any t1, t2 ∈ AG, we have
(t1 + t2)d = (t2)d(t1)d = (t1)d(t2)d
for any d ∈ D, so that (t1)d and (t2)d commute.
Proof. By the same token as in Proposition 6 of §3.2 of Lavendhomme [5].
As an easy corollary of this proposition, we can see, by way of example, that
(t1)d1d2 and (t2)d1d3 commute for any d1, d2, d3 ∈ D, for we have
(t1)d1d2(t2)d1d3 = (d2t1)d1(d3t2)d1 = (d3t2)d1(d2t1)d1 = (t2)d1d3(t1)d1d2
Proposition 3 For any t1, t2 ∈ AG, there exists a unique s ∈ AG such that
sd1d2 = (t2)−d2(t1)−d1(t2)d2(t1)d1
for any d1, d2 ∈ D.
Proof. By the same token as in pp.71-72 of Lavendhomme [5].
We will write [t1, t2] for the above s.
Theorem 4 The R-module AG endowed with the above Lie bracket [·, ·] is a
Lie algebra over R.
Proof. By the same token as in our previous paper [10].
2.2 Groupoids
Groupoids are, roughly speaking, categories whose morphisms are always invert-
ible. Our standard reference on groupoids is MacKenzie [6]. Given a groupoid
G over a base M with its object inclusion map id : M → G and its source and
target projections α, β : G → M , we denote by B(G) the totality of bisections
of G, i.e., the totality of mappings σ : M → G such that α ◦ σ is the identity
mapping on M and β ◦ σ is a bijection of M onto M . It is well known that
B(G) is a group with respect to the operation ∗, where for any σ, ρ ∈ B(G),
σ ∗ ρ ∈ B(G) is defined to be
(σ ∗ ρ)(x) = σ((β ◦ ρ)(x))ρ(x)
for any x ∈ M . It can easily be shown that the space B(G) is microlinear,
provided that both M and G are microlinear, for which the reader is referred
to Proposition 6 of Nishimura [10].
Given x ∈ M , we denote by AnxG the totality of mappings γ : D
n → G
with γ(0, ..., 0) = idx and (α ◦ γ)(d1, ..., dn) = x for any (d1, ..., dn) ∈ D
n. We
denote by AnG the set-theoretic union of AnxG’s for all x ∈ M . In particular,
we usually write AxG and AG in place of AxG and AG respectively. It is easy
to see that AG is naturally a vector bundle over M . A morphism ϕ : H → G
of groupoids over M naturally gives rise to a morphism ϕ∗ : AH → AG of
vector bundles over M . As in §3.2.1 of Lavendhomme [5], where three distinct
but equivalent viewpoints of vector fields are presented, the totality Γ(AG) of
sections of the vector bundle AG can canonically be identified with the totality
of tangent vectors to B(G) at id, for which the reader is referred to Nishimura
[10]. We will enjoy this identification freely, and we dare to write Γ(AG) for
the totality of tangent vectors to B(G) at id. Given X,Y ∈ Γ(AG), we define
a microsquare Y ∗X to B(G) at id to be
(Y ∗X)(d1, d2) = Yd2 ∗Xd1
for any (d1, d2) ∈ D
Given γ ∈ An+1G and e ∈ D, we define γie ∈ A
nG (1 ≤ i ≤ n+ 1) to be
γie(d1, ..., dn) = γ(d1, ..., di−1, e, di, ..., dn)γ(0, ..., 0, e
, 0, ..., 0)−1
for any (d1, ..., dn) ∈ D
n. For our later use in the last section of this paper,
we introduce a variant of this notation. Given γ ∈ An+2G and e1, e2 ∈ D, we
define γi,je1,e2 ∈ A
nG (1 ≤ i < j ≤ n+ 2) to be
e1,e2
(d1, ..., dn) = γ(d1, ..., di−1, e1, di, ..., dj−2, e2, dj−1, ..., dn)γ(0, ..., 0, e1
, ..., e2
, 0, ..., 0)−1
Given γ ∈ A2G, we define τ1γ ∈ A
2G to be
τ1γ (d1, d2) = γ(d1, 0)
for any (d1, d2) ∈ D
2. Similarly, given γ ∈ A2G, we define τ2γ ∈ A
2G to be
γ (d1, d2) = γ(0, d2)
for any (d1, d2) ∈ D
2. Given γ ∈ A2G, we define Σγ ∈ A2G to be
(Σγ)(d1, d2) = γ(d2, d1)
for any (d1, d2) ∈ D
Any γ ∈ A2G can canonically be identified with the mapping e ∈ D 7→ γ1e ∈
AG, so that we can identify A2G and (AG)D. As is expected, this identification
enables us to define γ2 −
γ1 ∈ A
2G for γ1, γ2 ∈ A
2G, provided that γ1(0, ·) =
γ2(0, ·). Similarly, we can define γ2 −
γ1 ∈ A
2G for γ1, γ2 ∈ A
2G, provided that
γ1(·, 0) = γ2(·, 0). Given γ1, γ2 ∈ A
2G, their strong difference γ2
− γ1 ∈ AG
is defined, provided that γ1 |D(2)= γ2 |D(2). Lavendhomme’s [5] treatment of
strong difference
− in §3.4 carries over mutatis mutandis to our present context.
We note in passing the following simple proposition on strong difference
−, which
is not to be seen in our standard reference [5] on synthetic differential geometry.
Proposition 5 For any γ1, γ2, γ3 ∈ A
2G with γ1 |D(2)= γ2 |D(2)= γ3 |D(2), we
− γ1) + (γ3
− γ2) + (γ1
− γ3) = 0
2.3 Differential Forms
Given a groupoid G and a vector bundle E over the same space M , the space
Cn(G,E) of differential n-forms with values in E consists of all mappings ω
from AnG to E whose restriction to AnxG for each x ∈ M takes values in Ex
satisfying the following n-homogeneous and alternating properties:
1. We have
ω(a ·
γ) = aω(γ) (1 ≤ i ≤ n)
for any a ∈ R and any γ ∈ AnxG, where a ·
γ ∈ AnxG is defined to be
γ)(d1, ..., dn) = γ(d1, ..., di−1, adi, di+1, ...dn)
for any (d1, ..., dn) ∈ D
2. We have
ω(γ ◦Dθ) = sign(θ)ω(γ)
for any permutation θ of {1, ..., n}, where Dθ : Dn → Dn permutes the n
coordinates by θ.
3 Connection
Let π : H → G be a morphism of groupoids over M . Let L be the kernel of
π with its canonical injection ι : L → H . It is clear that L is a group bundle
over M . These entities shall be fixed throughout the rest of the paper. Thus
we have an exact sequence of groupoids as follows:
0 → L
A connection ∇ with respect to π is a morphism ∇ : AG → AH of vector
bundles over M such that the composition π∗ ◦ ∇ is the identity mapping of
AG. A connection ∇ with respect to π shall be fixed throughout the rest of the
paper. If G happens to be M ×M (the pair groupoid of M) with π being the
projection h ∈ H 7→ (α(h), β(h)) ∈ M ×M , our present notion of connection
degenerates into the classical one of infinitesimal connection.
Given γ ∈ An+1G, we define γi ∈ AG (1 ≤ i ≤ n+ 1) to be
γi(d) = γ(0, ..., 0, d
, 0, ..., 0)
for any d ∈ D. As in our previous paper [11], we have
Theorem 6 Given ω ∈ Cn(G,AL), there exists a unique d∇ω ∈ C
n+1(G,AL)
such that
((d∇ω)(γ))d1...dn+1
{(ω(γi0))d1...bdi...dn+1((∇γi)di)
−1(ω(γidi))−d1... bdi...dn+1(∇γi)di}
(−1)i
for any γ ∈ An+1G and any (d1, ..., dn+1) ∈ D
4 A Lift of the Connection ∇ to microsquares
Let us define a mapping A2G → A2H , which shall be denoted by the same
symbol ∇ hopefully without any possible confusion, to be
∇γ(d1, d2) = (∇γ
)d2(∇γ
0 )d1
for any γ ∈ A2G.
It is easy to see that
Proposition 7 For any γ ∈ A2G and any a ∈ R, we have
∇(a ·
γ) = a ·
∇(a ·
γ) = a ·
Corollary 8 For any γ1, γ2 ∈ A
2G, we have
∇(γ2 −
γ1) = ∇γ2 −
∇γ1 provided that γ1(0, ·) = γ2(0, ·);
∇(γ2 −
γ1) = ∇γ2 −
∇γ1 provided that γ1(·, 0) = γ2(·, 0).
Proof. This follows from the above proposition by Proposition 10 of §1.2 of
Lavendhomme [5].
Proposition 9 For any t ∈ A1G, we define εt ∈ A
2G to be
εt(d1, d2) = t(d1d2)
Then we have
(∇εt)(d1, d2) = (∇t)(d1d2)
for any d1, d2 ∈ D.
Proof. It suffices to note that
(∇εt)(d1, d2) = (∇(d1t))(d2) = (d1∇t)(d2) = (∇t)(d1d2)
Theorem 10 For any γ1, γ2 ∈ A
2G with γ1 |D(2)= γ2 |D(2), we have
− γ1) = ∇γ2
Proof. Let d1, d2 ∈ D. We have
(∇(γ2
− γ1))(d1d2)
= (∇ε
)(d1, d2)
[By Proposition 9]
= (∇((γ2 −
))(d1, d2)
[By Proposition 7 of §3.4 of Lavendhomme [5]]
= ((∇γ2 −
∇γ1)−
∇τ2γ1)(d1, d2)
[By Corollary 8]
= ((∇γ2 −
∇γ1)−
)(d1, d2)
(d1, d2)
[By Proposition 7 of §3.4 of Lavendhomme [5]]
= (∇γ2
−∇γ1)(d1d2)
[By Proposition 9]
Since d1, d2 ∈ D were arbitrary, the desired conclusion follows at once.
5 The Curvature Form
Proposition 11 For any γ ∈ A2G, there exists a unique t ∈ A1L such that
ι(td1d2) = ((∇γ
0)d1)
−1((∇γ1d1)d2)
−1(∇γ2d2)d1(∇γ
for any d1, d2 ∈ D.
Proof. Let η ∈ A2H to be
η(d1, d2) = ((∇γ
0 )d1)
−1((∇γ1d1)d2)
−1(∇γ2d2)d1(∇γ
for any d1, d2 ∈ D. Then it is easy to see that
η(d, 0) = η(0, d) = idα(η(0,0))
Therefore there exists unique t′ ∈ A1H such that
t′d1d2 = η(d1, d2)
Furthermore we have
π(η(d1, d2))
= π(((∇γ20 )d1)
−1)π(((∇γ1d1)d2)
−1)π((∇γ2d2 )d1)π((∇γ
0 )d2)
= ((γ20)d1)
−1((γ1d1)d2)
−1(γ2d2)d1(γ
= γ(d1, 0)
−1(γ(d1, d2)γ(d1, 0)
−1)−1γ(d1, d2)γ(0, d2)
−1γ(0, d2)
= idα(η(0,0))
Therefore there exists a unique t ∈ A1L with ι(t) = t′. This completes the
proof.
We write Ω(γ) for the above t. Now we have
Proposition 12 The mapping Ω : A2G → A1L consists in C2(G,AL).
Proof. We have to show that
Ω(a ·
γ) = aΩ(γ) (1)
Ω(a ·
γ) = aΩ(γ) (2)
Ω(Σγ) = −Ω(γ) (3)
for any γ ∈ A2G and any a ∈ R. Now we deal with (1), leaving a similar
treatment of (2) to the reader. Let d1, d2 ∈ D. We have
ι(Ω(a ·
γ))d1d2
= ((∇(a ·
γ)20)d1)
−1((∇(a ·
γ)1d1)d2)
−1(∇(a ·
γ)2d2)d1(∇(a ·1
γ)10)d2
= ((∇γ20 )ad1)
−1((∇γ1ad1)d2)
−1(∇γ2d2)ad1(∇γ
0 )d2
= ι(Ω(γ))ad1d2
= ι(aΩ(γ))d1d2
Now we deal with (3). We have
ι(Ω(Σγ))ι(Ω(γ))d1d2
= {((∇γ10 )d2)
−1((∇γ2d2)d1)
−1(∇γ1d1)d2(∇γ
0)d1}{((∇γ
0)d1)
−1((∇γ1d1)d2)
−1(∇γ2d2)d1(∇γ
0)d2}
= idα(γ(0,0))
Thie completes the proof.
We call Ω the curvature form of ∇.
Proposition 13 For any γ ∈ A2G, we have
Ω(γ) = Σ∇Σγ
Proof. As in the proof of Proposition 8 of §3.4 of Lavendhomme [5], let us
consider a function l : D2 ∨D → H given by
l(d1, d2, e) = (∇γ
)d2(∇γ
0)d1Ω(γ)e
for any (d1, d2, e) ∈ D
2∨D. Then it is easy to see that l(d1, d2, 0) = (∇γ)(d1, d2)
and l(d1, d2, d1d2) = (Σ∇Σγ)(d1, d2). Therefore we have
(Σ∇Σγ
−∇γ)e
= l(0, 0, e)
= Ω(γ)e
This completes the proof.
Now we deal with tensorial aspects of Ω. It is easy to see that
Proposition 14 Let X,Y ∈ Γ(AG). Then we have
∇(Y ∗X) = ∇Y ∗ ∇X
Now we have the following familiar form for Ω.
Theorem 15 Let X,Y ∈ Γ(AG). Then we have
Ω(Y ∗X) = ∇[X,Y ]− [∇X,∇Y ]
Proof. It suffices to note that
Ω(Y ∗X)
= Σ∇Σ(Y ∗X)
−∇(Y ∗X)
[By Proposition 13]
= ∇Σ(Y ∗X)
− Σ∇(Y ∗X)
[By Proposition 6 of §3.4 of Lavendhomme [5]]
= ∇(Σ(Y ∗X)
−X ∗ Y )− (Σ∇(Y ∗X)
−∇(X ∗ Y ))
[By Proposition 5]
= ∇(Y ∗X
− Σ(X ∗ Y ))− (∇(Y ∗X)
− Σ∇(X ∗ Y ))
[By Proposition 6 of §3.4 of Lavendhomme [5]]
= ∇(Y ∗X
− Σ(X ∗ Y ))− (∇Y ∗ ∇X
− Σ(∇X ∗ ∇Y ))
[By Proposition 14]
= ∇[X,Y ]− [∇X,∇Y ]
[By Proposition 8 of §3.4 of Lavendhomme [5]]
6 The Bianchi Identity
Let us begin with the following abstract Bianchi identity, which traces back to
Kock [4], though our version is cubical, while Kock’s one is simplicial.
Theorem 16 Let γ ∈ A2G. Let d1, d2, d3 ∈ D. We denote points β(γ(0, 0, 0)),
β(γ(d1, 0, 0)), β(γ(0, d2, 0)), β(γ(0, 0, d3)), β(γ(d1, d2, 0)), β(γ(d1, 0, d3)), β(γ(0, d2, d3))
and β(γ(d1, d2, d3)) by O, A, B, C, D, E, F and G respectively. These eight
points are depicted figuratively as the eight vertices of a cube:
ւ ↓ ւ ↓
↓ ւ ↓ ւ
Theorem 17 For each pair (X,Y ) of adjacent vertices X,Y of the cube, PXY
denotes the following arrow in H, while PY X denotes the inverse of PXY :
POA = (∇γ
0,0)d1
POB = (∇γ
0,0)d2
POC = (∇γ
0,0)d3
PAD = (∇γ
PAE = (∇γ
PBD = (∇γ
PBF = (∇γ
PCE = (∇γ
PCF = (∇γ
PDG = (∇γ
d1,d2
PEG = (∇γ
d2,d3
PFG = (∇γ
d1,d3
For any four vertices X,Y, Z,W of the cube rounding one of the six facial squares
of the cube, RXY ZW denotes PWXPZWPY ZPXY . Then we have
PAOPDAPGDRGFBDRGECFRGDAEPDGPADPOAROCEAROBFCROADB
= idO
Proof. Write over the desired identity exclusively in terms of PXY ’s, and
write off all consective PXY PY X ’s.
Now we are ready to establish the second Bianchi identity in familiar form.
Theorem 18 We have
d∇Ω = 0
Proof. Let γ, d1, d2, d3, O,A,B,C,D,E, F,G be the same as in the previous
theorem. By the very definition of Ω, we have
ROADB = Ω(γ
0)−d1d2
ROBFC = Ω(γ
0)−d2d3
ROCEA = Ω(γ
0)d1d3
Now we have the following three calculations:
PAOPDAPGDRGDAEPDGPADPOA
= PAORAEGDPOA
= ((∇γ
0,0 )d1)
−1Ω(γ1d1)d2d3(∇γ
0,0 )d1
PAOPDAPGDRGECFPDGPADPOA
= PAOPDAPGDPEGPCERCFGEPECPGEPDGPADPOA
= PAORAEGDPEAPCERCFGEPECPAERADGEPOA
= ROCEAPCOPECPAERAEGDPEAPCERCFGEPECPAERADGEPEAPCEPOCROAEC
= Ω(γ20 )d1d3((∇γ
0,0)d3)
−1{((∇γ
−1(∇γ
)d3Ω(γ
)d2d3((∇γ
−1(∇γ
Ω(γ3d3)d1d2{((∇γ
−1(∇γ
)d3Ω(γ
)−d2d3((∇γ
−1(∇γ
0,0)d3Ω(γ
0)−d1d3
= Ω(γ20 )d1d3{((∇γ
0,0)d3)
−1Ω(γ3d3)d1d2(∇γ
0,0)d3}Ω(γ
0)−d1d3
[By Proposition 2, cf. Figure (4)]
= ((∇γ
0,0 )d3)
−1Ω(γ3d3)d1d2(∇γ
0,0)d3
[By Proposition 2, cf. Figure (5)]
C ((∇γ
−1(∇γ
)d3Ω(γ
)−d2d3((∇γ
−1(∇γ
−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−→
Ω(γ3d3)d1d2 ↓ 	 ↓ Ω(γ
)d1d2
−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−→
−1(∇γ
)d3Ω(γ
)−d2d3((∇γ
−1(∇γ
)d1 C
O ((∇γ
0,0 )d3)
−1Ω(γ3d3)d1d2(∇γ
0,0)d3
−−−−−−−−−−−−−−−−−−−−−−−−−→
Ω(γ20)−d1d3 ↓ 	 ↓ Ω(γ
0)−d1d3
−−−−−−−−−−−−−−−−−−−−−−−−−→
0,0 )d3)
−1Ω(γ3d3)d1d2(∇γ
0,0)d3 O
PAOPDAPGDRGFBDPDGPADPOA
= PAOPDARDGFBPADPOA
= ROBDAPBORBDGFPOBROADB
= Ω(γ30)d1d2{((∇γ
0,0)d2)
−1Ω(γ2d2)−d1d3(∇γ
0,0 )d2}Ω(γ
0)−d1d2
= ((∇γ
0,0)d2)
−1Ω(γ2d2)−d1d3(∇γ
0,0 )d2
[By Proposition 2, cf. Figure (6)]
O ((∇γ
0,0 )d2)
−1Ω(γ2d2)−d1d3(∇γ
0,0)d2
−−−−−−−−−−−−−−−−−−−−−−−−−−→
Ω(γ30)−d1d2 ↓ 	 ↓ Ω(γ
0)−d1d2
−−−−−−−−−−−−−−−−−−−−−−−−−−→
0,0 )d2)
−1Ω(γ2d2)−d1d3(∇γ
0,0)d2 O
Therefore we have
(d∇Ω(γ))−d1d2d3
= Ω(γ10)−d2d3{((∇γ
0,0)d1)
−1Ω(γ1d1)d2d3(∇γ
0,0)d1}
Ω(γ20)d1d3{((∇γ
0,0)d2)
−1Ω(γ2d2)−d1d3(∇γ
0,0)d2}
Ω(γ30)−d1d2{((∇γ
0,0)d3)
−1Ω(γ3d3)d1d2(∇γ
0,0)d3}
= {((∇γ
0,0)d1)
−1Ω(γ1d1)d2d3(∇γ
0,0)d1}{((∇γ
0,0)d2)
−1Ω(γ2d2)−d1d3(∇γ
0,0)d2}
{((∇γ
0,0)d3)
−1Ω(γ3d3)d1d2(∇γ
0,0)d3}Ω(γ
0)−d2d3Ω(γ
0)d1d3Ω(γ
0)−d1d2
[By Proposition 2, cf. Figures (7)-(12)]
= {((∇γ
0,0)d2)
−1Ω(γ2d2)−d1d3(∇γ
0,0)d2}{((∇γ
0,0)d3)
−1Ω(γ3d3)d1d2(∇γ
0,0)d3}
{((∇γ
0,0)d1)
−1Ω(γ1d1)d2d3(∇γ
0,0)d1}Ω(γ
0)d1d3Ω(γ
0)−d2d3Ω(γ
0)−d1d2
[By Proposition 2, cf. Figures (13)-(15)]
= {PAOPDAPGDRGFBDPDGPADPOA}{PAOPDAPGDRGECFPDGPADPOA}
{PAOPDAPGDRGDAEPDGPADPOA}ROCEAROBFCROADB
= idO
O ((∇γ
0,0 )d3)
−1Ω(γ3d3)d1d2(∇γ
0,0)d3
−−−−−−−−−−−−−−−−−−−−−−−−−→
Ω(γ10)−d2d3 ↓ 	 ↓ Ω(γ
0)−d2d3
−−−−−−−−−−−−−−−−−−−−−−−−−→
0,0 )d3)
−1Ω(γ3d3)d1d2(∇γ
0,0)d3 O
O (∇γ
0,0)d2)
−1Ω(γ2d2)−d1d3(∇γ
0,0)d2
−−−−−−−−−−−−−−−−−−−−−−−−−→
Ω(γ10)−d2d3 ↓ 	 ↓ Ω(γ
0)−d2d3
−−−−−−−−−−−−−−−−−−−−−−−−−→
0,0)d2)
−1Ω(γ2d2)−d1d3(∇γ
0,0)d2 O
O ((∇γ
0,0 )d3)
−1Ω(γ3d3)d1d2(∇γ
0,0)d3
−−−−−−−−−−−−−−−−−−−−−−−−−→
Ω(γ10)−d2d3 ↓ 	 ↓ Ω(γ
0)−d2d3
−−−−−−−−−−−−−−−−−−−−−−−−−→
0,0 )d3)
−1Ω(γ3d3)d1d2(∇γ
0,0)d3 O
O ((∇γ
0,0 )d2)
−1Ω(γ2d2)−d1d3(∇γ
0,0)d2
−−−−−−−−−−−−−−−−−−−−−−−−−−→
Ω(γ20)d1d3 ↓ 	 ↓ Ω(γ
0)d1d3
−−−−−−−−−−−−−−−−−−−−−−−−−−→
0,0 )d2)
−1Ω(γ2d2)−d1d3(∇γ
0,0)d2 O
O ((∇γ
0,0 )d3)
−1Ω(γ3d3)d1d2(∇γ
0,0)d3
−−−−−−−−−−−−−−−−−−−−−−−−−→
Ω(γ20)d1d3 ↓ 	 ↓ Ω(γ
0)d1d3
−−−−−−−−−−−−−−−−−−−−−−−−−→
0,0 )d3)
−1Ω(γ3d3)d1d2(∇γ
0,0)d3 O
O ((∇γ
0,0 )d3)
−1Ω(γ3d3)d1d2(∇γ
0,0)d3
−−−−−−−−−−−−−−−−−−−−−−−−−→
Ω(γ30)−d1d2 ↓ 	 ↓ Ω(γ
0)−d1d2
−−−−−−−−−−−−−−−−−−−−−−−−−→
0,0 )d3)
−1Ω(γ3d3)d1d2(∇γ
0,0)d3 O
O ((∇γ
0,0)d1)
0,0)d1
−−−−−−−−−−−−−−−−−−−−−−−−−→
0,0)d2)
−d1d3
0,0)d2 ↓ 	 ↓ ((∇γ
0,0)d2)
−d1d3
0,0)d2
−−−−−−−−−−−−−−−−−−−−−−−−−→
0,0)d1)
0,0)d1 O
O ((∇γ
0,0)d1)
0,0)d1
−−−−−−−−−−−−−−−−−−−−−−−−−→
0,0)d3)
)d1d2(∇γ
0,0)d3 ↓ 	 ↓ ((∇γ
0,0)d3)
)d1d2(∇γ
0,0)d3
−−−−−−−−−−−−−−−−−−−−−−−−−→
0,0)d1)
0,0)d1 O
O Ω(γ10)−d2d3
−−−−−−−→
Ω(γ20)d1d3 ↓ 	 ↓ Ω(γ
0)d1d3
−−−−−−−→
Ω(γ10)−d2d3 O
This completes the proof.
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math.DG/0612067
http://arxiv.org/abs/math/0612067
	Introduction
	Preliminaries
	Synthetic Differential Geometry
	Groupoids
	Differential Forms
	Connection
	A Lift of the Connection  to microsquares
	The Curvature Form
	The Bianchi Identity
