2. nition 1.1. A groupoid G is a small category where all arrows are
isomorphisms. In other words G consists of two sets Ob(G) := V;Mor(G) :=
A (vertices/arrows), two maps s; t : A ! V (source/target), a map id : V !
A and an associative composition law
A V A = f(y; x) 2 A Aj s(y) = t(x)g ! A; (y; x)7! y x
satisfying the condition that idp is an identity for all morphisms starting or
ending at p, and any arrow has a two-sided inverse.
Remark 1.2. Let G be a groupoid, and pick vertices pi 2 V in any isomor-phism
class i 2 V= =
. Denote by 0(G; fpig) G the full sub category with
vertices pi.
This inclusion i is an equivalence of categories.
Let us analyse this a bit more closely, we claim that there is a functor
r : G ! 0(G; fpig) such that r i =
id0(G) and i r =idG. We can
arrange r in such a way the
3. rst isomorphism is the identity, but the second
one involves the choice of arrows xp : p ! pi for all p into some representative
pi.
If G is connected, i.e. all objects of G are isomorphic, there is only one
object pi = p to choose and 0(G; p) will be a group. So everyhting we do
with the goupoid will have an equivalent in group theory.
Nevertheless, working with groupoids we avoid choices (pi; p ! pi) which
will turn out to be very handy for our considerations.
De
4. nition 1.3. A linear representation of a groupoid G is a functor
F : G ! VectC; p7! Fp; (x : p ! q)7! x : Fp ! Fq:
In other words, we pick a vector space Fp for each vertex p and a linear
map x : Fp ! Fq (necesserily an isomorphism) for each arrow x : p ! q,
satisfying (y x) = y x, and (idp) = idFp . Linear representations form a
category in the obvious way, which we denote by RepC(G). More generally
we can de
5. ne a representation of G in an arbitary category in the same way.
Remark 1.4. Let 0(G; fpig) be a as in 1.2 the inculsion 0(G; fpig) G
induces an equivalence of categories
: RepC(G) ! RepC(0(G; fpig)):
This equivalence is non-canonical, as the inverse functor and the natural
isomorphism ! idRep(G) involve choices.
Date: May 2009.
1
7. nition 1.5. Let X be a topological space. The fundamental groupoid
(X) of X has the points of X as vertices, and as arrows [
] : p ! q (based-)
homotopy classes of paths starting at p and ending at q. The composition
of arrows is de
8. ned as concatenation of paths1
([
] : q ! r; [] : p ! q)7! [
] [] : p ! r:
The identities idp are the constant paths based at the points p, inverses are
provided by reversing the direction of the path.
Remark 1.6. The fundamental groupoid (X) is connected if and only if X
is path connected. In this case we pick a point x 2 X, then the fundamental
group 1(X; x) of X is just 0((X); x).
Example 1.7. To any covering space f : Y ! X there is a canonical Set-
representation of (X), de
9. ned by
p ! Yp = f1(fpg); ([
] : p ! q)7! T[
] : Yp ! Yq
where T
sends a point r in Yp to ~
(1) 2 Yq where ~
is a/the lift of
starting
at r.
2. Local systems
De
10. nition 2.1. A local system L of rank r on a topological space X is
a sheaf of C-vector spaces, locally isomorphic to the constant sheaf Cr.
A morphism of local systems is a morphism of the underlying sheaves of
vector spaces. In this way local systems form category, which we denote
by LocC(X). We can de
11. ne local systems of (free) abelian groups/k-vector
spaces/sets in the same way.
De
12. nition 2.2. Let L be a local system on X and
: p ! q a path in
X. The inverse image
1(L) is a local system on the unit interval [0; 1].
Note that the stalks
1(L)t and L
(t) are canonoically isomorphic and the
restriction maps induce isomorphisms
Lp H0([0; 1];
1(L)) ! Lq
We denote the composition by PTL
: Lp ! Lq and call it the parallel
transport map. If
0 is a homotopic path, then PTL
= PTL
0 . This can be
seen using the fact that local systems on [0; 1] [0; 1]R are trivial.
Proposition 2.3. Let X be locally simply connected, then the functor
PT : LocC(X) ! RepC((X));L7! PTL
where PTL maps p7! Lp and [
]7! PTL
has an inverse
Loc : RepC((X)) ! LocC(X); F7! Loc(F)
where Loc(F)(U)
Q
p2U Fp is f (vp) j 8
: p ! q in U : [
]v(p) = v(q) g,
such that there are canonical (!) isomorphisms: idLoc ! Loc PT and
PT Loc ! idRep.
1This convention diers form the usual one used by topologists; their
is our
. The Advantage of our de
13. nition is, that parallel-transport/monodromy have the
right associativity behaviour, i.e. become honest representations (as opposed to anti
representations).
14. GROUPOIDS, LOCAL SYSTEMS AND DIFFERENTIAL EQUATIONS 3
Proof. It is easy to see that Loc(F) and Rep(L) are in fact local systems,
resp. representation of the fundamental groupoid.
To construct the
15. rst isomorphism, we start with a representation F of the
fundamental groupoid (X). The stalk Loc(F)p is caononical isomorphic
to Fp for any p 2 X.
Indeed, elements of Loc(F)p are represented by (U; (sq)q2U) where U is a
neighbourhood of p in X, and sq 2 Fq staisfy the condition that [
](sq1) =
(sq2) for all paths
: q1 ! q2 contained in U. Two representatives are
equivalent if they agree on some small neighbourhood of p.
As we assume that any open neighbourhood contains a simply connected
one, we can restrict to simply connected neighbourhoods. But on a simply
connected U we can extend vectors sp 2 Fp uniquely to Loc(F)(U), and
hence the restriction Loc(F)(U) ! Fp is an isomorphism.
To get the desired natural isomorphism of representations Rep(Loc(F)) !
F we only need to check that the following diagramm commutes:
Loc(F)p
PTLoc(F)
/ Loc(F)q
Fp
[
]
/ Fq
Dividing
in smaller pices, we can assume
to be contained in a simply
connected neighbourhood. But there the statement is obvious.
Note that the construction did not involve any choices.
For the other isomorphism, we start with a local system L. We assign
to it the representation Rep(L) = (p7! Lp; [
]7! PTL
). Let v 2 L(U)
be a section of L. The images in the stalks vq 2 Lq for q 2 U satisfy
vq1 = vq2 for any path
: q1 ! q2 in U since we can pullback the section
to H0([0; 1];
1L) which restrics appropiately at the endpoints.
In this way we constructed a morphism idLoc ! Loc Rep. It is obvi-ous,
that this morpism is an isomorphism at the level of stalks, hence it
is an isomorphism. Note again, that we did not make any choices in the
construction.
Corollary 2.4. Let X be path connected, and locally simply connected, then
there are equivalences of categories:
Loc(X) ! Rep((X)) ! Rep(1(X; x0)):
This, more precise verison of the well known equivalence Loc(X) =
Rep(1(X; x0)), allows us to explicitly construct an inverse Functor:
Rep(1(X; xo)) ! Loc(X):
Namely, choose paths
p : x0 ! p to all points p 2 X, and use them to induce
from a given representation V of 1(X; x0) a representation (X) ! V ectC:
Set p7! Vp := V , and [
] := ([
q]1 [
] [
p]) : Vp ! Vq where [
] : p ! q
is a path and hence [
q]1 [
] [
p] : x0 ! x0 lies in the fundamental gruop.
Then use the equivalence of Rep((X)) to Loc(X) which we described in
detail above.
16. 4 HEINRICH HARTMANN
Remark 2.5. The Functors Loc and Rep are compatible with tensored prod-ucts
and inner Homs.
3. Vector bundles with Connection
There is jet another equivalence of categroies which will be important for
us. Let B be a Riemann surface.
De
17. nition 3.1. Recall that a connection on a holomorphic vector bundle
E over B is a C-linear map of sheaves
r : E !
B
E
satisfying the Leibnitz rule
8f 2 OB; s 2 E : r(fs) = df
s + f r(s)
Note that r is automatically
at, since we are on a Riemann surface. A
at morphism : (E;r) ! (E0;r0) between two vector bundles with con-nections
is a OB-linear map : E ! E0 commuting with the connections,
i.e.
r0 = (id
) r:
The category objects (E;r) and
at morphisms is denoted by FlatVect(B).
Proposition 3.2. There is a canonical equivalence of categories
FlatVect(B) ! LocC(B):
This equivalence is given by the functors:
FlatSect : (E;r)7! kerr E
and
L7! (OB
C L; r : f
s7! df
s):
Moreover the obvious natural isomorphisms
OB
C FlatSect ! idFlatVect; idLoc ! FlatSect(OB
C )
do not depend on choices.
De
18. nition 3.3. Let E be a vector bundle of rank E with connection r. A
global section 2 H0(B;E) is called locally cyclic vector at p if are locally
de
20. elds 1; : : : ; r1 2 Tp such that the derivatives
; r1; r2r1; : : : ;rr1 : : :r1
restrict to a basis on the
21. ber E(p) = Ep=mpEp. We call a cyclic vector if
it is locally cyclic everywhere.
Remark 3.4. As we are on a riemann surface this condition is equivalent to:
For all 2 T , with (p)6= 0, the iterated derivatives
; r; r2
; : : : ;rr1
restrict to a basis of the
22. ber E(p).
If we are given a global section 2 H0(B;E), we can ask:
Where is cyclic?
25. ne i := ri
. The set where sections
0; : : : ; k are lineary dependent is closed, and there will be a minimal k
rk(E) such that they are lineary dependent everywhere.
We would like to conclude, that there is a global linear relation between
these sections. But there are in general too few global holomorphic functions
for this to hold true. So we should at least allow meromorphic coecients.
Let M be the sheaf of meromorphic functions on B. This is a (non-constant)
sheaf of
26. elds, which contains the structure sheaf OB. Assume
that M
E is trivial, i.e. E is generated by meromorphic sections.2
Then H0(M
E) is a vector space of dimension rkE over the
27. eldM(B)
of global meromorphic functions. And there will be a relation
Xk
(1) aii = 0
i=0
with coecients in ai 2M(B).
Let U be the dense open subet of B where all ai are have no poles and
ak6= 0. A priori it is still possible that 0; : : : ; k1 are lineary dependent
at some points in U. The special choice of the sections allows us to say a
bit more. Linear independence means, that
Wr() = 0 ^ ^ k1 2 k(E)
does not vanish. There is an induced connection on k(E) which we can
apply to this section. Using the relation (1) we compute
rWr() = (r0) ^ ^ k1 + + 0 ^ ^ (rk1)
= 0 ^ ^ k2 ^ k
= (ak1=ak)Wr():
Now on the set V U where 6= 0 there is a dual-one form 2
B(V )
satisfying () = 1 and we can form the connection
r0 := r (ak1=ak)
id
of k(E). The equation above just says Wr() is parallel for r0. But if a
parallel section vanishes somwhere, it has to be zero. As Wr() is non-zero
at generic points, we conclude it is non-zero everywhere on V . We thus
proved the following proposition.
Proposition 3.5. Let E be a vector bundle generated by meromorphic sec-
tions, 2 H0(B;E) and 2 H0(B; TB); 6= 0. Let
Xk
i=0
ai i = 0
be a minimal relation with meromorphic coecients ai 2M(B) beween the
derivatives i = ri
.
Then 0; : : : ; k1 are linear independet over
U = f t 2 B j a0; : : : ; ak have no pole at t, ak(t)6= 0; (t)6= 0: g
2This is always the case in the algebraic situation where B is a smooth, variety of
dimension 1 over C, and E an algebraic vector bundle with connection.
28. 6 HEINRICH HARTMANN
Moreover is a cyclic vector for the vectorbundle F = OU 0; : : : ; k1
spanned by this sections.
4. Differential equations
Let B be a riemann surface.
De
29. nition 4.1. A dierential dierential on B of order n is a C-linear
morphism of sheaves of the from
(2) ai i(f)
D : OB ! OB; f7! n(f)
nX1
i=0
where ai 2 H0(B;OB) are holomorphic functions and 2 H0(B; TB) is
a nowhere-vanishing, holomorphic vector
30. eld, which acts by directional
derivative on functions.
Remark 4.2. If = f for another vector
31. eld , equation (2) changes into
fnn
nX1
i=0
bi i
for some functions bi, which can be determined by iterated use of the Leibitz
rule. We see, that we need to divide by fn in order to recover the form (2),
which is crucial for Cauchy's theorem below. This is why we insist to be
nowhere vanishing.
There are of course more general de
32. nitions of a dierential equation, see
for example [?], for our application B C P1 this generality is sucient.
Proposition 4.3 (Cauchy). The solutions to a dierential equation form a
local system. More precisely if D : OB ! OB is a dierential equation of
order n the kernel
B U7! LD(U) = f f 2 OB(U) j Df = 0 g
is a sheaf of C-vector spaces locally isomorphic to Cn
B.
Remark 4.4. Even if ai are algebraic functions, the solutions of a dierential
equation will not be algebraic. This can be seen at even the simplest example
@tf = f on B = C, which is solved by the exponential function. Hence
working with sheaves of holomorphic functions is crucial.
Remark 4.5. As LD OB parallel transport in LD along a path
de
33. nes
an analytic continuation of a local solution of D. It follows, that solutions
of dierential equations can be extended along arbitary paths.
There is also a
at vector bundle that we can associate to D.
De
34. nition 4.6. Let D = n
Pn1
i=0 ai i be a dierential equation. De
35. ne
ED := OB e0; : : : ; en1 to be the trivial vector bundle of rank n, and
together with the connection on E, de
36. ned by setting
rei :=
(
ei+1 for i n 1 Pn1
i=0 aiei for i = n 1
(3)
The relation between ED and LD is clari
38. GROUPOIDS, LOCAL SYSTEMS AND DIFFERENTIAL EQUATIONS 7
Proposition 4.7. Let D be a dierential equation. The local systems LD
and Loc(ED
_) are canonically isomorphic.
Proof. Given a
at co-section : ED ! OB, the function f = (e0) is a
solution of the dierential equation since
nf = (rn
nX1
e0) = (
i=0
aiei) =
nX1
i=0
ai(ei) =
nX1
i=0
ai(ri
e0) =
nX1
i=0
aiif:
Conversly, we de
39. ne a map
LD ! E_; f7!
nX1
i=0
(if)i
where i is the dual basis of the vector bundle E = OB e0; : : : ; en1 .
Flatness is checked easily:
nX1
r(
i=0
(if)i) =
nX1
i=0
(i+1f)i +
nX1
i=0
(if)(ri)
=
Xn
i=1
(if)i1 +
nX1
i=0
(if)(i1 ain1)
= (nf)n1
nX1
i=0
(if)ain1 = 0:
Given a dierential equation D, the section e0 2 H0(B;ED) is always
cyclic. We can also go in the other direction:
De
40. nition 4.8. Given a
at vector bundle (E;r) of rank n, a cyclic vec-tor
2 H0(B;E) and a nowhere vanishing vector
41. eld . The dierential
equation associated to this datum is
D(E;r; ; ) := n
nX1
i=0
ai i : OB ! OB
where ai are the coecient in the expansion n =
Pn1
i=0 aii, i := ri
.
Question: Does D(E;r; ; ) really depend on ?
