Three themes in the work of Charles Ehresmann:
Local-to-global; Groupoids; Higher dimensions. *
Ronald Browny
University of Wales, Bangor.
February 26, 2006
Abstract
This paper illustrates the themes of the title in terms of: van Kampen *
*type theorems for the
fundamental groupoid; holonomy and monodromy groupoids; and higher homotop*
*y groupoids.
Interaction with work of the writer is explored. 1
1 Introduction
It is a pleasure to honour Charles Ehresmann by giving a personal account of so*
*me of the major
themes in his work which interact with mine. I hope it will be useful to sugges*
*t how these themes are
related, how the pursuit of them gave a distinctive character to his aims and h*
*is work, and how they
influenced my own work, through his writings and through other people.
Ehresmmann's work is so extensive that a full review would be a great task, *
*which to a considerable
extent is covered by Andr'ee Ehresmann in her commentaries in the collected wor*
*ks [24]. His wide
vision is shown by his description of his overriding aim as: `To find the struc*
*ture of everything'. `To
find structure' is related to the Bourbaki experience and aim, in which he was *
*a partner. A description
of a new structure is in some sense a development of part of a new language: th*
*e aim of doing this
contrasts with that of many, who feel that the development of mathematics is ma*
*inly guided by the
solution of famous problems.
The notion of structure is also related to the notion of analogy. It one of *
*the triumphs of category
theory in the 20th century to make progress in unifying mathematics through the*
* finding of analogies
between the behaviours of structures across different areas of mathematics.
___________________________________*
Expansion of an invited talk given to the 7th Conference on the Geometry and*
* Topology of Manifolds: The Mathe-
matical Legacy of Charles Ehresmann, Bedlewo 8.05.2005-15.05.2005 (Poland)
yPartially supported by a Leverhulme Emeritus Fellowship 2002-2004
1KEYWORDS: Ehresmann, local-to-global; fundamental groupoid; van Kampen theor*
*em; holonomy and monodromy;
higher homotopy groupoids.
MSClass2000: 01A60,53C29,81Q70,22A22,55P15
1
This theme is elaborated in the article [15]. That article argues that many *
*analogies in mathemat-
ics, and in many other areas, are not between objects themselves but between th*
*e relations between
objects. Here we mention only the notion of pushout, which we use later in disc*
*ussing the van Kampen
Theorem. A pushout has the same definition in different categories even though *
*the construction of
pushouts in these categories may be wildly different. Thus a concentration on t*
*he constructions rather
than on the universal properties may lead to a failure to see the analogies.
Ehresmann developed new concepts and new language which have been very influ*
*ential in mathe-
matics: I mention only fibre bundles, foliations, holonomy groupoid, germs, jet*
*s, Lie groupoids. There
are other concepts whose time perhaps is just coming or has yet to come: includ*
*ed here might be
ordered groupoids, multiple categories.
In this direction of developing language, we can usefully quote G.-C. Rota [*
*37, p.48]:
"What can you prove with exterior algebra that you cannot prove without it*
*?" Whenever
you hear this question raised about some new piece of mathematics, be assu*
*red that you are
likely to be in the presence of something important. In my time, I have he*
*ard it repeated
for random variables, Laurent Schwartz' theory of distributions, ideles an*
*d Grothendieck's
schemes, to mention only a few. A proper retort might be: "You are right*
*. There is
nothing in yesterday's mathematics that could not also be proved without i*
*t. Exterior
algebra is not meant to prove old facts, it is meant to disclose a new wor*
*ld. Disclosing new
worlds is as worthwhile a mathematical enterprise as proving old conjectur*
*es.
2 Local-to-global questions
Ehresmann developed many new themes in category theory. One example is structur*
*ed categories,
with principal examples those of differentiable categories and groupoids, and o*
*f multiple categories. His
work on these is quite disparate from the general development of category theor*
*y in the 20th century,
and it is interesting to search for reasons for this. One must be the fact that*
* he used his own language
and notation. Another is surely that his early training and motivation came fr*
*om analysis, rather
than from algebra, in contrast to the origins of category theory in the work of*
* Eilenberg, Mac Lane
and of course Steenrod, centred on homology and algebraic topology. Part of the*
* developing language
of category theory became essential in those areas, but other parts, such as th*
*at of algebraic theories,
groupoids, multiple categories, were not used till fairly recently.
It seems likely that Ehresmann's experience in analysis led him to the major*
* theme of local-to-
global questions. I first learned of this term from Dick Swan in Oxford in 1957*
*-58, when as a research
student I was writing up notes of his Lectures on the Theory of Sheaves [40]. D*
*ick explained to me
that two important methods for local-to-global problems were sheaves and spectr*
*al sequences_he
was thinking of Poincar'e duality, which is discussed in the lecture notes, and*
* the more complicated
Dolbeaut's theorem for complex manifolds. But in truth such problems are centra*
*l in mathematics,
science and technology. They are fundamental to differential equations and dyna*
*mical systems, for
example. Even deducing consequences of a set of rules is a local-to-global pro*
*blem: the rules are
2
applied locally, but we are interested in the global consequences.
My own work on local-to-global problems arose from writing an account of the*
* Seifert-van Kampen
theorem on the fundamental group. This theorem can be given as follows, as fir*
*st shown by R.H.
Crowell:
Theorem 2.1 [19] Let the space X be the union of open sets U, V with intersect*
*ion W , and suppose
W, U, V are path connected. Let x0 2 W . Then the diagram of fundamental group *
*morphisms induced
by inclusions
ss1(W, x0)i__//ss1(U, x0)
j|| || *
* (1)
fflffl| fflffl|
ss1(V, x0)__//ss1(X, x0)
is a pushout of groups.
Here the `local parts' are of course U, V put together with intersection W and *
*the result describes
completely, under the open set and connectivity conditions, the nonabelian fund*
*amental group of the
global space X. This theorem is usually seen as a necessary part of basic algeb*
*raic topology, but one
without higher dimensional analogues.
In writing the first 1968 edition of the book [7], I noted that to compute t*
*he fundamental group
of the circle one had to develop something of covering space theory. Although t*
*hat is an excellent
subject in its own right, I became irritated by this detour. After some time, I*
* found work of Higgins
on groupoids, [25], which defined free products with amalgamation of groupoids,*
* and this led to a
more general formulation of theorem 2.1 as follows:
Theorem 2.2 [3] Let the space X be the union of open sets U, V with intersecti*
*on W , and suppose
W, U, V are path connected. Let X0 be a subset of W meeting each path component*
* of W . Then the
diagram of fundamental group morphisms induced by inclusions
ss1(W, X0)i__//ss1(U, X0)
j|| || *
* (2)
fflffl| fflffl|
ss1(V, X0)___//ss1(X, X0)
is a pushout of groupoids.
Here ss1(X, X0) is the fundamental groupoid of X on a set X0 of base points: so*
* it consists of homotopy
classes rel end points of paths in X joining points of X0\ X.
In the case X is the circle S1, one chooses U, V to be slightly extended se*
*micircles including
X0 = {+1, -1}. The point is that in this case W = U \ V is not path connected a*
*nd so it is not clear
where to choose a single base point. The day is saved by hedging one's bets, an*
*d using two base points.
The proof of theorem 2.1 uses the same tricks as to prove theorem 2.2, but in a*
* broader context. In
order to compute fundamental groups from this theorem, one can set up some gene*
*ral combinatorial
3
groupoid theory, see [7, 26]. A key feature of this theory is the groupoid I, t*
*he indiscrete groupoid
on two objects 0, 1, which acts as a unit interval object in the category of gr*
*oupoids. It also plays a
r^ole analogous to that of the infinite cyclic group C in the category of group*
*s. One then compares the
pushout diagrams, the first in spaces, the second in groupoids:
{0, 1}___//_{0} {0, 1}___//_{0}
| | | |
| | | | *
* (3)
fflffl| fflffl| fflffl| |fflffl
[0, 1]___//_S1 I ___u__//_C
to see how this version of the van Kampen Theorem gives an analogy between the *
*geometry, and the
algebra provided by the notion of groupoid. This kind of result is seen as `cha*
*nge of base' in [5].
The fundamental group is a kind of anomaly in algebraic topology because of *
*its nonabelian nature.
Topologists in the early part of the 20th century were aware that:
o the non commutativity of the fundamental group was useful in applications;
o for path connected X there was an isomorphism
H1(X) ~=ss1(X, x)ab;
o the abelian homology groups existed in all dimensions.
Consequently there was a desire to generalise the nonabelian fundamental group *
*to all dimensions.
In 1932 ~Cech submitted a paper on higher homotopy groups ssn(X, x) to the I*
*CM at Zurich, but it
was quickly proved that these groups were abelian for n > 2, and on these groun*
*ds ~Cech was persuaded
to withdraw his paper, so that only a small paragraph appeared in the Proceedin*
*gs [17]. We now see
the reason for the commutativity as the result (Eckmann-Hilton) that a group in*
*ternal to the category
of groups is just an abelian group. Thus the vision of a non commutative higher*
* dimensional version
of the fundamental group has since 1932 been generally considered to be a mirag*
*e.
Theorem 2.2 is also anomalous: it is a colimit type theorem, and so yields c*
*omplete information on
the fundamental groups which are contained in it, even in the non connected cas*
*e, whereas the usual
method in algebraic topology is to relate different dimensions by exact sequenc*
*es or even spectral
sequences, which usually yield information only up to extension. Thus exact seq*
*uences by themselves
cannot show that a group is given as an HNN-extension: however such a descripti*
*on may be obtained
from a pushout of groupoids, generalising the pushout of groupoids in diagram 3.
It was then found that the theory of covering spaces could be given a nice e*
*xposition using the
notion of covering morphism of groupoids. Even later, it was found by Higgins a*
*nd Taylor [27] that
there was a nice theory of orbit groupoids which gave models of orbit spaces.
The objects of a groupoid add a `spatial component' to group theory, which i*
*s essential in many
applications. This is evident in many parts of Ehresmann's work. Another view o*
*f this anomalous
success of groupoids is that they have structure in two dimensions, 0 with the *
*objects and 1 with the
4
arrows. We have a colimit type theorem for this larger structure, and so a good*
* model of the geometry.
Useful information on fundamental groups is carried by the fundamental groupoid.
It is therefore natural to seek for higher homotopy theory algebraic models *
*which:
o have structure in a range of dimensions;
o contain useful information on classical invariants, and
o satisfy van Kampen type theorems.
That is, we seek nonabelian methods for higher dimensional local-to-global prob*
*lems in homotopy
theory. We return to this theme in section 4, which gives an indication of some*
* of the motivation for
the writer for higher dimensional algebra.
3 Holonomy and monodromy
Once I had been led to groupoids by Philip Higgins, and having been told by G.W*
*. Mackey in 1967 of
his work on ergodic groupoids, [31], it was natural to consider topological gro*
*upoids and differentiable
groupoids, and to seek their properties and applications, [10, 11].
So I came across papers of Ehresmann, [21], and of Jean Pradines, [35], and *
*in 1981 I visited Jean
in Toulouse under British Council support, to try and understand something of h*
*is papers. I saw the
grand vision of the whole scheme of generalising the relation between Lie group*
*s and Lie algebras to
a relation between Lie groupoids and Lie algebroids, which has now become a lar*
*ge theory, see for
example [30]. We concentrated on his first paper [35], which states theorems bu*
*t gives no indication
of proofs. What I found remarkable was first of all the beautiful constructions*
* Jean explained, which
seemed to me clear in principle, and then the fact that he gave for holonomy an*
*d monodromy universal
properties: these are fairly rare in differential topology.
Jean was interested in the monodromy principle, which involves the following*
* situation:
M(G,;WF);
w
w w | FF 0
i0w p|| FfF
ww | F
w | F
w i fflffl| F##
W ___________//____________77_______________________*
*__________________GH
________________________________________________*
*____________________________________________________________
_____________________________________________*
*_____________________________________________________________________________*
*_____________________________________________________________________________*
*_____________________________________________________________________________*
*________________________
f
Here G is a topological groupoid, W is an open subset of G which contains the s*
*et of identities G0
of G, and i is the inclusion. The aim is to find a topological groupoid M(G), c*
*alled the monodromy
groupoid of G, with morphism of topological groupoids p : M(G, W ) ! G and a `l*
*ocal morphism'
i0: W ! M(G, W ) such that for any local morphism f : W ! H to a topological gr*
*oupoid H, f
5
`locally extends' to f0: M(G, W ) ! H. The existence such M(G, W ), p, i0with t*
*his universal property
is called the monodromy principle.
This idea was expressed in Chevalley's famous book on `Lie groups', [18], fo*
*r the case the groupoid
G is the trivial groupoid X x X on a manifold X, when M(G, W ) becomes, for suf*
*ficiently small W ,
the fundamental groupoid of X and p : M(G, W ) ! G is the `anchor map' as defin*
*ed in Mackenzie's
book [30]. Pradines is the first, I believe, to see how to extend it this notio*
*n to the case of a general
groupoid, as announced in [35].
We have to explain the term `local morphism'. Note that if u, v 2 W and uv i*
*s defined in G, this
does not necessarily mean that uv 2 W . So the algebraic structure that W inher*
*its from G is what is
called a `local groupoid structure'. A local morphism is therefore one that pre*
*serves the local groupoid
structure.
It is easy to construct algebraically the groupoid M(G, W ) and p : M(G, W )*
* ! G so that it has
the required universal property, algebraically, as follows. The source, target *
*and identity maps for G
induce on W the structure of a reflexive graph. So one forms F (W ), the free g*
*roupoid on this reflexive
graph, the reflexive condition ensuring that identities in W become identities *
*in F (W ). There is a
function j : W ! F (X) which sends w 2 W to the corresponding generator [w] 2 F*
* (W ) and then one
factors the groupoid F (W ) by the relations [u][v] = [uv] for all u, v 2 W suc*
*h that uv is defined in
W . This defines M(G, W ), and the function j induces the local morphism i0, wh*
*ich is injective since
it is determined by the inclusion i.
The problem is how to topologise M(G, W ) so that it becomes a topological g*
*roupoid for which
the monodromy principle is satisfied not only algebraically but also with regar*
*d to continuity, or
differentiability.
Pradines solved this by a beautiful holonomy construction, which he explaine*
*d to me in 1981
during my visit, and which develops Ehresmann's ideas.
He took the view that the pair (M(G, W ), W 0), where W 0= i0(W ), should be*
* regarded as a
`locally Lie groupoid', and that this raised the general problem of when a loca*
*lly topological groupoid
is extendible, i.e. is obtained from an open subset containing the identities o*
*f a topological groupoid.
(The term used in [35] is morceau d'un groupo"ide diff'erentiable.)
We therefore start again and consider a groupoid G and a subset W of G such *
*that W contains
the identities of G and W has the structure of topological space and even of a *
*manifold. We ask: what
conditions should be put on W which ensure that G can be given the structure of*
* topological or Lie
groupoid for which W is an open set? In short, we ask is the pair (G, W ) exten*
*dible?
This is a classical question in the case G is a group, and the answer is giv*
*en in books on topological
groups. A topology on G is obtained by taking as subbase the sets gU for all U *
*open in W and g 2 G.
The conditions for this to give a topological group structure on G are fairly m*
*ild, and are loosely
that the algebraic operations on G should be as continuous on W as can be expec*
*ted given that these
operations are defined only partially on W .
One of the reasons this works is that in a topological group, the left multi*
*plication operator
Lg : G ! G, given for g 2 G by u 7! gu, is a homeomorphism and so maps open set*
*s to open sets.
6
This property of Lg no longer holds when G is a topological groupoid, becaus*
*e Lg is not defined
on all of G. This reflects the considerable change in moving from groups to gro*
*upoids, that is, from
algebraic operations always defined to those only partially defined. This is no*
*t a loss of information:
the wider concept has greater powers of expression.
There is also a wide range of examples where (G, W ) is not extendible, many*
* coming from the
theory of foliations. For example, if F is the foliation of the M"obius Band M *
*given by circles going
once or twice round the band, and R is the equivalence relation determined by t*
*he leaves, then R is
a subset of M x M and as such is a topological groupoid. But R is not a submani*
*fold of M x M,
since it has self intersections. However the foliation structure does determine*
* a locally Lie groupoid
(R, W ). The argument for both these facts is spelled out in [14].
Given a groupoid ff, fi : G ' G0, then an admissible section s : G0 ! G of f*
*f satisfies ff O s = 1
and fi O s : G0 ! G0 is a bijection. We follow Mackenzie in [30] in calling s a*
* bisection. We can also
regard s as a homotopy 1 ' a where a : G ! G is an automorphism of G. This inte*
*rpretation has
intuitive value, and is suggestive for analogues in higher dimensions, as appli*
*ed in [12].
Now suppose G is as above but G0 is also a topological space. Then a local b*
*isection of G is a
function s : U ! G with U open in G0 such that ff O s = 1U and fi O s maps U ho*
*meomorphically to
its image which is also open in G0. There is an `Ehresmannian composition' s * *
*t of local bisections.
We first make clear that if g : x ! y and h : y ! z in G then their composition*
* in G is hg : x ! z.
So the composition of local bisections is
(s * t)(x) = (sfit(x))(tx)
for x 2 G0. This means that in general the domain of s * t is smaller than that*
* of t, and may even
be empty. This composition makes the set of local bisections into an inverse se*
*migroup. Recall that
this is a semigroup in which for each element s there is a unique element s0, c*
*alled a relative inverse
for s, such that s0ss0= s0, ss0s = s. Pseudogroups, a concept first defined by *
*Ehresmann in [20], give
examples of such structures. We write this inverse semigroup as (G): of course*
* it depends on the
topology of G0.
A left partial `adjoint' operation Lsof the local bisection s on G is define*
*d by Ls(g) = (sfig)g, g 2 G.
It is easy to prove that if G is a topological groupoid and s is a continuous l*
*ocal bisection, then Ls
maps open sets of G to open sets of G. Thus the important observation is that f*
*or adjoint operations
on topological or Lie groupoids it is not enough to rely on the elements of G: *
* we need the local
continuous or smooth bisections, which are kind of `tubes' rather than `element*
*s', to transport the
local structure of G.
Now suppose given a pair (G, W ) such that G is a groupoid, G0 W G, W is*
* a manifold,
and the groupoid operations are `as smooth as possible' on W . By a smooth loca*
*l bisection of (G, W )
we mean a local bisection s of G such that s takes values in W and is smooth. T*
*he set of smooth
local bisections forms a subset (r)(W ) where r denotes the class of different*
*iability of the manifold
M = G0.
Pradines' key definition is to form the sub-inverse semigroup (r)(G, W ) of*
* (G) generated by
(r)(W ). My interpretation is that an element of (r)(W ) can be thought of as*
* a local procedure and
7
an element of (r)(G, W ) can be thought of as an iteration of local procedures*
*. Thus an iteration of
local procedures need not be local, and this is one of the basic intuitions of *
*non trivial holonomy.
We say that (G, W ) is sectionable if for all w 2 W there is a smooth local *
*bisection s in W whose
domain includes ffw and with sffw = w.
The next step is to form from (r)(G, W ) the associated sheaf of germs J(r)*
*(G, W ): the elements
of this are written [s]x where s 2 (r)(G, W ) and x 2 dom s. The inverse semi*
*group structure
on (r)(G, W ) induces a groupoid structure on J(r)(G, W ). This contains a su*
*bgroupoid J0 whose
elements are germs [s]x such that fisx = x and there is a neighbourhood U of x *
*such that s|U 2
(r)(W ); in words, s is an iteration of local procedures about x which is stil*
*l a local procedure. It is a
proposition that J0(G, W ) is a normal subgroupoid of J(r)(G, W ). The holonomy*
* groupoid Hol(G, W )
is defined to be the quotient groupoid J(r)(G, W )=J0. The class of [s]x in the*
* holonomy groupoid is
written ~~x. There is a projection p : Hol(G, W ) ! G given by ~~~~x 7! s(x).
The intuition is that first of all W embeds in Hol(G, W ), by w 7! ffw, w*
*here f is a local smooth
bisection such that fffw = w, and second that Hol(G, W ) has enough local secti*
*ons for it to obtain a
topology by translation of the topology of W .
Let s 2 c(G, W ). We define a partial function oes : W ! Hol(G, W ). The do*
*main of oes is the set
of w 2 W such that fiw 2 dom(s). The value oesw is obtained as follows. Choose *
*a smooth bisection
f through w. Then we set
oesw = ~~~~fiwffw= ffw.
Then oesw is independent of the choice of the local section f. It is proved in *
*detail in [2] that these oes
form a set of charts for Hol(G, W ) making it into a Lie groupoid with a univer*
*sal property.
The books [30, 32] argue that the most efficient treatment of the holonomy g*
*roupoid of a foliation
is via the monodromy groupoid, which is itself defined using the fundamental gr*
*oupoid of the leaves.
However there are counter arguments.
One fact is that their arguments do not so far obtain a monodromy principle,*
* which is obtained
by the opposite route in [35, 13]. Thus there is loss of a universal principle,*
* with its potentiality for
enabling analogies.
The second problem is that the route through the fundamental groupoid is bas*
*ed on paths, and
so on the standard notion of a topological space, and its exemplification as a *
*manifold.
The Pradines' approach gives a clear realisation of the intuitive idea of it*
*eration of local procedures,
without requiring the notion of path to `carry' these procedures, as happens fo*
*r example in the usual
process of analytic continuation. It is possible that this idea would lead to *
*wider applications of
non abelian groupoid like methods for local-to-global problems. For example, th*
*e following picture
illustrates a chain of local procedures from a to b:
a
b
We would like to be able to define such a chain, and equivalences of such chain*
*s, without resource to
8
the notion of `path'. The claim is that a candidate for this lies in the constr*
*uctions of Ehresmann and
Pradines for the holonomy groupoid.
Here are some final questions in this area.
Can one use these ideas in other situations to obtain monodromy (i.e. analo*
*gues of `universal
covers') in situations where paths do not exist but `iterations of local proced*
*ures' do? It is even
possible that holonomy may exist in wider situations, but not monodromy.
How widely useful is the notion of `locally Lie groupoid' as a context for d*
*escribing local situations?
Is there a locally Lie groupoid (G, W ) where G is an action groupoid?
A further point is that Pradines was very keen on having a theory for germs *
*of locally Lie groupoids.
It is in these terms that the theorems in [35] are stated. Such work is not con*
*sidered in [2, 13].
One could consider other structures on W and ask for bisections which preser*
*ve these structures.
Simple examples for degrees of differentiability, due to Pradines [36], are giv*
*en in [2]. One might
consider other geometric structures, such as Riemannian, or Poisson. `I,cen an*
*d I, in looking for
notions of double holonomy, have considered in [12] a double groupoid G and lin*
*ear bisections.
4 Higher dimensions
After writing out the proofs of theorem 2.2 a number of times to make the expos*
*ition clear, it became
apparent to me in 1966 that the method of proof ought to extend to higher dimen*
*sions, if one had the
right gadgets. So this was an idea of a proof in search of a theorem. The first*
* search was for a higher
dimensional version of the fundamental groupoid on a set of base points.
It was at this point that I found Ehresmann's book, [22]. It was difficult t*
*o understand, but it
did contain a definition of double category, and a key example, the double cate*
*gory 2C of commuting
squares in a category C. What caused problems for me was to find any constructi*
*on of a fundamental
double groupoid of a space which really contained 2-dimensional homotopy inform*
*ation. In fact
a solution to this problem was not published till 2002, [9]. The construction *
*of higher homotopy
groupoids went in 1975 in a different direction, using pairs of spaces, and the*
*n filtered spaces, in work
with Philip Higgins, which is surveyed in [6].
As an intermediate step, I decided in 1970 to investigate the notion of doub*
*le groupoid purely
algebraically, to see whether the putative homotopy double groupoid functor wou*
*ld take values in a
category of some interest.
Work with Chris Spencer, [16], found a relationship between double groupoids*
* and the crossed
modules introduced by J.H.C. Whitehead to discuss the second relative homotopy *
*group and its
boundary @ : ss2(X, A.a) ! ss1(A, a). We found a functor from crossed modules *
*to a certain kind
of double groupoid, which was later called edge symmetric, in that the groupoid*
*s of vertical and of
horizontal edges are the same. The next question was what kind of double groupo*
*ids arose in this
way?
At the same time we were looking at conjectured proofs of a vaguely formulat*
*ed 2-dimensional van
9
Kampen type theorem. It was clear that we needed to generalise commutative squa*
*res to commutative
cubes in the context of double groupoids, in such a way that any composite of c*
*ommutative cubes is
commutative.
A square in a groupoid
__a_//_
___/2/
c|| |b| fflffl|
fflffl|fflffl|//_1
d
is commutative if and only if ab = cd, or, equivalently, a = cdb-1. We wanted a*
* similar expression to
the last in the case of a cube. If we fold flat five of the faces of a cube we *
*get a net such as the first
of the following: _____ _____
| |
|- | ||- ||
| 1 | || 1||
____|____|___ ____||__||____ __/2/_
| | | | | | | |
|-2 | | | |-2 | | | 1fflffl|
|___|____|___| |___|___|___|__
| | || ||
| | || ||
|____| ||__||_
which is not a composable set of squares. However in the second diagram we have*
* noted by double lines
adjacent pairs of edges which are the same. Therefore we must assume we have ca*
*nonical elements,
called connections, to fill the corner holes as in the diagram:
_________________|_|_________________
____| | ______
____|-1|| |____
____|___|_______
| | | |
|-2 | | |
|___|___|___|_______
____| | ____
____||_ | __|____
____|___|____________________________
The rules for connections include the transport law:
|________||
|__ | __| _____
|| | __| | __ |
|___|___|_= | | |
| | __| |____|
|| || | |
|___|___|_
borrowed from a law for path connections found in Virsik's paper [41].
Virsik was a student of Ehresmann. He left Czechoslovakia at the time of the*
* Russian invasion,
and went to Australia. There he met Kirill Mackenzie, and suggested to him Lie *
*groupoids and Lie
algebroids as a PhD topic. Hearing of Kirill's work, I managed to get British C*
*ouncil support for him
to visit Bangor in 1986, and this led to his continuing to work in the UK.
In the paper [41], Virsik introduces the notion of path connection on a prin*
*cipal bundle p : E ! B
as follows. Let G be the Ehresmann groupoid EE-1 ' B of the principal bundle. T*
*hus G(b, b0) can be
identified with the bundle maps Eb! Eb0of the fibres over b, b0respectively. Le*
*t B be the category
of Moore paths on B, and let G be the category of Moore paths on G. This may b*
*e combined easily
to give the structure of a double category.
10
Virsik defines a path connection for the bundle E to be a function : B ! *
* G satisfying
two conditions. One is invariance under reparametrisation, and the other is kno*
*wn as the transport
law. Chris Spencer and I were amazed when this law was found to be exactly righ*
*t as a law on the
connections we needed for the boundary of a cube, and these have been central i*
*n work on cubical
higher groupoids ever since. So we found an equivalence between crossed modules*
* and what we then
called special double groupoids with special connections, [16].
Ehresmann had earlier shown in [23] that a 2-category gave rise to a double *
*category of quintettes.
Spencer showed in [38] that this gave an equivalence between 2-categories and e*
*dge symmetric double
categories with connections. This has been generalised to all dimensions in [1]*
*, and there has been
recent further work on commutative cubes by Higgins in [28] and Steiner [39].
What has yet to be accomplished is to find relations between the higher orde*
*r connections in
differential geometry and the geometry of cubes.
One of the advantages of cubes for local-to-global questions is that they ha*
*ve an easy to define
notion of multiple composition. Analogous ideas in the globular case present co*
*nceptual and technical
difficulties. Multiple compositions allow easily an algebraic inverse to subdiv*
*ision, and this is a key to
certain local-to-global results, as outlined for example in [6]. However the al*
*gebraic relations between
the cubical, globular, and (in the groupoid case) crossed complexes, are an ess*
*ential part of the picture.
It is possible to dream of a unification of all these themes in a way which *
*I believe Ehresmann
would have favoured, but there still seems quite a way to go to realise this. W*
*ork is in progress with
Jim Glazebrook and Tim Porter on aspects of this, [8], and many others are work*
*ing on stacks, gerbes
and higher order groupoids.
References
[1]Al-Agl, F.A., Brown, R. and Steiner, R., Multiple categories: the equivale*
*nce of a globular and
cubical approach, Adv. in Math. 170 (2002) 711-118.
[2]Aof, M.E.-S.A.-F. and Brown, R., The holonomy groupoid of a locally topolog*
*ical groupoid, Top. Appl.
47, 1992, 97-113.
[3]Brown R., Groupoids and Van Kampen's theorem, Proc. London Math. Soc. (3) 1*
*7 (1967) 385-401.
[4]Brown R., From groups to groupoids: a brief survey, Bull. London Math. Soc.*
* 19 (1987) 113-134.
[5]Brown, R., Homotopy theory, and change of base for groupoids and multiple g*
*roupoids, Applied categor-
ical structures, 4 (1996) 175-193.
[6]Brown, R., Crossed complexes and homotopy groupoids as non commutative tool*
*s for higher dimensional
local-to-global problems, Proceedings of the Fields Institute Workshop on C*
*ategorical Structures for De-
scent and Galois Theory, Hopf Algebras and Semiabelian Categories, Septembe*
*r 23-28, Fields Institute
Communications 43 (2004) 101-130.
[7]Brown, R., Topology and groupoids, BookSurge LLC (2006).
[8]Brown, R., Glazebrook, J.F. and Porter, T., Smooth crossed complexes I : no*
*nabelian cohomology
of a cover, (in preparation).
11
[9]Brown, R., Hardie, K., Kamps, H. and Porter, T., The homotopy double groupo*
*id of a Hausdorff
space, Theory and Applications of Categories, 10 (2002) 71-93.
[10]Brown, R., and Hardy, J.P.L., Topological groupoids I: universal constructi*
*ons, Math. Nachr. 71
(1976) 273-286.
[11]Brown, R., Danesh-Naruie, G., and Hardy, J.P.L., Topological groupoids II: *
*covering morphisms
and G-spaces, Math. Nachr. 74 (1976) 143-156.
[12]Brown, R. and `Ic,en, `I., Towards a 2-dimensional notion of holonomy, Adva*
*nces in Math, 178 (2003)
141-175.
[13]Brown, R. and Mucuk, O., The monodromy groupoid of a Lie groupoid, Cah. Top*
*. G'eom. Diff. Cat. 36
(1995) 345-369.
[14]Brown, R. and Mucuk, O., Foliations, locally Lie groupoids, and holonomy, C*
*ah. Top. G'eom. Diff. Cat.
37 (1996) 61-71.
[15]Brown, R. and Porter. T., Category Theory: an abstract setting for analogy *
*and comparison, Advanced
Studies in Mathematics and Logic, volume on `What is category theory?', edi*
*ted G. Sica (to appear, 2006).
[16]Brown, R. and Spencer, C.B., Double groupoids and crossed modules, Cah. Top*
*. G'eom. Diff. 17 (1976)
343-362.
[17]~Cech, E., H"oherdimensionale homotopiegruppen, Verhandlungen des Internat*
*ionalen Mathematiker-
Kongresses Zurich, Band 2, (1932) 203.
[18]Chevalley, C.,Theory of Lie groups, Princeton University Press, 1946.
[19]Crowell, R. H., On the van Kampen theorem, Pacific J. Math., 9 (1959) 43-50.
[20]Ehresmann, C., Structures locales et structures infinit'esimales, C.R.A.S. *
*Paris 274 (1952) 587-589.
[21]Ehresmann, C., Cat'egories topologiques et cat'egories diff'erentiables, Co*
*ll. G'eom. Diff. Glob. Bruxelles,
1959, 137-150.
[22]Ehresmann, C., Cat'egories et structures, Dunod, Paris, (1965).
[23]Ehresmann, C., Cat'egories doubles des quintettes: applications covariantes*
*, C.R.A.S. Paris 256 (1963)
1891-1894.
[24]Ehresmann, C., Oeuvres compl`etes et comment'ees: Amiens, 1980-84, edited a*
*nd commented by Andr'ee
Ehresmann.
[25]Higgins, P. J., Presentations of groupoids, with applications to groups, Pr*
*oc. Cambridge Philos. Soc., 60
(1964) 7-20.
[26]Higgins, P. J., Categories and groupoids, Van Nostrand Mathematical Studies*
* 32, (1971), Reprinted in
Theory and Application of Categories Reprints (2005).
[27]Higgins, P.J. and Taylor, J., The fundamental groupoid and homotopy crossed*
* complex of an orbit
space, Category Theory Proceedings, Gummersbach, 1981, Lecture Notes in Mat*
*h. 962, edited K.H.Kamps
et al, Springer, Berlin, pp. 115-122, 1982.
[28]Higgins, P. J., Thin elements and commutative shells in cubical !-categorie*
*s with connections, Theory
and Applications of Categories, 14 (2005) 60-74.
[29]Mackenzie, K.C.H., Double Lie algebroids and second order geometry, II, Adv*
*. Math. 154 (2000) 46-75.
12
[30]Mackenzie, K.C.H., General theory of Lie groupoids and Lie algebroids, Lond*
*on Math. Soc. Lecture Note
Series 213, Cambridge University Press, 2005.
[31]Mackey, G.W., Ergodic theory and virtual groups, Math. Ann. 166 (1966) 187-*
*207.
[32]Moerdijk, I. and Mr~cun, J., Introduction to foliations and Lie groupoids, *
*Cambridge Studies in Ad-
vanced Mathematics 91, Cambridge University Press, 2003.
[33]Mucuk, O., Covering groups of non-connected topological groups and the mono*
*dromy groupoid of a topo-
logical groupoid, PhD Thesis, University of Wales, 1993.
[34]Phillips, J., The holonomic imperative and the homotopy groupoid of a folia*
*ted manifold, Rocky Mountain
J. Math., 17 (1987) 151-165.
[35]Pradines, J., Th'eorie de Lie pour les groupoides diff'erentiables, relatio*
*n entre propri'et'es locales et globales,
Comptes Rendus Acad. Sci. Paris, S'er A, 263 (1966), 907-910.
[36]Pradines, J., Private communication, 1982.
[37]Rota, G.-C., Indiscrete thoughts, Birkh"auser.
[38]Spencer, C. B., An abstract setting for homotopy pushouts and pullbacks, Ca*
*hiers Topologie G'eom.
Diff'erentielle, 18 (1977) 409-429.
[39]Steiner, R., Thin fillers in the cubical nerves of omega-categories, arXiv:*
*math.CT/0601386.
[40]Swan, R.G., The theory of sheaves, Notes by Ronald Brown, Oxford Mathematic*
*al Inst., 1958; Chicago
University Press 1964.
[41]Virsik, J., On the holonomity of higher order connections, Cah. Top. G'eom.*
* Diff. 12 (1971) 197-212.
13
~~