Galois theory and a new homotopy double groupoid
of a map of spaces
Ronald Brown* George Janelidzey
August 27, 2002
UWB Maths Preprint 02.18
Abstract
The authors have used generalised Galois Theory to construct a homotopy*
* double groupoid of a
surjective fibration of Kan simplicial sets. Here we apply this to constru*
*ct a new homotopy double
groupoid of a map of spaces, which includes constructions by others of a 2*
*-groupoid, cat1-group or
crossed module. An advantage of our construction is that the double groupo*
*id can give an algebraic
model of a foliated bundle.1
Introduction
Our aim is to develop for any map q : M ! B of topological spaces the construct*
*ion and properties of
a new homotopy double groupoid which has the form of the left hand square in th*
*e following diagram,
while the right hand square gives a morphism of groupoids:
___s___// i1q
æ2(q)______//ß1(M)oo___//_ß1(B) *
* (1)
||OO| t |O|O| |O|O|
||| ||| |||
||| ||| |||
||| ||| |||
ffflffl|flffl|fflffl|fflffl|fflffl|fflffl||s//_||
Eq (q)_______//Moo___q____//B
t
where:
ß1(M) is the fundamental groupoid of M;
Eq(q) is the equivalence relation determined by q; and
s, t are the source and target maps of the groupoids.
Note that qs = qt and (ß1q)s = (ß1q)t, so that æ2(q) is seen as a double groupo*
*id analogue of Eq(q).
*_____________________________________
Mathematics Division, School of Informatics, University of Wales, Dean St., *
*Bangor, Gwynedd LL57 1UT, U.K.
email: r.brown@bangor.ac.uk
yMathematics Institute, Georgian Academy of Sciences, Tbilisi, Georgia.
12000 Maths Subject Classification: 18D05, 20L05, 55 Q05, 55Q35
1
This double groupoid contains the 2-groupoid associated to a map defined by*
* Kamps and Porter
in [16], and hence also includes the cat1-group of a fibration defined by Loday*
* in [17], the 2-groupoid
of a pair defined by Moerdijk and Svensson in [19], and the classical fundament*
*al crossed module of
a pair of pointed spaces defined by J.H.C. Whitehead. Advantages of our constru*
*ction are:
(i) it contains information on the map q, and
(ii) we get different results if the topology of M is varied to a finer topolog*
*y.
In particular, we can apply this construction in the case M is foliated by repl*
*acing the topology on
M by a finer one so that ß1M is replaced by the fundamental groupoid of the fol*
*iation.
This applies in particular to the Möbius Band with its standard foliation b*
*y circles. We can extract
from this double groupoid a small version D(M) with only three vertices, and wh*
*ich seems to represent
well many properties of the Möbius Band. It has basic vertices, edges and squar*
*es as follows:
B _`__//_C ___/2/
fflffl|
j ||ff |,| 1
fflffl|fflffl|
AO__'_//AOOO
, ||fi |j|
| |
C _OE_//_B
Note that the vertical groupoid for O1 is an indiscrete groupoid, while the hor*
*izontal groupoid for O2
contains a copy of the infinite cyclic group, since there are compositions
a1O2a2O2. .O.2an
where the aiare alternately (-1ff) and fi.
The idea for this double groupoid arose from the Generalised Galois Theory *
*of Janelidze [14, 15],
which under certain conditions gives a Galois groupoid from a pair of adjoint f*
*unctors. The standard
fundamental group arises from the adjoint pair between topological spaces and s*
*ets given by discrete
and ß0, see for example [8]. The adjoint pair between simplicial sets and cros*
*sed complexes given
by nerve and ß1 was studied in [9] and shown to lead to a Galois double groupoi*
*d of a fibration
of simplicial sets. We are now giving a topological version of this constructi*
*on. We show that if
p : E ! B is a Serre fibration then the fundamental groupoid ß1(E) has an addit*
*ional compatible
groupoid structure arising from the equivalence relation Eq(p) defined by the m*
*ap p; these two groupoid
structures define a double groupoid which we write fl(p), since it is defined b*
*y methods of Galois theory.
The double groupoid æ(q) arises by pullback by i where q = pi is the usual fact*
*orisation of any map
through a homotopy equivalence i and a fibration p. However the proof of the re*
*lation of æ(q) with
classical notions and compositions is tricky, and so is given in some detail. A*
* further reason for this
detail is the possibility that a modification of this construction could be use*
*d in association with the
`thin fundamental groupoids' and their smooth structures in differential geomet*
*rical situations, as
exemplified by Mackaay and Picken in [18].
Here is some background to the search for higher groupoid models of homotop*
*ical structures (for
more detailed references, see [3]). Geometers in the early part of the 20th cen*
*tury were aware that in the
connected case the first homology group was the fundamental group made abelian,*
* and that homology
2
groups existed in all positive dimensions. Further, the fundamental group gave *
*more information in
geometric and analytic contexts than did the first homology group. They were th*
*erefore interested in
seeking higher dimensional versions of the non abelian fundamental group. E. ~C*
*ech submitted to the
1932 ICM at Zurich a paper on higher homotopy groups, using maps of spheres. Ho*
*wever these groups
were quickly proved to be abelian in dimensions > 1, and on this ground ~Cech w*
*as asked to withdraw
his paper, so that only a small paragraph appeared [12]. Thus the dream of thes*
*e topologists seemed
to fail, and was widely felt to be a mirage, although the abelian higher homoto*
*py groups became and
still are very important.
J.H.C. Whitehead in the 1940s introduced the notion of crossed module, usin*
*g the boundary of the
second relative homotopy group of a pair and the action of the fundamental grou*
*p. He and Mac Lane
showed that crossed modules classified (connected) homotopy 2-types. Crossed mo*
*dules are indeed
more complicated than groups, and they make a good candidate for `2-dimensional*
* groups'.
In the 1960s, Brown introduced the fundamental groupoid of a space on a set*
* of base points,
and the writing of his 1968 book on topology suggested to him that all of 1-dim*
*ensional homotopy
theory was better expressed in terms of groupoids rather than groups. This rai*
*sed the question of
the putative value of groupoids in higher homotopy theory. A relation of certai*
*n double groupoids to
crossed modules was worked out with C.B. Spencer in the early 1970s, and this s*
*howed that double
groupoids are indeed more complicated than groups. A definition of a homotopy d*
*ouble groupoid of
a pair of pointed spaces was made with P.J. Higgins in 1974, and exploited to o*
*btain a 2-dimensional
Van Kampen type theorem for this double groupoid, and hence for Whitehead's cro*
*ssed module of a
pair (see [4]). The double groupoid constructed in [4] is edge symmetric and ha*
*s a connection, and so
is not the same as that constructed here.
A classification of certain double groupoids is given in [11], but this doe*
*s not yield much information
for the double groupoid considered here. Thus there is still a way to go in the*
* understanding and in
the use of double groupoids.
Higher homotopy groupoids were defined by Brown and Higgins for a filtered *
*space in [6], and by
Loday for an n-cube of spaces in [17]; his catn-groups were shown there to mode*
*l connected homotopy
(n + 1)-types. These higher groupoid methods yield new calculations in homotop*
*y theory through
higher order Van Kampen theorems [6, 10], as well as suggesting new algebraic c*
*onstructions.
1 Galois groupoids
op
Later we will be considering the category C = Sets of simplicial sets and the *
*fundamental groupoid
functor I = ß1 : C ! X from the category C to the category X = Grpdof (small) g*
*roupoids. Further,
C, an internal category in C, will be the particular simplicial category (actua*
*lly groupoid) Eq(p) which
is the equivalence relation (in C) determined by p where p : E ! B is a surject*
*ive fibration of Kan
complexes. Here we give first the general result, using this notation.
Let I : C ! X be an arbitrary functor between categories C and X with pullb*
*acks, and let
i ____//_____//_ j
C = C2 ____//_//_C1//_C0oo_ *
* (2)
be an internal category in C. We recall
3
Proposition 1.1 Suppose the canonical morphisms
I(C1xC0C1) ! I(C1) xI(C0)I(C1) *
* (3)
I(C1xC0C1xC0C1) ! I(C1) xI(C0)I(C1) xI(C0)I(C1) *
* (4)
are isomorphisms. Then:
(a) i j
_____// ____//_
I(C) = I(C2)_____////_I(C1)//_I(C0)oo_ *
* (5)
is an internal category in X;
(b) if C is a groupoid, then so is I(C).
For a morphism p : E ! B in C, let Eq(p) =
____//_ ____//_
(E xB E) xE (E xB E) E xB E xB E____//_//_E xB_E//_Eoo_ *
* (6)
be the equivalence relation corresponding to p (=kernel pair of p) regarded as *
*an internal groupoid in
C. Applying Proposition 1.1 with C = Eq(p), we obtain:
Corollary 1.2 Suppose the canonical morphisms
I((E xB E) xE (E xB E)) ! I(E xB E) xI(E)I(E xB E) *
* (7)
I(E xB E) xE (E xB E) xE (E xB E)) ! I(E xB E) xI(E)I(E xB E) xI(E)I(E xB E*
*) (8)
are isomorphisms. Then I(Eq(p)) =
____//_ ____//_
I((E xB E) xE (E xB E)) I(E xB E xB E)___//_//_I(E xB_E)//_I(E)oo_*
* (9)
is an internal groupoid in X.
This fact, which goes back to A. Grothendieck's observation "the fundamental gr*
*oupoids are to be
defined as quotients of equivalence relations", is used in categorical Galois t*
*heory and its various
special cases (see [1], [4], and other references in [1]), to define the Galois*
* groupoid of (E, p) as
GalI(E, p) = I(Eq(p)). *
* (10)
In particular this applies to the following situation studied by the authors be*
*fore (see Proposition 3.5
in [2]):
*
* op
Proposition 1.3 Let I : C ! X be the fundamental groupoid functor from the cate*
*gory C = Sets
of simplicial sets to the category X = Grpdof (small) groupoids, and p : E ! B *
*a surjective fibration
of Kan complexes. Then the morphisms (6) and (7) are isomorphisms and so the Ga*
*lois groupoid (9)
is well defined. Since the internal groupoids in Grpd are the same as double gr*
*oupoids, it is a double
groupoid.
4
2 From simplicial sets to topological spaces
Consider the diagram
oRo__ __I_//_
Top ____//_Setsoopo_Grpd *
* (11)
dSdIIII OO H tt::
II |y ttt
r IIII||tttit
t
where
o Top is the category of topological spaces, R is the geometric realisation *
*functor, and S is its
right adjoint, usually called the singular complex functor;
o I ` H is the adjoint pair used in [2], i.e. I is the fundamental groupoid *
*functor, and H the nerve
functor;
o y is the Yoneda embedding, r and i are the restrictions of R and I respect*
*ively along y; explicitly,
r is the singular simplex functor and i carries finite ordinals to codiscr*
*ete groupoids on the same
sets of objects.
By the universal property of the Yoneda embedding, the two adjunctions of t*
*he row are uniquely
(up to isomorphisms) determined by r and i; let us also recall from [9] and [14*
*, 15]:
Proposition 2.1 (a) The composite IS : Top! Grpdcan be identified with the clas*
*sical (geometric)
fundamental groupoid functor ß1;
(b) For every topological space X, S(X) is a Kan complex;
(c) The S-image of a morphism p in Topis a Kan fibration if and only if p i*
*tself is a Serre fibration.
3 What is the Galois double groupoid of a Serre fibration?
Let p : E ! B be a Serre fibration of topological spaces. By Propositions 1.3 *
*and 2.1, the Galois
(double) groupoid GalI(S(E), S(p)) is well defined. Moreover, since the functor*
* S being a right adjoint
preserves pullbacks, we can write
GalI(S(E), S(p)) GalSI(E, p) Gali1(E, p) *
* (12)
and conclude that Gali1(E, p) also is a well-defined double groupoid. We write *
*this double groupoid
as fl(p) to indicate the relation with Galois theory, and now describe it expli*
*citly.
The underlying double graph has the following description:
o presented as an internal groupoid in Grpd, fl(p) is displayed as
____//_ ____//_
ß1((E xB E) xE (E xB E)) ß1(E xB E xB E)___//_//_ß1(E xB_E)//_ß1(E)*
*oo_(13)
from which we conclude:
5
o the set of objects in fl(p) = Gali1(E, p) is E;
o a horizontal arrow e ! e0is a morphism e ! e0in ß1(E), i.e. a homotopy cla*
*ss of a path from
e to e0(recall that such a homotopy h : [0, 1] x [0, 1] ! E is required to*
* have h(0, t) = e and
h(1, t) = e0for every t in [0, 1]);
o a vertical arrow e ! e0is just the pair (e, e0) provided p(e) = p(e0);
o a square
e1__OE//_e2 *
* (14)
| |
| u |
fflffl|fflffl|
e01___//_e02
OE0
is a homotopy class of a path from (e1, e01) to (e2, e02) in E xB E; its v*
*ertical domain OE and
vertical codomain OE0are homotopy classes of paths from e1 to e2 and from *
*e01to e02respectively,
and the horizontal are pairs (e1, e01) and (e2, e02) respectively.
Clearly there is no problem with the horizontal composition as well - since*
* we know that ß1(ExB E)
and ß1(E) are groupoids. The only non-trivial part of our construction is the v*
*ertical composition of
squares:
Given
e1_OE_//_e2___2// *
* (15)
| | fflffl|
| u | 1
fflffl|fflffl|
e01OE0_//_e02
| |
| u0 |
fflffl|fflffl|
e001__//_e002
OE0
we have to define uO1u0, which must be (an equivalence class of) a path from (e*
*1, e001) to (e2, e002) in ExB
E. Of course we should choose a representative f of u, a representative f0of u0*
*, a homotopy h between
the vertical codomain path of f and the vertical domain path of f0, paste them *
*together, and take the
homotopy class of the resulting path in ExB E. However, how do we know that suc*
*h an uO1u0does not
depend on the choices we made? The nice consequence of the results above is tha*
*t we do not need to
prove this. Indeed, since the morphism ß1((E xB E)xE (E xB E)) ! ß1(E xB E)xi1(*
*E)ß1(E xB E) is
an isomorphism, the pair (u0, u) determines a path v in (E xB E)xE (E xB E), an*
*d the desired vertical
composite u0u is nothing but the image of v under the functor ß1((ExB E)xE (ExB*
* E)) ! ß1(ExB E)
induced by the composition map (E xB E) xE (E xB E) ! E xB E of Eq(p). Moreover*
*, we do not
have to worry about associativity of the horizontal composition, which in fact *
*follows from
ß1((E xB E) xE (E xB E) xE (E xB E)) ! ß1(E xB E) xi1(E)ß1(E xB E) xi1(E)ß1(E*
* xB E)
being an isomorphism.
6
Example 3.1 Suppose for example that p : E = (B x F ) ! B is the projection of*
* a product, and
so is a fibration. Then Eq(p) = E xB E is homeomorphic to the product B x (F x*
* F ) and hence
ß1(Eq(p)) is easily determined, together with its double groupoid structure. Th*
*e vertical edges are
triples (b, f- , f+ ), b 2 B, f 2 F , the horizontal edges are pairs (fi, OE) *
*2 ß1B x ß1F and the squares
are triples
(fi, OE-, OE+) 2 ß1(B) x ß1(F ) x ß1(F )
with vertical boundaries given by @1 (fi, OE-, OE+) = (fi, OE ). The horizontal*
* composition O2 is that of the
fundamental groupoids; the vertical composition O1 reflects that of the product*
* groupoid B x (F x F )
in which B is the discrete groupoid and F x F is the codiscrete groupoid.
In the next sections we translate the construction for fl(p) where p is a f*
*ibration into results for
an arbitrary map q : M ! B by the usual factorisation process, and so relate th*
*ese ideas to classical
constructions.
4 The homotopy double groupoid of an arbitrary continuous map
If C is an internal category (or a groupoid) in a category X with pullbacks, an*
*d i : M ! C0 a mor-
phism into the object-of-object of C, we can always form the "induced internal *
*category (respectively
groupoid)" i*(C) =
____//_ ____//_
C2xC0xC0xC0(M x M x M) ____//_//_C1xC0xC0(M x_M)_//_Moo_ *
* (16)
which, in the case X = Sets, can be simply described as the category with objec*
*ts all elements of M,
and morphisms ä s in C". In particular, for a topological space E and a subspac*
*e M one defines the
relative fundamental groupoid ß1(E, M) as i*(ß1(E)), where i : M ! E is the inc*
*lusion map; in this
paper we do not consider ß1(E) as a topological groupoid, but the same construc*
*tion of ß1(E, M) can
be repeated internally to Top.
Let p : E ! B be a Serre fibration and i : M ! E an arbitrary map in Top. U*
*sing the morphism
ß1(i) : ß1(M) ! ß1(E) of fundamental groupoids, and having in mind that ß1(E) i*
*s the object-of-
objects of Gali1(E, p) (when it is regarded as an internal groupoid in Grpd), w*
*e can construct what
we are going to call the relative Galois double groupoid Gali1(E, p, M, i) as
Gali1(E, p, M, i) = ß1(i)*Gali1(E, p). *
* (17)
A good reason for introducing this new double groupoid is that any continuous m*
*ap q : M ! B can be
presented as a composite q = pi above with i being a homotopy equivalence, and *
*so any q determines
such a double groupoid equivalent (in an appropriate sense) to the Galois doubl*
*e groupoid of a Serre
fibration. It is this double groupoid we have displayed in diagram (1).
In the next sections we will identify this double groupoid in terms of homo*
*topy classes of certain
maps and so relate these ideas and the compositions obtained more clearly in te*
*rms of classical
homotopical ideas. One further reason for doing this is to allow the possibilit*
*y of generalisations of
these geometric constructions to situations where the appropriate Galois theory*
* is not yet obtained,
for example in terms of smooth structures and `thin fundamental groupoids', as *
*in [18]. This could
lead to smooth structures on variations of the constructions given here.
7
5 Geometric interpretation
We now start to interpret the previous results geometrically in terms of compos*
*itions related to classical
notions of relative homotopy groups. To this end, we use standard notation for *
*the cubical singular
complex KM of a space M. Here KnM is the set of singular n-cubes in M (i.e. con*
*tinuous maps
In ! M with K0M identified with M). There are standard face maps @i : KnM ! Kn-*
*1M and
degeneracy maps "i : Kn-1M ! KnM for i = 1, . .,.n (with "1 : K0M ! K1M written*
* simply ",
and giving for x 2 M the constant path "x at x). There are also for i = 1, . .,*
*.n compositions a Oib
defined for a, b 2 KnM such that @+ia = @-ib, and inversions -i: Kn ! Kn. Final*
*ly we shall later
use connections ffi: Kn-1M ! KnM for i = 1, . .,.n, ff = induced by the maps*
* flffi: In ! In-1
defined by
flffi(t1, t2, . .,.tn) = (t1, t2, . .,.ti-1, A(ti, ti+1), ti*
*+2, . .,.tn)
where A(s, t) = max(s, t), min(s, t) as ff = -, + respectively. The enrichment *
*with connections +ifor
the traditional cubical sets was introduced in [5]. The full properties of thes*
*e structures are set out in
for example [1]. Here we will assume only the obvious geometric properties in t*
*he range n = 0, . .,.3.
Let q : M ! B be a map of spaces. We recall the standard factorisation q = *
*pi where i : M ! Mq
is a homotopy equivalence and p : Mq ! B is a fibration. Here
Mq = {(x, ~) | ~ : I ! B, ~(0) = q(x)} M x BI
and p(x, ~) = ~(1), while i : x 7! (x, "(q(x)) where "(q(x)) is the constant pa*
*th at q(x) in B. A point
of MqxB Mq is a pair ((x, ~), (x0, ~0)) with ~(1) = ~0(1) as in the following p*
*icture:
x o
q(x) o
|~|
fflffl|oOO
|~0|
q(x0)o|
x0 o
So Mq xB Mq is homeomorphic to the space M0 of triples (x, ~, x0) 2 M x K1B x M*
* such that
~(0) = q(x), ~(1) = q(x0). Hence we have a groupoid structure on M0 with object*
* set Mq, where the
source and target maps s0, t0send (x, ~, x0) to (x, ~1), (x0, -~2) where ~1, ~2*
* 2 K1B are respectively
the rescaled forms of the first half and the second half of ~. The composition *
*O0in this groupoid is,
when defined, given by
(x, ~, x0) O0(x0, ~0, x00) = (x0, ~1O1~2, x00)
where O1 is here the usual composition of paths. Of course this composition is *
*defined if and only if
~2 = -1~1.
Let j : M xB M ! M0 be given by (x, x0) 7! (x, "(qx), x0). The set of paths*
* I ! M0 with end
points in Im j can be identified with the subset R2(q) of K1M x K2B x K1M of tr*
*iples (f, ff, g)
8
such that qf = @-1ff, qg = @+1ff and @-2ff, @+2ff are constant paths. Thus an e*
*lement of R2(q) may be
pictured as:
_______f
|q
____|?_
||| ||||
|||| ||||
||||ff||||||| ___-|2
||||| ||||| |?
_______||||| 1
|6q
_______|
g
where the dotted lines show constant paths. Thus R2(q) fits in the following di*
*agram:
______//_
R2(q)______//_K1Moo_OOOO *
* (18)
|||| |||
|||| |||
|||| ||| ___/2/
|||| ||| fflffl|
fffflffl|fflffl|lffl|fflffl|fflffl||//_|1
Eq(q)________//Moo_
The boundary maps are given by:
@-1(f, ff,=g)f,
@+1(f, ff,=g)g,
@-2(f, ff,=g)(f(0), g(0)),
@+2(f, ff,=g)(f(1), g(1)).
The degeneracy maps "1 : K1M ! R2(q), "2 : Eq(q) ! R2(q) are given by:
"1(f)= (f, "1(qf), f),
"2(x, x0)= ("(x), "21(q(x)), "(x0)),
for (f, ff, g) 2 R2(q), (x, x0) 2 M xB M.
Clearly (18)can be considered as a diagram of reflexive graphs. We now exam*
*ine compositions on
R2(q).
The set R2(q) has two partial compositions. The composition O2 is determin*
*ed by the usual
composition of paths and squares in this direction:
(f, ff, g) O2(f0, ff0, g0) = (f O1f0, ff O2ff0, g O1g0).
The composition O1 in the direction 1 is given by
(f, ff, g) O1(g,=fi,(h)f, ff O1fi, h). *
* (19)
9
Note that this definition generalises a construction by Kamps and Porter in*
* [16, Section 4.1],
in which they assume f(0) = g(0), f(1) = g(1) whereas in our situation we have *
*only qf(0) =
qg(0), qf(1) = qg(1). Hence they end up with a 2-groupoid, and we end up with a*
* double groupoid.
Their method of proving the properties of their homotopy 2-groupoid is to assum*
*e first that p is
a fibration, and then apply this case to an arbitrary map q by converting it to*
* a fibration p = ~q.
This is analogous to our methods, except that we have used Galois theory, where*
*as they use directly
properties of fibrations.
We now form the quotient of diagram (18)by taking homotopy classes rel vert*
*ices of R2(q) and of
K1M to yield the diagram: _______//
æ2(q)______//ß1(M)oo_OOOO *
* (20)
||| |||
||| |||
||| |||
||| |||
ffflffl|flffl|fflffl|fflffl||//_|
Eq (q)_______//Moo_
where ß1(M) is the fundamental groupoid of M. It is clear that the horizontal c*
*omposition O2 on R2(q)
is inherited by æ2(q). Our main result of this section is a direct verification*
* that the composition O1
is also inherited, without going through the simplicial Galois theory of the pr*
*evious sections. We also
have to show that this composition is related to that derived from the equivale*
*nce relation structure
on K1M0.
In fact we prove a stronger result. The set
R2(q) @+1x@-1R2(p)
is the domain of composition of O1 on R2(q). A homotopy rel vertices on this se*
*t is a continuous family
((fu, ffu, gu), (gu, fiu, ku)), 0 6 u 6 1 of elements of this set such that fu(*
*0) = f0(0), ku(1) = k0(1), 0 6
u 6 1. We use the notation ßv0for the set of homotopy classes rel vertices.
Theorem 5.1 The natural map
: ßv0(R2(q) @+1x@-1R2(q)) ! æ2(q) @+1x@-1æ2(q) *
* (21)
defined by the projections, is a bijection.
For the proof we use properties of the connections, and we use the followin*
*g notation from [21].
We write:
__ __j+1// + __j+1// -
| fflffl|for j; __| fflffl|for j
j j
__/j+1/ __ __j+1//
|| fflffl|for"j; __ fflffl|for"j+1.
j j
Thus the thick lines denote degenerate faces. We shall use inversions applied *
*to connections, for
example
__ ___2//
-2| , fflffl|,
1
10
__
and write this also as |since it coincides with -1__|.
This notation allows us to write some compositions as for example that invo*
*lving 3-cubes x, y with
@+3x = @-3y as ~ __~
___/3/
A = ||x|y 2fflffl|
which is an abbreviation for ~ ~
"2@-2x +2@-2y
x y
and makes it transparent what are the faces of A. The direction arrows are omit*
*ted when convenient.
Proof of theorem 5.1 We define an inverse for .
We use square brackets [ ] to denote homotopy classes. Let ([f, ff, g], [h,*
* fi, k]) 2 æ2(q) @+1x@-1æ2(q).
Then there is a homotopy rel vertices of paths , : g ' h : I2 ! M. We set
([f, ff, g], [h, fi, k]) = [(f, ff, g), (g, (q,) O1fi, *
*k)](22)
and have to prove is well defined and an inverse to .
Suppose we are given homotopies
~ : (f, ff, g)(f0, ff0, g0) *
* (23)
~ : (h, fi, k)(h0, fi0, k0) *
* (24)
,0: g0' h0. *
* (25)
Then ~, ~ are given by three component homotopies rel vertices
~1 : f ' f0, ~2 : ff ff0, ~3 : g ' g0, *
* (26)
~1 : h ' h0, ~2 : fi fi0, ~3 : k ' k0, *
* (27)
with the properties that
q~1 = @-1~2, q~3 = @+1~2, *
* (28)
q~1 = @-1~2, q~3 = @+1~1. *
* (29)
We now use the fact that all homotopies are rel vertices and that the maps *
*ff, ff0, fi, fi0: I2 ! B are
constant on the edges @-2, @+2. So in the following picture, the dotted lines r*
*epresent constant paths,
and i is a hollow cube not yet filled in, but has four faces well defined. Not*
*e also that the maps
I2 ! B given by @2 (~2) and @2 (~2) are constant maps, by our definition of hom*
*otopies.
11
_f___
________~1________________________________________2*
*//_,,XXXXXXXX
___________________________________________________*
*__________________f0__3fflffl|
___ _______________________________________________*
*______1
____ff____ ________
________~2_______ff0_______________________________*
*__________
__g________________________________________________*
*_~3______________________________
___ _____________________________________0
__,___________________i_______g__________
___h______________,0_______________________________*
*_____________________________
_________________________________~1________________*
*________________________________________________________________h0___
_________________________fi________________________*
*__________
________~2________fi0______________________________*
*_____________
___________________________________________________*
*______________k______________________
___________________~3___________________k0__
The maps ,, ,0, ~3, ~1 define a map
(I2 x `I) [ (I`x I2) ! E
(where I`= {0, 1}) given by (s, t, 0) 7! ,(s, t), (s, t, 1) 7! ,0(s, t) on I2 x*
* `I, and by (0, t, u) 7!
~3(t, u), (1, t, u) 7! ~1(t, u) on I`x I2 respectively. By the rel vertices co*
*ndition, these maps can
be extended by the constant map over I x {0} x I, which is @-2(i). So we now ha*
*ve maps defined on 5
faces of I3 and agreeing on their common edges, and so these extend to a map i *
*: I3 ! E. However,
while this map does agree with the other homotopies, the result will not be a h*
*omotopy of the type
required since i1 = @+2(i) = (i|(I x {1} x I) is not constant as would be requi*
*red. So we have to make
a modification to get a homotopy between representatives of the original classe*
*s. Intuitively, we move
the face i1 of i to the right and down of our composite picture. This modificat*
*ion will also change
~2, ~3, but this does not matter for our purposes, since we need to show only t*
*hat a homotopy of the
required type exists.
We let ", "1, "2 denote degenerate elements - the element they act on in th*
*e following formulae will
be clear from the context, in order to make the compositions properly defined.
Our new homotopy
(f O ", ff O2"2, g O "), (g O ", ((q,) O1fi) O2"2, k O ")) (f0O ", ff0O2"2, g*
*0O "), (g0O ", ((q,0) O1fi0) O2"2, k0O "))
will be given by
~1O2" : f O'"f0O ", *
* (30)
~2O2"2 : ff O2"2 ff0O2"2, *
* (31)
~3O2" : g O'"g0O ", *
* (32)
~02: ((p,) O1fi) O2"2' ((p,0) O1fi0) O2"2, *
* (33)
~3O @+2i : k'OkÖ0 ", *
* (34)
where ~ __~ ___2//
~02= qi~ | fflffl|
2|| 1
12
__ + *
* __
where | is given by (s, t, u) 7! (@2 i)(min(s, 1 - t), u). Note that the combi*
*nation of | and " = ||
in the second column of the matrix has the effect of pushing the non constant f*
*ace @+2i of i down to
be able to combine with ~3.
It is clear that compositions with degenerate elements in direction 2 do no*
*t change homotopy
classes, and so this completes our geometric proof that is well defined.
Next we must prove = 1, = 1.
Considering the formula (22)for , we see that we can set
([f, ff, g], [g, fi, k]) = [(f, ff, g), (g, (q,) O1fi, *
*k)]
where now , can be chosen to be a constant homotopy ". It is then easily seen t*
*hat
[(f, ff, g), (g, (q") O1fi, k)] = [(f, ff, g), (g, fi, k*
*)],
and so that = 1.
To prove = 1 it is sufficient to show that if , : g ' h is a homotopy re*
*l vertices, then
(g, (q,) O1fi, k) (h, "1(q,) O1fi, k). Such a homotopy is given by (,, ~, "3(*
*k)) where
~ ~ ___/3/
~ = "__| fflffl|.
3(fi) 1
*
* 2
Corollary 5.2 The composition O1 on R2(q) is inherited by æ2(q) so that æ(q) be*
*comes a double
groupoid.
Proof The composition O1 on æ2(q) is the composition of the maps
æ2(q) @+1x@-1æ2(q) -! ßv0(R2(q) @+1x@-1R2(q)) ! æ2(q)
where the second map is induced by the composition on R2(q). It is easy to see *
*that the structure O1
gives a groupoid structure on æ2(q).
Thus the only part remaining is the interchange law. However we easily find*
* that a double com-
position can be given as
~ ~ ~ ~ ~ __/~2/_
[f, ff, g][f0, ff0, g0] 0 ff ff0 ff0lffl|
[h, fi, k][h0, fi0,=k0]f O f , (q,) O1fi(q,0) O1fi0, k O1k
where , : g ' h, ,0: g0' h0. So the interchange law follows from that for singu*
*lar squares. 2
Finally, we have to show the relation between the composition O0on K1M0 and*
* the composition
O1 above. Let (f, ff, g), (g, fi, h) 2 R2(q). We first note that, analogously t*
*o the existence of identities
in the fundamental groupoid,
[f, ff, g] = [f, ff O1("1(qg)), g], [g, fi, h] = [g, ("1(qg))*
* O1fi, h].
Hence
[f, ff O1fi,[h]f=, ff, g] O1[g, fi, h]
=[f, ff O1"1(qg), g] O1[g, "1(qg) O1fi, h]
=[(f, ff O1"1(qg), g) O0(g, "1(qg) O1fi, h)]
as required.
13
6 Examples
In order to study the double groupoid æ(q) we need to have examples of double g*
*roupoids with which
to compare it, in addition to the product fibration of Example 3.1. As we shall*
* see, there are some
sub-double groupoids of æ(q) which are familiar, but it is interesting that we *
*have little information
about the most general form of double groupoids. For example, the methods of [1*
*1] give an equivalence
between double groupoids satisfying some filler conditions and what are there c*
*alled core diagrams,
but these do not seem to be helpful in this case.
Here we suggest various examples and comparisons for further investigation.
Example 6.1 Let i : M ! B be the inclusion of a subspace M of B. Then the equi*
*valence relation
Eq(i) is discrete, and so æ(i) is a 2-groupoid. Further, if m 2 M then the natu*
*ral map
j : ß2(B, M, m) ! æ2(i)
is injective.
Proof We represent ß2(B, M, m) by maps ff : I2 ! B such that the face @-1ff map*
*s into M and the
other three faces map to the base point m. The homotopy classes of ff which yie*
*ld an element [ff] of
ß2(B, M, m) are through maps of the same type. Then ff also yields an element <*
*ff> of æ2(i), but there
the homotopies allow @+1ff to vary in M. We have to prove that the map j : [ff]*
* 7! is injective.
Suppose then [ff-], [ff+] 2 ß2(B, M, m) and = 2 æ2(i). Let h : I3 *
*! B be a homotopy
determined by this equality, so that
@-3(h) = ff-, @+3(h) = ff+,
@2 (h) maps to m. Let ` = @+1(h). The problem is that ` is not constant. So we *
*change h to `move' `
to the top face and still give a homotopy h0: ff0-' ff0+where [ff ] = [ff0]. We*
* can take
~ ~ ___2//
h0= |||h__| 1fflffl|
so that the two ends of this homotopy are
~__ ~
@3 (h0)= |_|ff|__||__|,
__
where |_|denotes a double identity, as required. *
* 2
Note that æ(i) is the homotopy 2-groupoid of a pair discussed by Moerdijk a*
*nd Svensson in [19],
and is also recovered from the work of [16].
Example 6.2 The double groupoid æ(q) contains a 2-groupoid
____//_ ____//_
æ~(q)___//_ß1(M)___//_M
where ~æ2(q) is the subset of æ2(q) of elements u such that @-2u, @+2u are dege*
*nerate, that is consist
of pairs (x, x). This is essentially the homotopy 2-groupoid of a map discussed*
* by Kamps and Porter
14
in [16]. This 2-groupoid contains various cat1-groups of the form considered by*
* Loday in [17]. The
crossed module of groupoids associated to this 2-groupoid is of the form C ! ß1*
*(M) where for each
point x 2 M we have C(x) is isomorphic to ß1(Fx, ~x), the fundamental group of *
*the homotopy fibre
Fx of q over q(x) at the base point ~xdetermined by x. If M is a subspace of B *
*and q is the inclusion
then C(x) is isomorphic to the familiar relative homotopy group ß2(B, M, x) and*
* the crossed module
C(x) ! ß1(M, x) is essentially that first studied by J.H.C. Whitehead. However*
* we do not have
a reconstruction method for æ(q) from ~æ(q), whereas the 2-groupoid can be reco*
*nstructed from the
crossed module of groupoids it contains, as shown in [7]. *
* 2
Example 6.3 Foliations Let F be a foliation on a space M. Thus the leaves of t*
*he foliation define
an equivalence relation R = R(F). Let q : M ! B be a map of spaces. The folia*
*tion defines
a finer topology than that given on M to give a space MF in which all leaves of*
* the foliation are
open components. So we also have a map qF : MF ! B and hence may define the hom*
*otopy double
groupoid æ(qF ). Where this differs from æ(q) is that in æ(qF ) the `horizontal*
*' paths, and the homotopies
of paths, all lie in leaves of the foliation.
An illustrative example is the Möbius Band M with its projection q : M ! S1*
* and foliation F by
circles of which the centre one goes once round the Band and the other circles *
*go twice round. Then
æ(qF ) contains the double groupoid D(M) explained in the Introduction, and whi*
*ch seems to be a
good discrete algebraic model of the foliated Möbius Band. *
* 2
Acknowledgements
This work was partially supported by the following grants:INTAS 93-436 `Algebra*
*ic K-theory, groups
and categories', 97-31961 `Algebraic Homotopy, Galois Theory and Descent', `Alg*
*ebraic K-theory,
Groups and Algebraic Homotopy Theory'; with Bielefeld, an ARC Grant 965 `Global*
* actions and
algebraic homotopy', and by the London Mathematical Society fSU Scheme.
The first author is also grateful to the Erwin Schrödinger Institute of Mat*
*hematical Physics and
a Leverhulme Emeritus Fellowship for support to attend a Workshop on Foliations*
* in August, 2002.
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