... Ã } is the fut L sub- d luble groupoid of C with point set {x}, and 7' is a trpp double groupoid. A an intf resting application of this c'Jrollary, vte deduce properties ...
C AHIERS DE TOPOLOGIE ET GÉOMÉTRIE DIFFÉRENTIELLE CATÉGORIQUES
RONALD B ROWN C HRISTOPHER B. S PENCER Double groupoids and crossed modules Cahiers de topologie et géométrie différentielle catégoriques, tome 17, no 4 (1976), p. 343-362.
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Article numérisé dans le cadre du programme Numérisation de documents anciens mathématiques http://www.numdam.org/
CAHIERS DE
TOPOLOGIE
Vol. X VII -4
(1976)
ET GEOMETRIE DIFFERENTIELLE
DOUBLE GROUPOI DS AND CROSSED MODULES
by Ronald BROWN
and
Christopher
B. SPENCER*
INTRODUCTION
The notion of double category has occured often in the literature
( see for example (1 , 6 , 7 , 9 , 13 , 14] ). cular
algebraic object theory
connection. The
which
we
call
a
In this paper
special
we
double
study
a more
parti-
groupoid with special
be called «2-dimensional grou-
of these
objects might poid theory ». groupoid theory derives much of its technique and motivation from the fundamental groupoid of a space, a device obtained by homotopies from paths on a space in such a way as to permit cancellations. The 2-dimensional animal corresponding to a groupoid should have features derived from operations on squares in a space. Thus it should have the algebraic analogue of the horizontal and vertical compositions of squares ; it should also permit cancellations. But an extra feature of the 2-dimensional theory is the existence of an analogue of the Kan fibres of semi-cubical theory. This analogue is provided by what we call a special The
is that
reason
connection . In this paper
We define
we
cover
the main
DG
of
algebraic theory
special
double
of these
objects.
with
special D such that objects subcategory DG! D0 is a point. We prove in Theorem A that D§ is equivalent to the well-known category of crossed modules [ 5, 10, 12] , while Theorem B gives a description of connected objects of DG. Theorem A has been available since 1972 as a part of [4]. It was explained in [4] that the motivation for these double groupoids was to find connection,
*
and
a
a
The second author
cil under
a
category
of
full
was
groupoids
supported at Bangor in 1972 by the Science Research CounB/RG/2282 during a portion of this work.
research grant
343
proofs of putative forms of a 2-dimensional van Kampen Theorem. Such applications have now been achieved in [3], which defines the homotopy double groupoid p (X, Y,Z) of a triple and applies it, with Theorem A of this paper, to obtain new results on second relative homotopy groups. an
algebraic object
The
which could express
structure
of this paper is
as
and
statements
follows. In parag. 1
we set
up the
machinery of double categories in the form we require. We do this complete generality for two reasons : 1° we expect applications to topology of general double groupoids, 2° we expect applications to category theory of the general notion of
basic
connection for
a
in
a
double category.
However the notion of connection is the
In section 2 oids. In section 3
we
we
real
only
novelty
of this section.
show how crossed modules arise from double group-
define the category
fD§
and prove Theorem A. Section
4 discusses retractions and proves Theorem B. We also discuss «rotations» for
of DG .
objects We
are
grateful
to
Philip Higgins
for
a
number of very
helpful
com-
ments.
1. Double categories.
Usually
a
double category is taken
equivalently,
category structures, or,
small
categories.
We shall
use
a
to
be
category
a
set
with
object
two
commuting
in the category of
another, but again equivalent, definition.
sextuple (H, P, d0,d1, m, u) in which //. P are sets, d0, d1 :H- P are respectively the initial and final maps, "1 is the partial composition on H , and u: P -- H is the unit function; these data are to satisfy the usual axioms. By
By
as
a
a
part ial ly
category is
meant a
double category D is
shown in the
meant
diagram
344
four related
categories
and
satisfying
the rules
be called squares, of of P
points .
We will
and this allows
to
(1.1- (1.5 ) given
H, V horizontal assume
below. The elements of C will
and vertical
edges respectively,
the relations
represent
a
pictured
as
square
a
as
having bounding edges pictured
as
while the
’Vae will
so
edges
assume
that the
are
the relations
identities 1a, °b
and
for squares have boundaries
345
We also
require
and this square is written e,
f
if
for all
no
confusion will arise.
We
assume two
(:3 , y E
a ,
( 1.5) the
More
B, y , BE
a , f3 satisfy
ular array
we
= E0y,
generally,
we
(aij),
simply
1
o .
Similarly, ex ,
fx
are
written
further relations :
defined, and
are
law
C and both sides
It is convenient
and if E1 a
or
C such that both sides
interchange
whenever a ,
Thus if
ox
to use
defined.
are
matrix notation for
a, a = d0B,
we
compositions
of squares.
write
write
define
i
a
m,
subdivision cf
1, i ’n
a
square
a
of squares in C
in C
to
be
a
rectang-
satisfying
and
such that
We call
composite of the array (aii) and write a =[aij]. The interchange law then implies that, if in the array represented by the subsequent matrix, we partition the rows and columns into blocks Bkl and compute the composite (3 kl of each block, then a = [Bkl]. a
the
346
For
example
We
we
may compute :
give
now
two
examples
of double
categories
in
Topology.
topological space and Y, Z subspaces of X . The double category A2 A2 (X; Y, Z ) will have the set C of squares» to be the continuous maps F : [0 , r]x (0 , s ] - X for some r, s > 0 , satisfying Let X be
EXAMPL E 1.
a
=
0 , s] - Z , s > 0 , and the vertical consist of the maps [ 0, r] - Y, r > 0 , while P will be Y n Z . The obvious horizontal and vertical compositions of squares, together with the usual multiplication of paths and the obvious boundary,
edges >> H edges V will
The « horizontal
will consist of the maps
give us a double category. Note that in this example it is convenient to keep H , V, P as part of the structure, rather than just regard A2 as C with two commuting partially defined compositions. zero
and unit functions will
EXAMPLE
which
tions
Let
2.
we mean
are
that
T = (H , P ,d0 , d1 , . , f) H,
are
topological
gical space X . the elements of
AH,
obtain
a
topological
set
of Moore
category
structure
paths
on a
topolo-
double category A T whose squares
whose horizontal
edges
and vertical
edges
are
(by
func-
are
H, A P
points are P . The category structure on A P is the usual multiplication of Moore paths, while the two compA H are the multiplication of Moore paths and the addition +
respectively, that given by on
we
a
spaces and all the
continuous). Let A X denote the Then
ositions
P
be
and whose
347
composition o on H . This last example arises in the theory of connections in differential geometry. In order to explain this, suppose given as in the beginning of this Section a double category D . By a connection for D we mean a pair (T ,y) in which y : V - H is a functor of categories and F: V - C is a induced
by
the
function such that :
(1.6
by
the
the
bounding edges diagram
of
y (a)
and
h (a) ,
for
a : x -
y in
V,
are
given
and the transport law
(1.7)
holds,
viz if a, b
E
V and
We call 1’ the transport of the connection and y the nection. The EX AMP L E
for this
terminology
is the
I.et T be the
topological
category of
reason
3.
double category of
paths
on
T . A connection
a.
b is defined, then
holonomy
of the
con-
following Example
(1,y)
2 and A T the
for A T then consists
holonomy y : A P- H . The transport essentially equation ( 10) of the Appendix to [11] (where 1 is « path-connections).
of the transport r : A P > A H and the
law>> is
called
a
2. Doubl e
groupoids.
groupoids, that is double categories in which each of the four underlying categories is a groupoid. In this section we -consider simple aspects of their homotopy theory, in parFrom
ticular their
now on our
interest will be in double
«homotopy groups >>.
,
348
Let D be Then
groupoid.
double category
a
have
we
We also have has
object
set
P,
and
The is
a
oid
square /3
a
in Section 1 such that D is
171 D
for
a
double
of components of D , namely
two fundamental groupoids Mh1D, Mv1 D .
classes of elements of there is
two sets
as
example
has
y the
arrows x -
where a, b in
H (x, y) bounding edges
whose
are
Each of these
H (x, y) given by
are
equivalence equivalent if
multiplication in rr1 D is induced by +, and the verification that Mhl groupoid is easy. Similarly, we obtain the « vertical » fundamental group-
Mv1 D
and
so
for each x in P
The second of squares of the
of D
a
o
have
two
coincide and
are
they
are
rr 7 D,
groups
of D is the
a
point
x
or
f..
It is
restricted
to
a
set
consequence
M2 (I), x ) ,
the op-
abelian.
rr2 (D, x), x E P , form local system Mv1D. Suppose for example that
The groups
groupoids
fundamental
homotopy group M2 (D, x) at whose bounding edges are ex law that, when
interchange
erations + ,
we
a
over
each of the
We then define
and this
gives
If the
on
operation
a, bE H (x,y)
cotnmon
of H
are
on
the
family i
equivalent by
subdivision
a
square
(3,
and
349
so
ya, yb have ya - yb .So,
then
we
obtain
operation
an
of
there is
Similarly,
an
ope-
ration of We
now
show how
obtain from
to
a
double
groupoid
families of
two
crossed modules. We recall
[5, 10, 12]
A,B,
operation
groups
an
a morphism d: A - B
and
that
of B
a
on
crossed module the
right
of groups. These
satisfy
B’ of groups such that
f: A , A’, g : B -
phisms
morphism
operator
we
must
have
a
with respect to g , i.
category (F
Let D be
a
ga
=
of
the conditions :
of crossed modules consists of
A map
So
of the
(A , B , d) consists group A , written
d’ f
and
f
mor-
is
an
e.
of crossed modules.
double
and let x E P. We define the groups be the sets of squares of D with bounding
groupoid,
M2 ( D, H , x), M2 ( D, V, x)
to
edges given respectively by
a E H, b E V respectively, and with group structures respectively. Clearly we have morphisms of groups
for
some
+,
o ,
PROPOSITION
then
we
1.
If
D is
a
double
groupoid
have crossed modules
350
as
above and
x
induced from
a
point of D
PROOF. It is sufficient
to
( i) and (ii). L et b E H{x},
It is trivial
verify
to
define the
and
=
b,
then
that this is
If D is is
a b and -B+a+B
so ab=-B+a+B.
chosen), A
we
a
A similar
morphism f:
to
verify
the axioms
operation and that condition ( i ) for a proof of condition (ii) we note that if
an
have the
common
proof holds
pointed double groupoid ( by
abbreviate
and
a E M2(D, H, x) .We define
crossed module is satisfied. For the
E (B)
operations
y ( D, x )
to
for
subdivision
y’ ( D , x ) .
which
we mean a
base
point x
y(D). categories D, D’
D - D’ of double
consists of four
functions
commuting with the category structures. So there is a category of double categories. Similarly there is a category of double groupoids and their morphisms, and of pointed double groupoids (in which a base point is chosen for each double groupoid and morphisms preserve base point). Clearly y defines
3.
a
functor from this last category
the category of crossed modules.
Specicrl double groupoids. By
as
to
a
special
double
in Section 2 but with the
groupoid extra
we
shall
mean a
double
groupoid D
condition that the horizontal and vertical
category structures coincide. These double groupoids will, from now on, be our sole concern , and for these it is convenient to denote the sets of
points, edges in
D1
and squares
by
will be written ex,
or
D0 , D1 simply
and e .
351
D2 respectively. The
boundary
The identities
maps
D1 - D0
will
be written
80 , 8 1 .
By
a
morphism f:
D - E of
double
special
groupoids
is
meant a
triple
of functions
which
commute
A
with all three
special
connection for
(F y )
connection
groupoid
with
structures.
groupoid D will mean a equal to the identity D1 - D1.
double
special
a
holonomy
map y
simply T. A morphism f : D - E of spespecial connections T,A is said to preserve
Such a connection will be written
cial double
groupoids with i f 12 r = A f1.
the connection
category DG
pairs ( D, T) of special double groupoid D with special connection, and arrows the morphisms of special double groupoids preserving the connection. The full sub-category of -Tg on objects (D,T) such that D has only one point will be written DG!. If (i), T ) is an object of DG! then we have a crossed module y (D) by Proposition 1. Clearly y extends to a functor y from Tgl to the category of crossed modules. Our main result on double groupoids is : The
TIIEOR EM A. PROOF.
Let
oid with
The
has
the
DG!-C is inverse E : C - D§’
functor
We define
objects
y:
an
equivalence of categories.
t
an
( ,1 , B ,d) special connection E be
a
to
y .
crossed module. We define =
E( A, B ,d )
as
a
special double
follows.
First,
group-
Eo
is
to
single point 1 say. Next E 1 = B , with its structure as a group (regarded as a groupoid with one vertex). The set E2 of squares is to consist of quintuples consist of
a
such that a -
..f,
a,
b,
c,
d E R and
boundary operators on 0 are given by possible conventions for this boundary -
The
the
8
the convention chosen is the
352
following diagram (there
are
most
convenient in
tion of
It is
terms
squares).
of
The addition and
straightforward
to
expressions for addition and composicomposition of squares are given by
and the
signs
check that these
operations
are
well-defined, i.e.
that with the above data
(for which condition (i) of
crossed module is needed). It is also easy
a
to
check that each of these
E2
with the initial, final and
and for
o
operations
zero
maps for +
groupoid structure being respectively a
on
being
The verification of the
interchange
law
module, but again is routine and is left The
defines
special connection
requires condition ( ii ) to
T : E1 - E 2
for
a
crossed
the reader.
for E is given by
The verification of the transport law is trivial.
This
completes
the
description of E
353
=
E (A , B, d)
and it is clear that
e
extends
functor E: C- DG!
to a
It is immediate that
yE :C -C
is
nat-
urally equivalent to the identity. We now prove that ey is naturally equivalent to the identity. Let (D, I° ) be an object of DG!. Let E = Ey (D) . Then E 0 = DO’ El = D1. We define n : E - D to be the identity on Eo and E1 and on E2 by
(for (a) = a-I b c d
which
clearly
E2 - D2 , serves
-1 )
has the
so to
as
shown in the
correct
prove 77 is
an
diagram
bounding edges. Clearly 77 is a bijection : isomorphism it suffices to prove that 17 pre-
+ , o and connections.
For +
we
have, by the definition of
(3 d..l
and
equation ( 1 ) :
For
o
while
we
on
have, using the notation of equation (2),
the other hand
354
using
the notation of
equality of (3) and (4) follows from the fact that by the transport right hand sides of both (3) and (4) have the common subdivision
The the
17 preserves the connection since
Finally
Since the
proved
law
naturality
that 7y is
a
category DG!
of rj in the
natural
equivalence
from
Ey
to
is
the
clear,
identity
we
have
now
functor.
4. Retractions.
groupoids a key role is played by the retractions ([8], 47, 92 and [2], 6.7.3 and 8.1.5 ). The object of this section is to set up similar results on double groupoids to those for groupoids ( as in the last section, double groupoid here will mean special double groupoid with speIn the
theory
of
cial connection ).
groupoid. Then (as in Section 1) the set 170 G is the set of components of the groupoid (G1, G0) . We say that G is connected if uo G is empty or has one element. We say a subdouble groupoid G’ of G is representative in G if G’ meets each component of G ; we say G’ is full in G if the groupoid (G’1 , G¿) is full in (G1, GO) and also the set of squares with given boundary edges in G’ is the same for G’ as for G. Let G be
a
double
groupoid T To there is exactly
A double x, y
in
and for each a
in
T2
quadruple (x,
is called one
y, z,
edge
w)
of
with
355
a
tree
a
in
points
double
T1 of
groupoid
if for each
with
To
there is
exactly
one
For
example if X
is any set, there is
a tree
groupoid T(X)
double
in which
and
with the boundaries of
given by
i
groupoid structures 2nd connection make T (X) a double groupoid
The
lllEOREM
double
let G’ be
B.
a
then
are
uniquely
defined
subdouble
full, representative
so
as
to
groupoid of the
G. Then
groupoid
( i ) there i s a retraction r G-G’ T (ii) if f: G 11 is a rnorf phism of double groupoids such that fo is injective, then there is a lull, representative subdouble groupoid H’ of 11 -
and
a
pushout
f’ isthertacion f andsi aretacion.
in which
I
,square
(iii ) iI
G0
where T i s
a
ROOF F.
ihis is
(i)
of
each ,
E
possible since G’
Since
belongs
G’
to
there is
singleton, then douhlo groupoi d.
ns a
is
G’1. So
full in
BBe
Go is
choose
an
for any
have r1 :
G1
>
isomorphism
edge Ox
representative
(;1’
an
G1
T,
such that
in G’ . Define
edge a
in
(J1 defined,
356
in
g : G - G’ X
G,
the
edge
ar d r1 is the usual
retrac-
tion defined for Now let
We define
groupoids.
G2
a E
have boundaries
r2 (a )
to
be
(it is convenient
to
denote
given by
the
diagram
check that r2 preserves + and that r2 preserves connections we have to prove that
It is
straightforward
to
o .
To prove
Now
However it is
a
simple
consequence of the transport law that
(That r2 preserves T
It follows that
follows from
Proposition
1
of [3].)
357
also
(ii) Let 11’
Then
we
have
in which
i,j
a
commuta- ive
groupoid of H
on
d-’a2ram
f’’ is the restriction of r. edges 0x, x E G0 , and so r: G-G’ as ii (i).
are
We choose
ub30ub1:
l
the
the
mclusions and
F ,)r
each
y in H0, let (in which
3.n:1 the
is
case x
unique ),
edges 0 (y), y E H0
and otherwise let
O(y)
with the connection A
=
ey . .
for H
Then o
= f1 0
determine
a re-
traction s:H-’. We prove (*) is com uta ive. If x E G0, then
If a:x-y in G1, then
Finally,
if
a
E G2,
and tÎ1e fact that To prove
(*)
f a
then
,s f (a )
--
fr (a)
. s
immediate frcm the Jefirition
preserves the connection.
pushout,
suppose
given
a
commutative diagram f double
groupoids
lr there so
is
there is
a
morphism
at ’nost one
1V:
H’ -) K such that
such it; . On the other
358
w s -
v , then "/1
hand,
let w = v " Thep
=
.
wsj
=
vj,
so
7.L’
is
v o (y),
which is
H2
on
and
so
only w s l’. If vEH , then ó(y), and hence also identity for )- ?-, f l’ x ) for anx x , whi le if y = f ( then
need verify an
again
=
It follows that
identity.
an
the relation
the
proof
of
fined above. Then there is such that
o
we
obtain h at.
Let T = a
T( Go,
or
H2,
unique morph
be the sm
identity. Let r: G - C ’ edges 0(x) with
on
/i
.
Finally
w s =
vi
i, ,
f:
tree
This
groupoid dedouble groupoids
double
G - T of
be the retraction defined
is the
(r) by choices of
= v
( ii).
Go{xo} .
( ii i Let
vjs
follows from the f,tct that if y = f ( t )> then
ws = v
frcm the definition of s
completes
u’s =:
as
in
D r ine g : G - G’ X T to have component s r and f respectively. Clearly go is a bijection, and it is a standard fact about groupoids that g is a bi-. je ion. But g2 is also a bijection, since it has inverse
wh,- re
and B
Therefore g is
an
COROLLARY. RY.
If
then tiiere is d luble
has
is3morphism. G is
a
connected double G z G{ xl
is ,morphism groupoid of C with point A
an
all
intf
set
resting application
X
groupoid
x
is
a
point of G,
1 where r; f Â} is the
{x}, and of this
359
and
7’ is
a
c’Jrollary,
trpp
vte
double
deduce
fut L subgroupoid.
properties
of
rotation»
a
tions of squares in
a
Let G be
a
interchanges double groupoid. double groupoid,
associated with h is the function
edges
and
of
and
a
(a )
r
r (a) is defined
THEOREM C.
are
to
related
r
is
The rotation
a
T:
composi-
with connection P . The rotation
G2 -+ G2
such that if
a
E G2
r
then the
by
be
(i) (ii)
(v)
the horizontal and vertical
which
r
r
satis fies : whenever
a
whenever
a o
+B
is
y is
defined, defined,
bijection.
Clearly (iv) is a consequence of (iii) which also implies that r9 is a bijection ; ( v) follows easily. For the proofs of (i), (ii) and (iii) we use Theorem A and B (iii), which imply that it is sufficient to prove the theorem for the case of a double groupoid G E(A, B, a ) determined by a crossed module (for the theorem is clearly true for tree double groupoids, is true for a product if true for each factor, and is true in one groupoid if true in an isomorphic one). However, in the notation of the proof of Theorem A , a straightforward P ROO F .
=
calculation shows that :
360
From this it is easy
But
by
This
to
deduce
the definitions of +
completes
REMARKS.
the
proof
( i ) and ( ii ) . Note also that
and
o ,
of Theorem C .
10 We have been able
to
find direct
Theorem C
proofs
using only the laws for double groupoids. A has been found by P. J. Higgins ; this proof involves rotation Q , and checking that
2° The methods of this paper
topy double of
(i) and ( ii )
of
similar an
(c
proof
for
of
(iii)
anti-clockwise »
applied to p ( X , Y, Z) , the homotriple described in [3] , to prove the existence
groupoid of a homotopies between various
can
be
maps
361
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362
