Double groupoids and crossed modules

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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.

© Andrée C. Ehresmann et les auteurs, 1976, tous droits réservés. L’accès aux archives de la revue « Cahiers de topologie et géométrie différentielle catégoriques » implique l’accord avec les conditions générales d’utilisation (http://www.numdam.org/legal.php). Toute utilisation commerciale ou impression systématique est constitutive d’une infraction pénale. Toute copie ou impression de ce fichier doit contenir la présente mention de copyright.

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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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Computer Science of North Wales College University BANGOR LL57 2UW, Gwynedd, G.-B. and

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362