The equivalence of -groupoids and cubical T-complexes - Numdam

C AHIERS DE TOPOLOGIE ET GÉOMÉTRIE DIFFÉRENTIELLE CATÉGORIQUES

RONALD B ROWN P HILIP J. H IGGINS The equivalence of ω-groupoids and cubical T -complexes Cahiers de topologie et géométrie différentielle catégoriques, tome 22, no 4 (1981), p. 349-370.

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3e COLLOQUE

CAHIERS DE TOPOLOGIE E T GEOMETRIE DIFFERENTIELLE

SUR LES CATEGORIES

DEDIE A CHARLES EHRESMANN

Vol. XXII-4 (1981)

Amiens, Juillet

1980

THE EQUIVALENCE OF 03C9-GROUPOIDS AND CUBICAL T-COMPLEXES

by Ronald BROWN

and Philip J. HIGGINS

INTRODUCTION.

The Seifert-van Kampen Theorem involves the category spaces with

base-point,

the category

Group

of groups and the fundamental

Jap * -> Group ;

the theorem group functor II1 : preserves certain special colimits.

The generalisation of this theorem to

answers

three immediate

generalisations tor

to

Jop*,

that the

asserts

functor 171

all dimensions thus what

questions, namely,

of the category

5"a-p. * of

the category

the

are

Group

requires

appropriate

and the func-

11’1 ? In

[4,5 ,

where such

generalised Seifert-van Kampen Theorem is the first question is simple: a space with basea

proved, the answer given to point is replaced by a filtered such that each

loop

in

X0

space

is contractible in

However, the second question has answer,

even

embarrassingly

which shows the known

egories generalising

so.

Xi .

a

surprisingly

rich and varied

The situtation is summarised in the diagram

explicit equivalences

between five

algebraic

cat-

the category of groups

The category of crossed

special role. A homotopy functor 17: ( filtered spaces) -> (crossed complexes) is easily defined and is a fairly well known construction in homotopy theory ( see [2, 5, 12]). This

complexes plays

349

a

functor

in

generalised Seifert- van Kampen Theorem. Since colimits of crossed complexes can be completely described in terms of group presentations (see [5] ), this is the natural computational

replaces 77.

version of

one

our

form of the theorem.

By contrast, the proof of this theorem (see [3] for

tion)

is carried

in the category of

mainly equivalence

out

essential way the

of

(w-groupoids)

a

w-groupoids

sketch

and

exposi-

uses

in

an

categories

H

(crossed complexes)

and also the transition r :

both of which to

(m -groupoids ) - (cubical T-complexes),

were

established in

show that the transition

T

[4].

is part

The

object

of the present paper is

of an equivalence (in fact

an

isomor-

p hi sm ) of categories. For the in its relation

topologist,

an

familiar ideas. The

to some

w-groupoid p X of the cubical complex K X,

the interest of this

purely algebraic result lies proofs in [5] use the homotopy

filtered space X . Here p X is related to the singular and the fact that p X is also a T-complex highlights

feature of

intriguing

singular complexes. singular complex K X of a space X is a property that is usually expressed succintly as «every box

It is well known that the

complex -

Kan

fillers. However, this property of KX is based on the existof retractions r : In ->Jna,i , where Jna,i is a box in jn , and so is a

in K X has ence

a

a

property of the models rather than the particular space X . In particular, the fillers can be chosen simultaneously for all X so that they are natural with respect

to

maps of X .

Further,

the retraction

r: In -> Jna,i

is

unique

This suggests the potential usefulness of Kan homotopy rel Jna,i complexes in which boxes have canonical fillers satisfying suitable con-

up

to

.

ditions. Such tain elements

an

idea is realised in the notion of

are

designated

«thin» and

axioms :

350

are

a

required

T-complex to

in which

cer-

satisfy Keith Dakin’s

T 1)

Degenerate elements

T 2 ) Every

box has

a

thin; unique filler;

T 3 ) If every face but maining face.

are

thin

one

of

a

thin element is

thin,

then

so

is the

re-

We prove here that every cubical

T-complex admits the structure of an wsimple axioms for a T-complex contain a groupoid ; wealth of algebraic information, Our proof uses a refinement of the notion of collapsing which we call reduction, and the methods formalise techniques implicit in Kan’s fundamental paper [ 10]. Keith Dakin’s original definition of T-complex [7] was in terms of simplicial sets, and was conceived as an abstraction of properties of the nerve of a groupoid. Nicholas Ashley proved in [1] that the category of simplicial T-complexes is also equivalent to the category of crossed complexes. (At present, this seems to be the most difficult to prove of the equivalences in the diagram above. It generalises the equivalence between simplicial Abelian groups and chain complexes proved by Dold and Kan [8, 11].) A fifth algebraic category, of 00 -groupoids, will be defined in [6], and proved also equivalent to the category of crossed complexes. (This seems to be the easiest of the equivalences. ) Thus we have an example, possibly unique, of five equationally defined categories of (many-sorted) algebras which are non-trivially equivalent. From an algebraic point of view the equivalences provide a useful method of transferring concepts and constructions from crossed complexes, where they are easily formulated, to the other four types of structure, which are less well understood. (See, for example, [9].) this shows that the

particularly appropriate that this paper, and its sequel [6] , should appear in the Proceedings of a conference in memory of Charles Ehresmann, as he initiated the study of double and multiple categories and felt clearly that these notions should have important applications in Geometry and Algebra. It is

354

T-comPLEXES.

1.

Our methods involve

complicated filling processes in special kinds of Kan complexes, that is, cubical complexes satisfying Kan’s extension condition [10]. We adopt the following conventions. A cubical set (= semi-cubical complex) K is a graded set (Kn)n >0 with face maps

and

degeneracy maps

satisfying the usual cubical relations. But we shall also use cubical sets without degeneracies and, in particular, jn will denote the free cubical set without degeneracies generated by a single cube cn of dimension n. Subcomplexes A of jn will again be without degeneracies and a (cubical) map f: A -> K, where K is a cubical set, will be a graded map preserving face operators. If A, B are subcomplexes of Im , In , respectively, then A X B will denote the cubical set ( without degeneracies) such that (A X B)n is the disjoint union of A p X B q for p + q = n , and the face operators on A p X Bq are given by Then

Im x In

is

isomorphic

to

1’+’

and

we

shall

identify

these whenever

convenient.

jn generated by all faces a ( 03B2 = 0,1 ; j = 1 , 2, ... , n ) of cn except aa cn . For any cubical set K, a cubical map b; In K is called a box in K , and an extension f: In -> K Let

Jna,i

of b is called

by

a set

a

be the

filler

subcomplex

of the box b .

of

Equivalently,

the box b is determined

of 2n-1 elements

satisfying A filler

f

of b is then determined

by

an

352

element

x ( = f ( cn ) ) of Kn

sat-

isfying The Dakin

following

axioms

(in

simplicial context)

a

are

due

to

Keith

[7].

T-complex is a cubical set K having in each dimension n > 1 a set Tn C Kn of elements called thin and satisfying the axioms: T 1 ) Every degenerate element of K is thin ; T 2 ) Every box in K has a unique thin filler; DEFINITION. A

T 3) If t

thin,

is

thin element of K and all its faces

a

except

one are

then this last face is also thin.

Recall that a

dBjt

a

complex

Kan

is

filler. Kan showed in [10] how

the n-th

homotopy

his methods

to

group

of

an oj-groupoid w-groupoid later;

as

defined in

groupoid

7 and

n >

We shall

[4].

to

to construct

the

We here extend structure

the full definition of

an

say that the definition

2

re-

n

T 1, T2 ... , Tn : K n -> Kn+1

laws. Our first aim is

complex K,

a

+, +,... , + in dimension n and 1

also «connections»

x E Kn-1.

give

enough

structures

for such

then it carries the

T-complex

a

for the present it is

admit

quires that K

for

in which every box has

set

define,

one can

IIn (K, x)

show that if K is

cubical

a

operations

satisfying

a

number of

+j and prove their basic

propertie s.

construction, let x, Y E K1 satisfy d11 Let t f T 2 be the unique thin filler of the box

As motivation for the

and let

If

e = E d11y.

we now

groupoid requires

define x + y =

with the

K0 use

as

its

d02t,

set

ments

x, y, z,

not

hard

to see

of vertices. (The

proof

that

K1

=

becomes

a

of the associative law

of thin elements in dimension 3 . ) The laws for +

are

1

conveniently proved if

more

it is

x a0 y,

w

of

K1

one

also considers

which fit

together

353

as

arbitrary quadruples in the diagram

of ele-

and for which there is

a

thin element t c T with x, y, z ,

fact, precisely those for generalise to dimension n .

quadruples

are, in

idea which

we

which x +1 w

w =

as

faces. (Such

z +1 y . ) It is this

subcomplex Ij-1 X I2 X In-j of In+1 . Thus S n is generated by four cubes of dimension n , namely, daj cn+1 for a 0, 1, and a a a in cubical is meant i = j, j+1. By socket set K cubical map 5: Snj -> K a is and four elements x, y, z , U’ for some n , j . Such socket specified by j of Ken satisfying Let Sn

be the

=

as

expressed

in the

and is written s

c omposite

of

dai

=

picture

’s and

An element u

( where

For any cubical

sj (x, y ; z , w ) . f. J ’s)

of Kn + 1

we

imply

is said

that in this

any

abbreviate

x, y, z , w E Kn satisfy ( 1 ) )

The cubical laws

operator (b ( i, e.,

to

if

case

354

span the socket

s = s j (x, y ; z , w)

and Note also that u spans s if and that

û ( cn + 1 ) Let

of

m ent t

(3)

s

= u

is

extension of s :

an

cept

possibly

map

û; l n+1 ->

K such

S nj -> K. A

Kin.

plug fo r s

is

an

ele-

operator

corresponding

to

a

cube of

then 0 t is thin.

expressing ( 3 ) ( ii )

those which

are

is that t and all its

x, y, z, w and their

Not every socket admits as we

unique

socket in

(i) t spans s, and ( ii ) if A is any cubical

intuitive way of

more

a

the

such that

K n+ 1

I n + 1 BS nj, A

be

Sj(x,y;z, w)

=

only if

a

plug,

faces,

are

faces,

ex-

thin.

but when it does the

shall show. Furthermore any socket which admits

a

plug is unique, plug is uniquely

by any three of the faces x, y , z , w. This crucial result is most easily proved by refining the notion of collapse for subcomplexes of I n to that of « reduction » for pairs of subcomplexes. This, we do in the next

determined

section. We

ly

note

in standard

here that

form

a

cubical operator

of I n + 1

can

be written

unique-

as

where

and that A

corresponds

to a

cube of ,

2. REDUCTION.

pair of subcomplexes of j" we mean an ordered pair ( A, B) of subcomplexes such that A D B . We say that a pair ( A, B ) is an el ementary reduction o f ( A’, B ’) , written ( A’, B’) -> e ( A, B ) , if there is a p-cube By

a

355

x

in A’

(p 2 1 )

and

B n x = x B{y},

and (ii) B n x = B’,

subcomplex generated by x, and x = xB{ x}. Thus for elementary reduction, B is an elementary collapse of B’ with free face

where an

y of x such that

( p- i )-face (i) A u x = A’, A n x = x a

denotes the

x

y, but A ((remembers)) the free face. We say that

able from

(A, B) is

(A’, B’) by

a

a

reduction

finite number

if

of (A’, B’)

(possibly 0)

of

(A, B )

is obtain-

elementary reduc-

tions. Then B’

collapses to B, while A « remembers) all the free faces of this collapsing. We write ( A ’, B’)B(A, B). Before explaining the relevance of this to T-complexes, we give a crucial example.

I. If (A’, B’) reduces to (A, B), then ( A’x I, B’xl) (A XI, B xl) and (I x A’, I x B’) reduces to (I X A , I X B ).

PROPOSITION

duces

to

P ROOF. It is

sufficient

to

prove that if

( A " B ’ )Be ( A , B) ,

re-

then

So let

Then X

=

(x,

c1)

is

a

cube of B’XI with Y

delete X and Y from B’ X I

Clearly

( and X

from

=

A’ X l )

(y, c1)

as

a

face. If

we

obtain

we

pair can be reduced to (Axl, B X I ) in two steps, first (x, 0) and (y,o), then by deletion of (x, 1) and (y, 1). c

this last

b y d ele tion

of

COROLLARY 2. 1f 1 j n, a = 0, 1, i = 1 , 2, then ""’7

P ROO F.

This is immediate from

Proposition

1 since

( I2 , I2) Be ( j2, J2a,i). *

pair ( A, B) of subcomplexes of In, we say that B supports A in dimension n if, given any T-complex K and any cubical map g: B -> K , there is a unique extension g: In -> K of g whose values on In, A are all thin. D EFINITION. Given

a

356

3. If (In, In)B(A, B), then B supports A in dimension In supports ¡n , it is enough to show that B supports A

P ROPOSITION P ROOF.

Since

dimension n whenever

( A’, B’) Be ( A, B )

sion n. In this

have

case we

cube x and face y , If

n.

in

and B supports A’ in dimen·

given cubical map, then g is defined on the box xB{ y} but not on x or on y. Any extension g of g which takes thin values outside A must map x to a thin element and, by axiom ( T 2 ), this determines g (x) uniquely, giving a unique extension for

some

g : B ’ -> K .

A’, g Since h (x)

as

required,

unique extension h: In -> K 9(x) is thin, h is thin outside a

2 and

Proposition 3 that the subcomplex dimension n + 1 by any of the four subcom-

Corollary

supported in plexes Ii-1 X J2a,i X In-i ( a =0,1, is

of sockets and

=

a

o

It follows from

Snj of In + 1

is

has

Since B’ supports

which is thin outside A’ .

A,

g: B -> K

i

=

1, 2 ) . We

restate this fact in

terms

plugs.

T-complex and suppose that we are given any three of x, y, z , w E Kn satisfying two of the conditions (1) o f Section 1 (namely the two which are meaningful). Then there is a unique fourth elePROPOSITION 4.

m ent

L et K be

a

such that:

(i ) x, y, z, w form a socket s sj( x, y ; z , w) , and ( ii ) there is a plug t t,(x, y ; z , w) spanning s . Furtherncore this plug t is uniquely determined by the three given =

=

elements.

3. THE COMPOSITIONS + . .

i

following proposition, which defines compositions + j T-complex K , is an immediate consequence of Proposition 4. The

PROPOSITION 5. L et K be

Then there is

a

a T-complex

unique element

z

=

x+j y

357

and l et x,

of Kn

y E Kn satisfy

such that

in any

(i) there is a socket s sj(x, e; z, y) with and (ii) this socket has a plug. Further, the (partial) operation +j satisfies left and right cancellation laws =

that is,

if

x, Y,

ermin e the third

related

are

z

by

z

=

x +j y, then any

of

two

x, y,

det-

z

uniquely.

properties of these compositions depend on some elementary plugs in K . Vie assume that x, y, z , w c Kn .

Further

properties

of

P ROPO SITIO N

P ROO F.

G. I f

=

a

cube of

case

i

> j +

spans the

no

7.If

so

dBj

t

is

plug,

a

then

=

that or

given socket. Suppose

ik is j -1 or j .

dBj+1. Hence Vdajt

a

plug for

that i

and

j

corresponding

to

Then the standard form is thin

as

required.

The

o

tj(X’y; z,w)

Certainly fit spans i

no

is

aq t

cubical operator in standard form

a

1 is similar.

PROPOSITION

PROOF.

is

In + 1 BSnj-1,

of Vdaj contain s

tj( x, y; z , w)

=

aq t

Certainly

da1i1da2ai ... darir

V

t

the

is

a

plug,

then

fit

given socket. Suppose

cubical operator of

358

In+2BSn+1j+1] j

is

a plug for

that

i j

in standard

and

form,

that

is

no ik is j+1 or j + 2. If no ik is i, then 0 Ei therefore 0394 Ei t is thin by axiom (T 1 ). If is =0 i then so

and since

A

Ei,

The

is

is -1 i is + 1

cubical operator of

a

case

and

i j i

> j

+ 1 is similar.

different and need

not concern

Applying Propositions we obtain immediately: P ROPOSITION 8. L et

A t this stage

and no

In + 1 BSnj . i

( When us. )

ik is j + 1 or j + 2, we see that Thus VEi. t is thin, as required.

= j+1,

6 and 7

to

the definition of + in

also prove th at + has

The thin element t

in all dimensions

faces of t

faces of

those which

are

f

particular, f

+j y

=

y. In

We

=

now

make

PROPOSITION 10.

our

Ej + 1 y are

d aj + 1 t

=

is

slightly

Proposition

Then

appropriate left identitie

s.

f=Ejz.

f +jf = f for any f of the form

P ROO F.

ci t

0

i

Also

the form of

x,y£Kn satisfy d1j x = d0jy.

we c an

and

degenerate

spans the socket

degenerate ( and y Therefore t ·

+j f = f for

any

first and crucial

Composites under

+

i

359

element f use

therefore =

and

so

form f Ej z .

o

=

( T 3 )·

elements

The

thin), except

tj( f, e; y, y)

of the

of axiom

o f thin

s j ( f , e; y, y) .

are

thin.

t

=

t , except z

is thin.

possibly z ,

are

plugs,

then

plugs,

then

plugs

Then all the n dimensional faces of

Ejaj y.

thin. Since t is thin, axiom ( T 3 ) implies that

t

( For illustrations,

t’ is

+

j+1

and

defined,

proposition, which gives rest of the proof.

next

and is vital for the

PROPOSITION 11

be

=

this result in the

use

composing plugs

are

are

e

and y are thin. Let

n

We

are

where

tj ( x, e ; z , y ) ,

n , where x

+j y in dimension

PROOF. Let z = x

t + t’

is

a

1

a

the

see

s

is

a

page). (i) If

plug for

plug for

elements in

let i t j. If the four composite

then

next

rules for

socket and has

PROOF. (i) By Proposition 8,

t" = t

as

plug

+ t’ spans the socket s . Also t" is j+1

thin, by Proposition 10. Repeated applications of Proposition 8 show for any composite A of face operators At" has one of the forms for suitable

In+1BSnj

corresponds to a cube of aplugfor s. ( ii ), ( iii ): The proofs are similar,

so

if 11

P ROP OSITIO N 12.

sion n

gives

a

structure

on

then V t " is thin. Hence t 11 is

-E-

i

( j = 1 , 2, ... , n ) in dimen-

(Kn , Kn-1 )

360

k;

o

( i ) Eaclz composition

groupoid

that

with

d0j, al

and

fj

as

361

initiul, final and identity maps resp ectively, (ii) The interchange law holds, that is, i f i t j,

whenever both sides P ROO F.

are

There is

(i)

initial and final maps,

then

defined.

graph structure on (Kn, Kn-1) with and the composition x + y is defined if a

9q, d1j and

as

only if

i

al x aqy. =

To prove the associative law suppose also that

d1jy d0jz and =

write

Then

Since

by Proposition 10 (i )

e

+ i

e

=

e

there is

a

( by Proposition 9 ),

plug

this

implies

that

right cancellation ( Proposition 5 ) and existence of left identities ( Proposition 9). It follows easily that +j is a groupoid structure and that c, is its identity map. ( ii ) In proving the interchange law we may suppose that i j . Let satisfy We

now

have

associativity,

left and

Write

362

p ropos ition

11

gives

us a

But

as

by p ropos irion

required,

plug

8. So

c

R EM ARK. It is easy to a

plug

if and

only if x

see

+

i

w

that =

z

an

arbitrary

socket s

=

s j(x, y; z ,w )

has

+ y.

i

4. w -GROUPOIDS.

[4] that an (0-groupoid K is a cubical set with the algebraic structure. Firstly, for each n > 1 , the pair: n groupoid structures each with objects Kn-1 and arrows

We recall from

following

extra

(Kn, Kn_i )

has

363

Kn . For i written

=

1 , 2, ... , n

the

groupoid

degenerate

cubes

such that

and the

f’ix

has the

following

E Kn

and

di x

=

degeneracy operators Fritsch. ) Suppose

have

a

operations identity ele-

Here - is the inverse op-

these

operations operations + of

ones

which

T-complex

in the

are a

the

has conn ecti ons

following faces :

rules hold :

(It is curious that the

we

and its

each of the connections satisfies the transport

(6 ) x, y

by

directions has

a!

c

already been established for the

previous section Secondly, an a)-groupoid

j-th

Ej y ( y Kn-1 ) ,

eration for -E- . The laws satisfied have

«in the

it has initial and final maps

+j, -j ;

ments are

the

now

a7 y, then

9q, of

a

( see illustrations

Tj together

satisfy

simplicial set;

that is

a

T-complex.

relation

364

this

on

the

laws, namely, next

if

page)

the rules for the face and was

For any

pointed

out

by

R.

,

So

we

have

a

plug

r i.- Kin - Kn+1

the

tj (x, e; x, e) . j-th

where We write

TjX = tj(x, e; x, e)

connection in dimension

36

n

+1 .

and call

THEOREM A.

K i s

an

With the

definitions o f

+j,

given above, the T-complex

Tj

w-groupoid.

PROOF. We have

verified the laws for +j

already

alone,

and

we

have

to

(4), (5) and (6).

prove

Now

follows from

(4) ( i ) and ( ii ) follow from

P roposition (6 ) ( i ),

To prove

the

definitions,

while

( 4 ) (iii)

6. suppose that x + y is defined. We have where where

and

by Proposition 9. Application ( ii) gives immediately

Ej+1 y = tj ( e, f; Y, Y )

11, parts ( i )

and

The other

equality in (6) ( i ) position 1 2 (ii) ) to

using

( ii)

the fact that

is

an

by applying

identity

for the

the

by

a

Proposition

interchange

operation

single application of Proposition prove ( 5 ) we need the following

is obtained

To

Ej y

follows

of

11

law (Pro-

i + , Similarly, ( 6 ) ( ii).

L EMM A.

P ROO F.

If p q , then

T p Eq = Eq + 1 T p,

( with i = q -+ 1, j = p ) gives: follows from equations (4) ( iii ). The case

Proposition

and the result

p > q is similar. If p = q we let

have Ep

d1pe=e,

7

x E Kn

and write

e

=

6

d 1 p x.

so

by Proposition 9. However, fp + 1

e

=

Ep e

366

and

also, by ( 4) ( ii ),

Then

we

The result is therefore

true

Now, consider the

Write

u

=

Ti

when p

rule (

=

q,

5 ). It is enough

To show that

Tj x.

we must

are

prove that

by

definition

to

show that

where

prove :

and ( c ) the faces of u in all dimensions

which

to

is ,

So

o

faces of

are

thin,

daj+1 u or a j +2u.

some

equations ( a ) follow directly from (4) ( i )

The

those

possibly

except

and

( 4 ) (iii).

Also

by (4) (ii) and (4) (iii),

and this is

equal

to

f by the

d1j+1 u = f , while for i = j

lemma. When

we

the

j,

same

gives

argument

h ave

and

This proves (

b). Finally, to

prove

(c),

we

observe that since where ,

,

all faces of or

u

by

the

of

one

tj ( , ; , ).

Lemma. ) of the

However,

thin except

possibly

those which

Each of these four faces is either

dai+1 u.

of the form

are

(Note that

e

= Eid1iTjx

It follows that any face of

special

it is easy

r. x

u

are

or

faces of e

dai u

and is therefore

is of the form

which is

some

not

TjEkd1kx

thin is

a

face

faces

to

verify

that each of these

367

special

faces is

a

face of

This proves ( c ) and of Theorem A.

proof

completes

the

n

5. THE ISOMORPHISM OF CATEGORIES.

If

is any

T-complex ( K being the underlying cubical set and T the collection of thin elements), the construction described above gives an w-groupoid o (K, T ) ( K, +,r;). Conversely, given an m-group-

(K, T)

=

oid

(K, +j, r.),

plex

structure

it

proved

was

in Section 7 of

in which the thin elements

are

[4] that K

all

carries

composites

a

T-com-

under the oper-

ations + of elements of the form

j

We abbreviate this last element

T-complex by

T (K, +j, Tj ).

to

( ± ) Tm y ,

and

Both constructions

Q

denote the

resulting and r are clearly func-

we

torial. THEOREM B.

Let denote the category of T-complexes

of w-groupoids. v ers e isomorphisms. egory

Then the

fun ctors o : F -> G and let

P ROO F. Let

be

r

where T’ consists of all

( K, T ) a (K, T) = (K, T’)

o (K, T) = (K, +j, Tj). Then composites of elements fir

(± )Tm y.

By definition of T-complex, EiY E T. By definition of a T-complex, 1rn y E T . Also, if t f T n , we have

and in

a T-complex

and

and G the catT : G -> F are in-

Tm

11

(since

+j gives

has

plug.

a

follows 10

a

groupoid

Since this

plug

that -t c Tn , by

implies

structure on

Kn )

and

so

the socket

and all its n-f aces other than - t

axiom

( T 3 ).

that T ’ C T. But this

Hence (

implies

± ) Tm y

that T’

=

T;

c

si( t, e . e, -tj) are

T and

for if

in

Tn

it

Proposition

t f

Tn+1

and

b is any box consisting of all n-faces of t except one, then b has a unique filler in T’ ( see [4], Proposition 7.2), whence t E T’n+1 . This proves that

T a

is the

identity functor on 5 . 368

Now let

where T’ is defined show

must

( i)

that,

be any

(K, +j, T j) as

above. To show that arK, T’)

for x, y c

f-ix is

a

plug

given w -groupoid and

a

Now

plug

in

r.1x

certainly

in

for the socket

(K, T’)

k, 0’,

d1j x = d0jy, At

c

spans

s j ( x, e;

x,

e)

where

where f

and hence

w-groupoids imply

VTj x E T’.

the element

the

proof

for any cubical operator

s j .(x, e ; x, e ) . Also,

of Theorem B.

for

that

This proves

( i ). Similarly,

spans

T’’ since it is of the form

completes

we

( K, T’). the laws of

suitable

(K, +j,1.)

K n’

then the socket , has

=

let

,

,

a

369

This proves

( ii )

if and

and

REFERENCES. 1. N.

ASHLEY, T-complexes and crossed complexes, Ph. D. Thesis, University

of

Wales, 1978. BLAKERS, Some relations between homology of Math. 49 (1948 ), 4 28 - 461.

2. A.L.

and

homotopy groups, Ann.

Higher dimensional group theory, in Low dimensional topology, ed. Thickstun, London Math. Soc. Lecture Notes 48, Cambridge, 1982.

3. R. BROWN, Brown &

4. R. BROWN &

P. J. HIGGINS,

On the

algebra

of

cubes, J.

Pure

Appl. Algebra,

21 (1981), 233- 260. 5. R. BROWN & P. J. HIGGINS, Colimit theorems for relative Pure Appl. Algebra, 22 (1981), 11-41.

homotopy

groups, J.

6. R. BROWN & P.J. HIGGINS, The equivalence of ~-groupoids and crossed plexes, This same issue.

DAKIN, Kan complexes and multiple groupoid structures, Ph. University ofWales, 1977.

7. M.K.

8.

D.

com-

Thesis,

A. DOLD, Homology of symmetric products and other functors of complexes, Ann, of Math. 68 (1958), 54-80.

9.

J. HOWIE, Pullback functors Diff. XX-3 (1979), 281-296.

and crossed

complexes, Cahiers Topo.

et

Géom.

KAN, Abstract homotopy theory I, Proc. Nat. Acad. Sci. U.S.A. 41 (1955), 1092-1096.

10. D.M.

11. D.M.

KAN,

Functors

involving

c. s. s.

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87

(1958),

330 - 346.

12. J.H.C. WHITEHEAD, Combinatorial homotopy II, Bull. A.M.S. 496.

R. BROWN: School of Mathematic s and computer Science University College of North Wales

BANGOR LL57

2UW, Gwynedd, U.K.

P. J. HIGGINS: Department University of Durham

of Mathematics

Science Laboratorie s South Road DURHAM DH1

3LE, U. K.

370

55 (1949), 45 3-