(a,1) (a,x1 x2 )

G A — π, A — π — π G A — π.

A

This triple is also denoted by π. The category of π-algebras, Aπ , consists of sets equipped

G A, that is, right π-sets. A group object

with unitary, associative π-structures A — π

in Aπ consists of a group object in A, i.e., an ordinary group, G, with a π-structure G — π

G G which is compatible with the multiplication in G: (g1 g2 )x = (g1 x)(g2 x). Such an

object is called a right π-group. An abelian group object in Aπ is then just a right-π-module

4

Editors™ note: At the time the thesis was written, compact meant compact Hausdor¬, under the

in¬‚uence of Bourbaki. Both earlier and later, this was not the standard usage.

41

TRIPLES, ALGEBRAS AND COHOMOLOGY

[Cartan & Eilenberg (1956), Mac Lane (1963), . . . ]. The cohomology groups H n (A, Y )

are de¬ned for a right π-set A and a right π-module Y . They can be calculated from

the nonhomogeneous complex (§2) which in this case becomes: C n (A, Y ) = all functions

GY,

f : A — πn

(a, x1 , . . . , xn+1 )(f d) = (ax1 , x2 , . . . , xn+1 )f

n

(’1)i (a, x1 , . . . , xi xi+1 , . . . , xn+1 )f

+

i=1

+ (’1)n+1 (a, x1 , . . . , xn )f xn+1

being the coboundary of an n-cochain f ∈ C n (A, Y ). This is identical with the Eilenberg-

Mac Lane complex [Eilenberg & Mac Lane (1947)], except that usually one takes A = 1,

the trivial right π-set (in which case terms of the form ax drop out). The groups H n (1, Y ),

relative to the triple π are usually written H n (π, Y ) and called the (Eilenberg-Mac Lane)

cohomology groups of π.

G Y (i.e.,

As we know, H 0 (A, Y ) is isomorphic to the group of right π-maps A

equivariant maps). When A = 1, this reduces to the invariant elements of Y (those

y ∈ Y such that yx = y for all x ∈ π). The interpretation of H 1 (A, Y ) is well known

when A = 1. Then H 1 classi¬es principal homogeneous π-sets for the π-module Y (see

[Serre (1965), I-56 ¬.]). When A = 1, a representative of an element of H 1 (A, Y ) can be

pictured as y

E operates ¬berwise

Y

y1

1

1s

p

1

A

E, p, A, Y are all right π-maps, and the operation of Y is consistent with π-structures.

The ¬bers are all non-canonically isomorphic with Y (via the section s which is not

preserved by maps), and A is the quotient set of the Y -operation.

H 1 can also be interpreted in terms of derivations. A 1-cocycle in the above complex

G Y such that (a, x1 x2 )f = (ax1 , x2 )f + (a, x1 )f x2 . If A = 1, f

is a map f : A — π

G Y and is a derivation in the usual sense. The

can be regarded as a function π

1-coboundaries are precisely the inner derivations.

EXAMPLE 4. Cohomology in the category of groups. We will consider the cohomology

groups H n (π, Y ) where π is a group and Y is an abelian group object in the category

of groups, G . As is well known [Brinkman & Puppe (1965), Eckmann & Hilton (1962),

for example], this just means that Y is an abelian group. In this setting π does not

operate on Y (or operates trivially). For the cohomology of π with coe¬cients in a π-

module Y , we refer to Example 5; we want to study this example ¬rst, in order to ¬lter

42 JONATHAN MOCK BECK

the complications. Of course, “cohomology” in G is not de¬ned except with reference to

an underlying object functor. For this we choose the usual underlying set functor U : G

G A, which we know is tripleable, by Example 1.

G

F

G

Ao (A = sets)

U

The standard cotriple G has G = U F as its functor, and the natural epimorphism of the

G π as its counit (π ∈ G ). H n (π, Y ) is the nth cohomology group of

free group π : πG

either of the following two complexes. The homogeneous complex (§2) is

G (πG2 , Y )

G (πG, Y ) G ··· G (πGn+1 , Y ) G ···

d d d d

0

n i i n’i

where d = i=0 (’1) (πG G , Y ) and the hom is in the category of groups. The

nonhomogeneous complex is

G (π, Y ) G (πT, Y ) G ··· G (πT n , Y ) G ···

0

G A is the underlying triple

where we have written π, Y for the underlying sets, T : A

(Example 1), and the hom is in the category of sets. The coboundary formula can be

found in §2.

GY.

Since U is tripleable, H 0 (π, Y ) is the abelian group of homomorphisms π

G π, as follows. The group

H 1 (π, Y ) classi¬es Y -principal groups over π, p: E

G π and θ: Y T G Y . The group

structures of π and Y will be written as ξ: πT

structure of the trivial principal group π — Y will be componentwise (§3)5 :

proj. G ξ—θ

G π — Y.

(π — Y )T πT — Y T

For example, if we take the word (x1 , y1 )(x2 , y2 ) ∈ (π — Y )T , its image in π — Y will give

the binary operation of multiplication:

(x1 , y1 )(x2 , y2 ) ’ ((x1 )(x2 ), (y1 )(y2 )) ’ (x1 x2 , y1 y2 ).

G π is any Y -principal group over π, we will have E π — Y as a set but its

If E

group structure will be of the form ξ — (θ + a), in the additive form of the notation in §3,

G Y is a 1-cocycle:

where a: πT

ξ—(θ+a)

G πT — Y T Gπ—Y

(π — Y )T

(w0 , w1 ) ’ (w0 ξ, w1 θ + w0 a)

5

Editors™ note: The second arrow label was simply θ in the original. We have changed it because it

seems correct to do so and also because the spacing in the original suggests that something was to have

been added by hand

43

TRIPLES, ALGEBRAS AND COHOMOLOGY

Thus ξ —(θ +a) instructs us to multiply in E (identifying E with π —Y as a set) according

to the rule

(x1 , y1 )(x2 , y2 ) = (x1 x2 , y1 + y2 + [(x1 )(x2 )]a)

where (x1 )(x2 ) is the formal product in πT . Evidently, Y -principal groups over π are just

group extensions of π by Y in the ordinary sense (with π operating trivially on Y ). We

have an isomorphism

G EM2 (π, Y )

H 1 (π, Y )

because of this classi¬cation of extensions (EM is the Eilenberg-Mac Lane theory). Using

the method of [Barr & Beck (1966)], one proves

G EMn+1 (π, Y )

H n (π, Y )

for n ≥ 0 (n = 0 was covered above).

GY

Some further remarks about principal groups: note that the cocycle a: πT

“twists” the whole group structure on π — Y , not just the multiplication. For example,

the new neutral element is (1, 0 + [( )]a). On the other hand, the cocycle a will always

satisfy [(x)]a = 0. (This is the relation X··a = 1, which appeared in the proof of Theorem

5.) The identity operation never is twisted.

G is also tripleable over the category (1, A), the comma category, better known as

the category of pointed sets. The free functor (X, x0 )F is the free group on the set X

modulo the single relation (x0 ) = 1. The whole example can be carried through similarly

in this setting. One ¬nds that the cocycle satis¬es [(x0 )]a = 0. Thus the underlying

pointed structure of the principal group over π, say, is untwisted. (The pointed structure

is the neutral element.) In any principal object, in this theory, the underlying category

structure remains that of a product. Only the added, T-, structure is twisted.

EXAMPLE 5. Cohomology of a group with coe¬cients in a module. A π-module means

an abelian group object in the comma category (G , π). We ¬rst show that this general

categorical de¬nition reduces to the ordinary notion of right π-module (or left, with a

di¬erent choice of product representation below). We shall prove:

ker G Right π ’ modules

π ’ Mod = Ab(G , π)

is an equivalence of categories.

G π into M = ker(Y G π) with right π-

This functor maps a π-module Y

G π is an abelian

operators de¬ned by conjugation. We shall give the details. Since Y

group object it must have a zero section (§3)

Yy

(sp = π)

p s

π

G π — M as a set. Explicitly, yσ =

which is a map in the category of groups. Thus Y

G π—M G Y.

(yp, y(yps)’1 ) and (x, m)„ = m(xs) de¬ne inverse isomorphisms Y

44 JONATHAN MOCK BECK

Viewing σ as an identi¬cation for simplicity, we now express the group multiplication in

GY

Y in terms of the product representation. Knowing that s, p, and the injection M

are group maps, we have

(x1 , 1)(x2 , 1) = (x1 x2 , 1)

(1, m1 )(1, m2 ) = (1, m1 m2 )

(x1 , m1 )(x2 , m2 ) = (x1 x2 , . . .)

and from the formula for „ ,

(1, m)(x, 1) = (x, m) .

Conjugation induces right π-operators in M , since the kernel is an invariant subgroup:

(x, 1)(1, m)(x, 1)’1 = (1, mx) .

Thus M is a right π-group (Example 3). The full multiplication table for Y now emerges:

(x1 , m1 )(x2 , m2 ) = (1, m1 )(x1 , 1)(1, m2 )(x2 , 1)

= (1, m1 )(1, m2 x1 )(x1 , 1)(x2 , 1)

= (1, m1 (m2 x1 ))(x1 x2 , 1)

(x1 x2 , m1 (m2 x1 )) ·

(1) =

Thus Y is isomorphic to the crossed product of π by M .

Up to this point we have only used the zero element of the module. Now we introduce

the addition which, being a binary operation, is represented by a map

GY

Y —π c

Y

c cc

cc

c1

π

Replacing Y by π—M , we have that (π—M )—π (π—M ) is universal for pairs of morphisms

(x1 , m1 ), (x2 , m2 ) such that x1 = x2 . Thus Y —π Y is isomorphic to π — M — M as a set,

and the above diagram becomes

Gπ—M

π—M —M

c cc

cc

cc

cc

1

π

G M is a group law on M as a right π-group.

The induced map of kernels M — M

Thus

M ∈ Gp(Right π ’ groups) = Gp Gp Aπ

= Ab Aπ

= Right π ’ modules ;

45

TRIPLES, ALGEBRAS AND COHOMOLOGY

G M must coincide with the group law in M (as a

moreover, the map of kernels M —M

subgroup of Y ) and be abelian. (These facts follow from [Brinkman & Puppe (1965)] and

[Eckmann & Hilton (1962)]. Notice that in this case there are no properly non-abelian

π-“modules”.) This proves that the kernel functor takes its values in the category of right

π -modules, as required.

The multiplication in Y can now be written as

(2) (x1 , m1 )(x2 , m2 ) = (x1 x2 , m1 + m2 x1 ) ,

the usual formula for multiplication in the split extension. Of course, for the inverse of

the kernel functor, take any right π-module and construct a π-module in our sense by

formula (2). Thus we have established the desired equivalence.

G π for a π-module, M for the

For the rest of this example we use the notation Y

kernel, and we identify Y with π — M with multiplication (2) when convenient.

The cohomology theory which arises is written H n (Z, Y )π (§3), and is de¬ned for Z

G π any group over π (object in the comma category (G , π) and Y G π a π-module.

It is the n-th cohomology group of either of the complexes

G (ZG2 , Y )π

G (ZG, Y )π G ··· G (ZGn+1 , Y )π G ···

0

G (Z, Y )π G (ZT, Y )π G ··· G (ZT n , Y )π G ···

0

The cochains are maps in (G , π) or (A, π) respectively (A = Sets), T is really an abbrevi-

ation for (T, π), the functor part of the triple induced on the comma category (§3), and

both of the displayed general terms are in dimension n.

G Y in (G , π). By formula (2) such a

H 0 (Z, Y )π is the abelian group of maps Z