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in the category of sets, A, by
(a,1) (a,x1 x2 )
G A Г— ПЂ, A Г— ПЂ Г— ПЂ G A Г— ПЂ.
A

This triple is also denoted by ПЂ. The category of ПЂ-algebras, AПЂ , consists of sets equipped
G A, that is, right ПЂ-sets. A group object
with unitary, associative ПЂ-structures A Г— ПЂ
in AПЂ consists of a group object in A, i.e., an ordinary group, G, with a ПЂ-structure G Г— ПЂ
G G which is compatible with the multiplication in G: (g1 g2 )x = (g1 x)(g2 x). Such an
object is called a right ПЂ-group. An abelian group object in AПЂ is then just a right-ПЂ-module
4
EditorsвЂ™ note: At the time the thesis was written, compact meant compact Hausdorп¬Ђ, under the
inп¬‚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 deп¬Ѓned for a right ПЂ-set A and a right ПЂ-module Y . They can be calculated from
the nonhomogeneous complex (В§2) which in this case becomes: C n (A, Y ) = all functions
GY,
f : A Г— ПЂn
(a, x1 , . . . , xn+1 )(f d) = (ax1 , x2 , . . . , xn+1 )f
n
(в€’1)i (a, x1 , . . . , xi xi+1 , . . . , xn+1 )f
+
i=1

+ (в€’1)n+1 (a, x1 , . . . , xn )f xn+1
being the coboundary of an n-cochain f в€€ 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 ПЂ-set (in which case terms of the form ax drop out). The groups H n (1, Y ),
relative to the triple ПЂ are usually written H n (ПЂ, Y ) and called the (Eilenberg-Mac Lane)
cohomology groups of ПЂ.
G Y (i.e.,
As we know, H 0 (A, Y ) is isomorphic to the group of right ПЂ-maps A
equivariant maps). When A = 1, this reduces to the invariant elements of Y (those
y в€€ Y such that yx = y for all x в€€ ПЂ). The interpretation of H 1 (A, Y ) is well known
when A = 1. Then H 1 classiп¬Ѓes principal homogeneous ПЂ-sets for the ПЂ-module Y (see
[Serre (1965), I-56 п¬Ђ.]). When A = 1, a representative of an element of H 1 (A, Y ) can be
pictured as y

E operates п¬Ѓberwise
Y


y1
1
1s
p
1

A
E, p, A, Y are all right ПЂ-maps, and the operation of Y is consistent with ПЂ-structures.
The п¬Ѓbers 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 Г— ПЂ
G Y and is a derivation in the usual sense. The
can be regarded as a function ПЂ
1-coboundaries are precisely the inner derivations.
EXAMPLE 4. Cohomology in the category of groups. We will consider the cohomology
groups H n (ПЂ, Y ) where ПЂ 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 ПЂ does not
operate on Y (or operates trivially). For the cohomology of ПЂ with coeп¬ѓcients in a ПЂ-
module Y , we refer to Example 5; we want to study this example п¬Ѓrst, in order to п¬Ѓlter
42 JONATHAN MOCK BECK

the complications. Of course, вЂњcohomologyвЂќ in G is not deп¬Ѓned 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 ПЂ as its counit (ПЂ в€€ G ). H n (ПЂ, Y ) is the nth cohomology group of
free group ПЂ : ПЂG
either of the following two complexes. The homogeneous complex (В§2) is

G (ПЂG2 , Y )
G (ПЂG, Y ) G В·В·В· G (ПЂGn+1 , Y ) G В·В·В·
d d d d
0
n i i nв€’i
where d = i=0 (в€’1) (ПЂG G , Y ) and the hom is in the category of groups. The
nonhomogeneous complex is
G (ПЂ, Y ) G (ПЂT, Y ) G В·В·В· G (ПЂT n , Y ) G В·В·В·
0
G A is the underlying triple
where we have written ПЂ, 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 В§2.
GY.
Since U is tripleable, H 0 (ПЂ, Y ) is the abelian group of homomorphisms ПЂ
G ПЂ, as follows. The group
H 1 (ПЂ, Y ) classiп¬Ѓes Y -principal groups over ПЂ, p: E
G ПЂ and Оё: Y T G Y . The group
structures of ПЂ and Y will be written as Оѕ: ПЂT
structure of the trivial principal group ПЂ Г— Y will be componentwise (В§3)5 :

proj. G ОѕГ—Оё
G ПЂ Г— Y.
(ПЂ Г— Y )T ПЂT Г— Y T

For example, if we take the word (x1 , y1 )(x2 , y2 ) в€€ (ПЂ Г— Y )T , its image in ПЂ Г— Y will give
the binary operation of multiplication:

(x1 , y1 )(x2 , y2 ) в‡’ ((x1 )(x2 ), (y1 )(y2 )) в‡’ (x1 x2 , y1 y2 ).
G ПЂ is any Y -principal group over ПЂ, we will have E ПЂ Г— Y as a set but its
If E
group structure will be of the form Оѕ Г— (Оё + a), in the additive form of the notation in В§3,
G Y is a 1-cocycle:
where a: ПЂT

ОѕГ—(Оё+a)
G ПЂT Г— Y T GПЂГ—Y
(ПЂ Г— Y )T

(w0 , w1 ) в‡’ (w0 Оѕ, w1 Оё + w0 a)
5
EditorsвЂ™ note: The second arrow label was simply Оё 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 Оѕ Г—(Оё +a) instructs us to multiply in E (identifying E with ПЂ Г—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 ПЂT . Evidently, Y -principal groups over ПЂ are just
group extensions of ПЂ by Y in the ordinary sense (with ПЂ operating trivially on Y ). We
have an isomorphism
G EM2 (ПЂ, Y )
H 1 (ПЂ, Y )
because of this classiп¬Ѓcation of extensions (EM is the Eilenberg-Mac Lane theory). Using
the method of [Barr & Beck (1966)], one proves
G EMn+1 (ПЂ, Y )
H n (ПЂ, Y )
for n в‰Ґ 0 (n = 0 was covered above).
GY
Some further remarks about principal groups: note that the cocycle a: ПЂT
вЂњtwistsвЂќ the whole group structure on ПЂ Г— 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 XО·В·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 п¬Ѓnds that the cocycle satisп¬Ѓes [(x0 )]a = 0. Thus the underlying
pointed structure of the principal group over ПЂ, 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 coeп¬ѓcients in a module. A ПЂ-module means
an abelian group object in the comma category (G , ПЂ). We п¬Ѓrst show that this general
categorical deп¬Ѓnition reduces to the ordinary notion of right ПЂ-module (or left, with a
diп¬Ђerent choice of product representation below). We shall prove:
ker G Right ПЂ в€’ modules
ПЂ в€’ Mod = Ab(G , ПЂ)
is an equivalence of categories.
G ПЂ into M = ker(Y G ПЂ) with right ПЂ-
This functor maps a ПЂ-module Y
G ПЂ is an abelian
operators deп¬Ѓned by conjugation. We shall give the details. Since Y
group object it must have a zero section (В§3)
Yy
(sp = ПЂ)
p s

ПЂ
G ПЂ Г— M as a set. Explicitly, yПѓ =
which is a map in the category of groups. Thus Y
G ПЂГ—M G Y.
(yp, y(yps)в€’1 ) and (x, m)П„ = m(xs) deп¬Ѓne inverse isomorphisms Y
44 JONATHAN MOCK BECK

Viewing Пѓ as an identiп¬Ѓcation 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 П„ ,
(1, m)(x, 1) = (x, m) .
Conjugation induces right ПЂ-operators in M , since the kernel is an invariant subgroup:

(x, 1)(1, m)(x, 1)в€’1 = (1, mx) .

Thus M is a right ПЂ-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 )) В·
(1) =

Thus Y is isomorphic to the crossed product of ПЂ 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 Г—ПЂ c
Y

c cc

cc
c1 
ПЂ
Replacing Y by ПЂГ—M , we have that (ПЂГ—M )Г—ПЂ (ПЂГ—M ) is universal for pairs of morphisms
(x1 , m1 ), (x2 , m2 ) such that x1 = x2 . Thus Y Г—ПЂ Y is isomorphic to ПЂ Г— M Г— M as a set,
and the above diagram becomes
GПЂГ—M
ПЂГ—M Г—M
c cc 
cc 
cc 
cc 
1 
ПЂ
G M is a group law on M as a right ПЂ-group.
The induced map of kernels M Г— M
Thus

M в€€ Gp(Right ПЂ в€’ groups) = Gp Gp AПЂ
= Ab AПЂ
= Right ПЂ в€’ modules ;
45
TRIPLES, ALGEBRAS AND COHOMOLOGY

G M must coincide with the group law in M (as a
moreover, the map of kernels M Г—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
ПЂ-вЂњmodulesвЂќ.) This proves that the kernel functor takes its values in the category of right
ПЂ -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 ПЂ-module and construct a ПЂ-module in our sense by
formula (2). Thus we have established the desired equivalence.
G ПЂ for a ПЂ-module, M for the
For the rest of this example we use the notation Y
kernel, and we identify Y with ПЂ Г— M with multiplication (2) when convenient.
The cohomology theory which arises is written H n (Z, Y )ПЂ (В§3), and is deп¬Ѓned for Z
G ПЂ any group over ПЂ (object in the comma category (G , ПЂ) and Y G ПЂ a ПЂ-module.
It is the n-th cohomology group of either of the complexes

G (ZG2 , Y )ПЂ
G (ZG, Y )ПЂ G В·В·В· G (ZGn+1 , Y )ПЂ G В·В·В·
0

G (Z, Y )ПЂ G (ZT, Y )ПЂ G В·В·В· G (ZT n , Y )ПЂ G В·В·В·
0
The cochains are maps in (G , ПЂ) or (A, ПЂ) respectively (A = Sets), T is really an abbrevi-
ation for (T, ПЂ), the functor part of the triple induced on the comma category (В§3), and
both of the displayed general terms are in dimension n.
G Y in (G , ПЂ). By formula (2) such a
H 0 (Z, Y )ПЂ is the abelian group of maps Z
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