Editors™ note: The editors believe that the “a” at the right end of the last line is correct, although

the original shows a ξ.

32 JONATHAN MOCK BECK

Hence

X· · a = X·· · Xµ · a

= X··((aT · θ) —¦(ξT · a))

= (X·· · aT · θ) —¦(X·· · ξT · a)

= (X· · a) —¦(X· · a)

In a group, this implies X· · a = 1.

G a is a map of 1-cocycles, then b˜’1 = X —

It must also be proved that if b: a

Y —¦ b: a˜’1 G a ˜’1 in PO(X, Y ). One must dissect the diagram

(X—Y —¦ b)T

G (X — Y )T

(X — Y )T

π π

XT — Y T XT — Y T

ξ—θa ξ—θa

GX —Y

X —Y —¦b

X—Y

as in the proof of associativity, and at the end invoke the coboundary relation a —¦ ξb =

(bT · θ) —¦ a . Also one has to use the fact that X — Y —¦ b can be written as X — (πY —¦ πX b).

G Y in A to start with.)

(b is a map X

From all the foregoing we know that ˜’1 is correctly de¬ned with regard to T-

structures, both on objects and on maps. However, we do not quite know yet that

the values of ˜’1 are actually (Y, θ)-principal algebras. We have to verify that the T-

structure and the Y -operation on X — Y in a˜’1 are connected by the multiplicativity

G X — Y in A, that is, e = (e0 , e1 ) where

relation stated in Lemma 4. We take e: A

G X and e1 : A G Y , and recalling that Y operates on the left of the second

e0 : A

factor of X — Y , we have

(y —¦(e0 , e1 ))T · π(ξ — θa) = (e0 T, (y —¦ e1 )T )(ξ — θa)

= (e0 T · ξ, ((y —¦ e1 )T · θ) —¦(e0 T · a))

= (e0 T · ξ, (yT · θ) —¦(e1 T · θ) —¦(e0 T · a)),

(yT · θ) —¦(e0 , e1 )T · π(ξ — θa) = (yT · θ) —¦(e0 T · ξ, (e1 T · θ) —¦(e0 T · a))

= (e0 T · ξ, (yT · θ) —¦(e1 T · θ) —¦(e0 T · a)).

The multiplicativity condition to be satis¬ed by b˜’1 = X — Y —¦ b (Lemma 4) can be

similarly veri¬ed. This completes the proof of Theorem 5.

If there were an exact sequence in the ¬rst variable of the cohomology, all of this might

be avoided. (See [Barr & Rinehart (1966)], for such a possibility.)

33

TRIPLES, ALGEBRAS AND COHOMOLOGY

Modules and Extensions. We shall now consider a special case of the foregoing the-

ory, in which principal objects are interpreted as algebra extensions of a given algebra

by one of its modules. This specialization is the version of triple cohomology outlined in

[Barr & Beck (1966)]. In this part we limit ourselves to discussing tripleable adjoint pairs,

for brevity.

Recall that the “comma category” (AT , X), where X is a T-algebra, has maps Z

G X in AT as its objects, and commutative triangles

GZ

Zc

cc

cc in AT

c1

X

as its maps. This is also called the category of objects over X. (See [Lawvere (1966)] for

the general de¬nition of this useful notation. Nothing depends on our having T-algebras

in this de¬nition. X could be an object in any category. The same is true of the following

de¬nition.)

DEFINITION 5. An X-module is an abelian group object in the category (AT , X).

If AT has pullbacks (¬bered products) the addition in an X-module will be represented

by a binary operation

GY

Y —X cY

cc

cc

c1

X

(The ¬bered product is the ordinary cartesian product in (AT , X). It exists if ¬bered

products exist in A.) The identity map X is always terminal in (AT , X), since an object

G X admits the unique map

p: Z

p

GX

Zc

cc

c

p cc X

1

X

Since a terminal object is a 0-fold product, the nullary operation consisting of the zero

G X will be represented by a map of the terminal object:

element in an X-module p: Y

GY

s

Xc

cc

cc p

X c1

X

Thus an X-module always has a zero section, which splits the projection, sp = X, in the

category AT . In e¬ect, our procedure is to identify X-modules with split extensions.

34 JONATHAN MOCK BECK

Now let A F G AT G A be a tripleable adjoint pair. Then we get adjoint functors

U

on comma categories

(F,X) (U,X)

G (AT , X) G (A, X)

(A, X)

where X = (X, ξ) is a given T-algebra. (U, X) is the obvious forgetful functor. (F, X) is

the functor which on objects:

Z (ZT, Zµ)

CQ

p pT

X (XT, Xµ)

ξ

(X, ξ)

This is the usual free algebra functor lifted up to the comma categories. The composition

(F, X)(U, X) induces a triple (T, X) in (A, X) given by

(·,X) (µ,X)

G ZT G ZT

Zc ZT T

c

cc cc

c cc

p cc

pT T ·ξT ·ξ c1

pT ·ξ pT ·ξ

1

X X

p

G X. The adjoint pair (F, X)

on a typical object Z (U, X) is tripleable, that is, the

canonical functor

¦

(A, X)(T,X) o (AT , X)

cc

cc

cc

cc

cc (U,X)

(T,X)

cc

U

1

(A, X)

GX∈

is an isomorphism. For it is a triviality to verify that a (T, X)-structure on p: Z

|(A, X)|

GZ

θ

ZTc

cc

cc

p

pT ·ξ c1

X

p

G (X, ξ), i.e., an object of (AT , X).

is precisely equivalent to a T-algebra map (Z, θ)

We have a cohomology theory

n ≥ 0,

H n (Z, Y )X ,

35

TRIPLES, ALGEBRAS AND COHOMOLOGY

de¬ned for Z a given T-algebra over X (suppressing the algebra structures and the projec-

G X from the notation) and Y a given X-module. The cohomology is relative

tion Z

to the underlying object functor (U, X): (AT , X) G (A, X). The subscript X is put

in as a reminder that all algebras, . . . are being considered with given projections into

X. The complexes used to de¬ne these groups resemble the usual ones (homogeneous or

nonhomogeneous), but all cochains have to be maps over X.

By tripleableness, H 0 (Z, Y )X is the abelian group of maps

GY

Zc

cc

cc in (AT , X) ·

c1

X

G Z over X. We will return to this

H 1 (Z, Y )X classi¬es Y -principal T-algebras E

in a moment, giving a separate interpretation to the case Z = X. First we point out that

there is a sort of co-Shapiro lemma by which one can in e¬ect always assume that Z = X,

that is, there is a cohomology isomorphism

H(Z, Y p’1 )Z G H(Z, Y )X ·

This results from the adjoint pair

(AT ,p)

G

T

(AT , X)

(A , Z) o

p’1

where if Y is an algebra over X, Y p’1 is the pullback (existing, as remarked above, if

A has pullbacks). The coadjoint is just composition with p, the ¬xed structural map Z

G X. Since p’1 is an adjoint, it preserves abelian group objects:

±

Y p’1 • • • •G Y

Z-module X-module

GX

Z

p

This explains the appearance of Y p’1 as coe¬cients above. Once everything is de¬ned,

the proof of the co-Shapiro lemma is trivial (the two complexes involved are isomorphic

under the above adjointness). In this sense it is su¬cient to consider H 1 (X, Y )X , which

evidently provides a cohomology classi¬cation for the objects described in the following

de¬nition:

G X (in the category (AT , X))

DEFINITION 6. An extension of X by the X-module Y

G X)-principal T-algebra over the terminal object X G X, which is trivial-

is a (Y

ized relative to the underlying object functor (U, X): (AT , X) G (A, X).

36 JONATHAN MOCK BECK

Explicitly, such an extension consists of the following data:

p

G X.

(1) A T-algebra map E

The de¬nition insists on our giving a T-algebra in (AT , X) together with a projection

map p into the terminal object:

p

GX

Ec

cc

cc in AT ·