map is the same thing as a function f : Z

(z1 z2 )f = z1 f + (z2 f )z1 .

G M , where M is treated as a right Z-module via the ¬xed

Thus f is a derivation of Z

G π. We have

map Z

H 0 (Z, Y )π Der(Z, M ) .

G

This isomorphism depends on the fact that the underlying set-over-π functor (G , π)

G A, as remarked in §3.

(A, π) is tripleable; this is a consequence of tripleableness of G

G π of π by the π-module

The cohomology group H 0 (π, Y )π classi¬es extensions E

G π. Using the given set section s: π G E we get an isomorphism E π — M in

Y

G

G (x, 0). The group structure of E will be determined

which s corresponds to the map x

G Y in (A, π), and it is su¬cient to know the M -component of

by a 1-cocycle a: πT

G M . The addition in the π-module being

a. We regard the cocycle as a map a: πT

carried out in the M -component, we ¬nd that the multiplication in E π — M is:

(x1 , m1 )(x2 , m2 ) = (x1 x2 , m1 + m2 x1 + [(x1 )(x2 )]a)

46 JONATHAN MOCK BECK

(by the remarks after Theorem 6, ψ = θ + pT · a). These are the usual extensions of the

group π by the π-module M . In general, maps of extensions will not commute with the

product representations, which depend on the sections. A map of two extensions will be

G G M is the 0-cochain whose coboundary

G (x, m + xb). Then b: π

of the form (x, m)

puts the two extensions into the same 1-cohomology class. One conjectures as a result

(taking dimension 0 considered above into account):

Der(π, M ), n=0

G

H n (π, Y )π n+1

EM (π, M ), n > 0

(EM being the Eilenberg-Mac Lane theory). This is proved by an acyclic models method

in [Barr & Beck (1966)].

We shall now give a series of examples involving linear algebras.

EXAMPLE 6. Associative K-algebras with identity. K being a commutative ring, we

let A denote the category described. A has few group objects. Indeed, if Y ∈ AbA , then

G Y.

the zero operation in Y must be represented by a map of the terminal object 0

Since this map must preserve identity elements, 1 = 0 in Y ; therefore Y = 0. To get

abelian group objects, we must consider categories of modules. Let Λ be a K-algebra.

Then

ker G Λ-Λ-Bimodules

Λ-Mod = Ab(A , Λ)

is an equivalence of categories.

G Λ be a Λ-module. There is a zero section Λ G Y in A , so we can write

Let Y

Λ • M as a K-module, and its multiplication will be of the form

Y

(»1 , m1 )(»2 , m2 ) = [(»1 , 0) + (0, m1 )][(»2 , 0) + (0, m2 )]

= (»1 , 0)(»2 , 0) + (»1 , 0)(0, m2 ) + (0, m1 )(»2 , 0) + (0, m1 )(0, m2 )

= (»1 »2 , »1 m2 + m1 »2 + m1 m2 ) .

The ¬rst component must be »1 »2 since the projection is an algebra map, »1 m2 is de¬ned

as (»1 , 0)(0, m2 ) and is an element of the kernel, m1 »2 similarly, and m1 m2 appears because

G Y in (A , Λ) will be

the kernel is multiplicatively closed. The addition map Y —Λ Y

G Λ • M . This

given by addition in the kernel, i.e., it is equivalent to Λ • M • M

addition must be an algebra map. We leave to the reader the easy task of showing that

this imposes the condition m1 m2 = 0. Thus, we obtain that Y is equivalent to the split

extension of Λ by the two-sided Λ-module M , with multiplication

(»1 , m1 )(»2 , m2 ) = (»1 »2 , »1 m2 + m1 »2 )

47

TRIPLES, ALGEBRAS AND COHOMOLOGY

Some cohomology theories in A come from the following underlying object functors:

U5

GA

A

u T ss 5

TT ss

u˜

U4 uu ˜˜ TT sssU2

u˜

uu TT sss

uu ˜˜˜

zu 6

TT

˜ TT A

A4 ˜˜˜ U3 U0 2

U1 T

TT

˜

˜˜ TT

˜ T&

”˜˜

GA o

A A

3 0 1

all of which are tripleable. A2 is K-modules, and the coadjoint F2 U2 is the tensor

algebra functor AF2 = K + A + A —K A + · · · with juxtaposition as multiplication. A T2

G A is equivalent to an associative-algebra-with-1 structure on the K-

structure ξ: AT2

U1 is given by AF1 = (A —Z K)F2 ,

module A. A1 is the category of abelian groups. F1

coadjoint of a composition being the composition of the coadjoints. A0 is the category of

sets. AF0 is the polynomial K-algebra with the elements of the set A as noncommuting

variables. A4 is the category of rings. U4 forgets K-structure, F4 puts it back in; AF4 =

A —Z K. A3 is the category of monoids. U3 remembers only the multiplication and the

multiplicative identity. AF3 is the K-monoid algebra (free K-module with basis A and the

G A, so that

obvious multiplication.) Note that a T3 -structure is a monoid map AT3

a T3 -algebra keeps its original multiplicative structure and receives the new operations of

addition and K-scalar multiplication; similar comments apply for all the other cases.

These triples give cohomology theories, of which we shall only consider the groups

Hin (Λ, Y )Λ for i = 0, . . . , 5. These are the cohomology groups of, for instance, the homo-

geneous complexes

G (Λ(Gi )2 , Y )Λ )

G (ΛGi , Y )Λ ) G ···

0

where the cotriples Gi = Ui Fi are being used to build up free resolutions of varying depths

of freeness, as it were.

The theories arising when i = 2 or 0 are known. Relative to U2 the extensions clas-

si¬ed are K-linearly split, and their K-algebra structures are only twisted with regard

G Λ relative to U2 is isomorphic to

to multiplication. Thus an extension of Λ by Y

Λ • M as a K-module and has a multiplication given by (3) plus a bilinear function of

two variables with values in M :

(»1 , m1 )(»2 , m2 ) = (»1 »2 , »1 m2 + m1 »2 + (»1 , »2 )a) .

Thus the extensions and, one supposes, the whole cohomology theory agree with that

de¬ned by [Hochschild (1945)]. In fact, an isomorphism

Der(Λ, M ), n=0

G

n

H2 (Λ, Y )Λ

Hochn+1 (Λ, M ), n > 0

G Λ). In the U0 -cohomology

is obtained in [Barr & Beck (1966)], where M = ker(Y

theory, the extensions are isomorphic to Λ—M only as sets. All three structures - addition,

48 JONATHAN MOCK BECK

K-scalar multiplication, and multiplication - are twisted. These extensions have been

classi¬ed by a cohomology theory devised by Shukla [Shukla (1961)]. Barr has shown

n

[Barr (1967)] that the H0 groups are isomorphic to the Shukla groups, with the same

degree 0 value and shift in dimension as above. Note that whatever underlying category

we descend to, if A is tripleable over it, H 0 will always be the hom functor in (A , Λ),

which is derivations, by formula (3).

n n n

The theories H1 ,H3 ,H4 have not been studied. In dimension 1 they classify exten-

sions which are additively, multiplicatively, and both additively and multiplicatively split,

respectively.

We have refrained from speaking until now about the cohomology theory given by the

underlying object functor U5 , which is a little bizarre. We take A5 to be the category of

K-Lie algebras. As a K- module ΛU5 is the same as Λ, and has the Lie algebra operation

[»1 , »2 ] = »1 »2 ’ »2 »1 . In [Lawvere (1963)] it is proved that such “algebraic functors”

always have coadjoints. In the triple context one proves that they are in fact tripleable.

Thus we get a cohomology theory H5 in A whose 1-dimensional part classi¬es algebra

n

extensions which are split with respect to their Lie algebra structures. This theory is

nonzero, because when applied to a commutative K-algebra, and a commutative module

(see below), it gives the ordinary K-split commutative theory.

We conclude Example 6 with a few remarks about exactness in the category of

G A, or better,

Λ-modules and the choice of the underlying object functor U : A

G (A, Λ). Recall that a sequence of Λ-modules

(U, Λ): (A , Λ)

GY GY GY G0

0 cc

c

c

c

(3) c

c1

Λ

is (U, Λ)-exact in Λ-Mod = Ab(A , Λ) if

G (A, Y )Λ G (A, Y )Λ G (A, Y )Λ G0

0

G Λ in the underlying

is an exact sequence of abelian groups for every object A

category (A, Λ). ( ( , )Λ denotes the hom functor in (A, Λ), and strictly speaking we

should introduce Y U ™s into the above sequence.) It is in this situation that a long exact

sequence arises in the U -cohomology (see §2, Theorem 2). For convenience identify Y

G Λ regarded as a Λ-Λ-bimodule, and the

with Λ • M where M is the kernel of Y

same for Y , Y (see the computation of Λ-Mod at the beginning of this example). Then

(3) gives rise to the sequence of Λ-Λ-bimodules

GM GM GM G0

(4) 0

G A as above, and interpret some of the resulting (Ui , Λ)-

Let us now take U = Ui : A i

exactness. We shall ¬nd that (3) is (Ui , Λ)-exact ⇐’ (4) is an ordinary exact sequence of

Λ-Λ-bimodules and has an additional splitting property relative to the underlying category

Ai .

49

TRIPLES, ALGEBRAS AND COHOMOLOGY

G A , the category of K-modules. Letting A G Λ be an

First consider U2 : A 2

object of (A2 , Λ) with zero projection, and choosing A variously, one sees that (4) must

G Λ one ¬nds a map Y o

be exact in the usual sense; taking A = Y Y that splits

G Y in (A , Λ). Thus (3) is (U2 , Λ)-exact ⇐’ (4) is a K-split exact sequence of

Y 2

Λ-Λ-bimodules. This is the type of coe¬cient sequence usually considered in Hochschild

cohomology, (cf. [Mac Lane (1963), p. 287]).

Similarly, (3) is (U1 , Λ)-exact (A1 being the category of abelian groups) ⇐’ (4) is

a Z-split exact sequence of Λ-Λ-bimodules, and (3) is (U0 , Λ)-exact (A0 = sets) ⇐’

(4) is exact in the ordinary sense; here no additional splitting condition enters, except

G M is onto. (U4 , Λ)-exactness (A = rings)

set-theoretically, in showing that M 4

is equivalent to M = M • M as bimodules. We leave the formulation of the curious

(U3 , Λ)- and (U5 , Λ)-exactness to the reader.

The same study of exactness can be carried out in all the other examples, but we omit

it.

EXAMPLE 7. Lie Algebras. We include this divertissement as an elementary demon-

stration of the fact that our theory does not discriminate against non-associative systems.

Let L be the category of K-Lie algebras, A the category of K-modules, K a commutative

G A be the usual underlying. The free Lie algebra functor F

ring, and let U : L U

is described in [Cartan & Eilenberg (1956), p. 285]. This generates a triple T in A as

G XT as x ’ (x), so that the

usual, with T = F U . If X ∈ A, we write the map X

symbol (x) obeys (x0 ) + (x1 ) = (x0 + x1 ), k(x) = (kx). Other elements of XT are written

in terms of Lie algebra operations applied to the generators (x). For example, [(x0 ), (x1 )]

is in XT , while ([(x0 ), (x1 )]) and

W0 = [([(x0 ), (x1 )]), ((x2 ))],

W1 = [([(x0 ), (x2 )]), ((x1 ))] + [((x0 )), ([(x1 ), (x2 )])]

G XT ,

are in XT T . Under the triple multiplication XT T

W0 ’ [[(x0 ), (x1 )], (x2 )],

W1 ’ [[(x0 ), (x2 )], (x1 )] + [(x0 ), [(x1 ), (x2 )]],

and these are equal as elements of the free Lie algebra XT .

Now let (X, ξ) be a T-algebra. X is a K-module, and we introduce a possible Lie

bracket in X by de¬ning

[x0 , x1 ] = [(x0 ), (x1 )]ξ

G XT ,

(cf. Example 1 on groups). Clearly, under ξT : XT T

W0 ’ [([x0 , x1 ]), (x2 )],

W1 ’ [([x0 , x2 ]), (x1 )] + [(x0 ), ([x1 , x2 ])].

Thus we have

[[x0 , x1 ], x2 ] = W0 · ξT · ξ

50 JONATHAN MOCK BECK

W0 · Xµ · ξ

=

W1 · Xµ · ξ

=

W1 · ξT · ξ

=

= [[x0 , x2 ], x1 ] + [x0 , [x1 , x2 ]].

Since ξ is associative, [ , ] satis¬es Jacobi™s identity (!).

G AT . Modules are (antisymmetric) as usual. H 1

Of course the canonical ¦: L

classi¬es K-split extensions [Mac Lane (1963)]. L is also tripleable over sets with exten-

sions as in [Dixmier (1957)].

Other types of linear algebras are also tripleable and have triple cohomology, for

example, the algebras in [Eilenberg (1948)], Jordan algebras [McCrimmon (1966)], Lie

triple systems [Harris (1961)] (obviously there is no restriction to binary systems), . . . .

I am indebted to Michael Barr for showing me the following pathological case.