topics will be needed in §3.

Non-abelian cohomology. We will de¬ne H 0 (X, Y ) and H 1 (X, Y ) when Y is a group

object in B, more or less as is usually done ([Serre (1965), p. I-56 and ¬.], for example).

G A with left adjoint F , and Y a group object in |B| means

We are in the situation U : B

that every hom set (X, Y ) where X ∈ |B| has a group structure which is (contravariantly)

natural in X. The group operations in the hom set will be written

±

G Y (product)

y0 —¦ y1 : X

y0 ,y1 ,y

G Y (inverse)

GY CQ y ’1 : X

X

G Y (neutral element)

1 = 1X : X

We will construe the 1-cocycles and the 1-coboundaries as objects and maps in a category

Z 1 (X, Y ). A (non-abelian) 1-cocycle is a map a: XG2 G Y such that 2 a —¦ 0 a = 1 a

G Y (regarding i : XG3 G XG2 , i = 0, 1, 2). b: a G a is a 1-

as maps XG3

G Y such that a —¦ 0 b = 1 b —¦ a .

coboundary or a map of 1-cocycles if b is a map XG

G a is given by the neutral map 1XG : XG G Y , and

The identity map a

b—¦b

Ga Ga CQ Ga

b b

a a

that is, composition in Z 1 (X, Y ) is induced by multiplication in the group of maps

G Y . Z 1 (X, Y ) is therefore a category. In fact, it is a groupoid, a category in

XG

19

TRIPLES, ALGEBRAS AND COHOMOLOGY

which every map is an isomorphism. The objects of Z 1 (X, Y ), i. e., the 1-cocycles, there-

fore fall into equivalence classes, and one can speak of the automorphisms of any one

1-cocycle in this category. We de¬ne

H 1 (X, Y ) = the set of isomorphism classes of 1-cocycles,

H 0 (X, Y ) = the automorphism group of the trivial 1-cocycle 1XG2 ∈ |Z 1 (X, Y )|.

H 1 (X, Y ) is a set with distinguished element [1XG2 ], the isomorphism class of the

neutral 1-cocycle, and H 0 (X, Y ) is a group. Clearly if Y is an abelian group object, both

H 1 and H 0 are abelian groups and coincide with the cohomology groups de¬ned earlier.

The non-abelian cohomology shares the properties of the abelian theory to the degree

that it is de¬ned. H 1 (AF, Y ) = 1, and there is a six-term “exact” sequence. Theorem 3

also continues to hold.

Nonhomogeneous complex. This cochain complex C(X, Y ) = (C n (X, Y )), n ≥ 0, is

derived by adjointness from the standard (homogeneous) complex (XG— , Y ). The cochains

in the nonhomogeneous complex will be maps in the underlying category A. The word

“nonhomogeneous” refers to the varied forms of the terms occurring in the cobound-

ary formula. It will be evident that the nonhomogeneous complex exists whenever one

GB G A. However in stating the formulas we shall con-

has adjoint functors A

¬ne ourselves to the tripleable case (B = AT ) which is the only one we will need. We

will then prove that the adjointness isomorphism ±T gives an isomorphism of complexes

G (X(GT )— , Y ).

C(X, Y )

Let X = (X, ξ) be a T-algebra and (Y, θ) an abelian group object in the category of

T-algebras. (From now on we often suppress the algebra structures from the notation for

brevity). We de¬ne

n ≥ 0,

C n (X, Y ) = (XT n , Y ),

G C n+1 (X, Y ) is given by d = (’1)i di , 0 ¤ i ¤ n + 1,

and the coboundary d: C n (X, Y )

where

ξT n · a, i=0

XT i’1 µT n’i · a, 1 ¤ i ¤ n

adi =

aT · θ, i=n+1

G Y (a map in A). This de¬nes C(X, Y ). We now have that

for any n-cochain a: XT n

G ((X, ξ)Gn+1 , (Y, θ)),

±

n ≥ 0,

C n (X, Y )

is an isomorphism of cochain complexes of abelian groups.

(Superscript T will be omitted for the time being, ± = ±T , · · ·. The above form of

words is not accurate since C(X, Y ) is not yet known to be a cochain complex. But if the

±™s are additive isomorphisms and commute with the coboundary operators, as we shall

show, then C(X, Y ) is a complex and as such isomorphic to the homogeneous complex).

20 JONATHAN MOCK BECK

Indeed,

C n (X, Y ) = (XT n , Y )

= (XF (U F )n’1 U, (Y, θ)U )

±G

(X(F U )n F, (Y, θ))

= ((XT n+1 , XT n µ), (Y, θ))

= ((X, ξ)Gn+1 , (Y, θ))

for n ≥ 0. Moreover each diagram

G ((X, ξ)Gn+2 , (Y, θ))

C n+1 (X, Y )

y y

±

( i ,(Y,θ))

di

G ((X, ξ)Gn+1 , (Y, θ))

C n (X, Y ) ±

commutes, 0 ¤ i ¤ n + 1. Recalling that = Gi Gn’i , we note

i

ξT n+1 , i=0

(X, ξ) i =

, 1¤i¤n+1

i’1 n’i+1

XT µT

G Y is a nonhomogeneous n-cochain, then2

If a: XT n

a± · ((X, ξ) 0 , (Y, θ)) = ξT n+1 · aT · θ

= (ξT n · a)T · θ

= (ad0 )±

which checks the square for i = 0; we omit the veri¬cation for 1 ¤ i ¤ n + 1.

As a sample, here are the coboundaries of a ∈ C 1 (X, Y ), b ∈ C 0 (X, Y ):

(a)d = ξT · a ’ Xµ · a + aT · θ,

(4)

(b)d = ξb ’ bT · θ

Finally, we need the nonhomogeneous category of (non-abelian) 1-cocycles, Z 1 (X, Y )

(same notation as in the homogeneous case). Con¬ning ourselves to the tripleable case, a

G (Y, θ) can be replaced using adjointness by a nonhomogeneous

1-cocycle a: (X, ξ)G2

G Y . Similarly, a 1-coboundary b: a G a in the category of 1-

1-cocycle a: XT

G (Y, θ), can now be thought of as a map b: X

cocycles, formerly a map b: (X, ξ)G

G Y in A. The homogeneous cocycle and coboundary conditions now have to be

translated into nonhomogeneous terms using the formulas from the foregoing discussion

G Y is a 1-cocycle, and b: a G a is a map of

of C(X, Y ). We ¬nd that a: XT

G Y in A) if the following formulas hold:

1-cocycles (where b: X

(aT · θ) —¦(ξT · a) = Xµ · a,

(5)

= (bT · θ) —¦ a

a —¦ ξb

2

Editors™ note: In the original thesis, the third line in the display was (ad0 )± but the subscripted 0

seemed inconsistent with previous notation

21

TRIPLES, ALGEBRAS AND COHOMOLOGY

Abelianized, these formulas read: (a)d = 0, (b)d = 0 (cf. (4) above). Of course, in the last

G Y , XT GY

two formulas we have used multiplication in the group of maps XT 2

in A. We assumed to start with that (Y, θ) ∈ |GpAT | (the category of group objects in

AT ); since adjoints preserve groups as well as abelian groups, the image of (Y, θ) under

AT G A, namely Y, is a group object in A.

3. Interpretation of Cohomology in Dimensions 0 and 1

GB G A there is a map of the interpretation into

For a general adjoint pair A

the cohomology (at least in the dimensions we will consider, 0 and 1). The main result

in this section is that this map is an isomorphism if the adjoint pair is tripleable. Here

“interpretation” means the hom set in dimension 0 and a concept of principal object for

the coe¬cient group in dimension 1. The latter can be made to specialize to extensions of

algebras of the type usually classi¬ed by Ext1 or H 2 . In this sense the triple cohomology

generalizes Ext (it always classi¬es extensions in dimension 1).

G B, U : B G A, ±: F

From now on we assume F : A U a ¬xed adjointness.

G = U F is the functor part of the standard cotriple in B used to de¬ne cohomology with

respect to the underlying object functor U .

In dimension 0 a natural map

G H 0 (X, Y )

(X, Y )

G Y , consider the

where ( , ) denotes the hom set in B, is de¬ned as follows. If y: X

diagram

G

0 y

XG2 GX GY

G XG

1

Then y is in Z 0 (X, Y ) since 0 = 1 . The desired map sends y =’ [ y], the correspond-

ing 0-dimensional cohomology class.

THEOREM 4.

G H 0 (X, Y )

(X, Y )

is an isomorphism if the adjoint pair F U is tripleable.

Proof. In this case we write X = (X, ξ), Y = (Y, θ) in AT (replacing B by AT to which

G AT ). The above diagram becomes, in this case,

it is equivalent via ¦: B

ξT

G ξ y

2 G (X, ξ) G (Y, θ)

G (XT, Xµ)

(XT , XT µ)

Xµ

The ¬rst three terms constitute a coequalizer diagram, so the result follows. (Given any

G (Z, ζ) such that ξT ·z = Xµ·z then X· ·z: (X, ξ) G (Z, ζ) is uniquely

z: (XT, Xµ)

determined by its satisfying the equation z = ξ(X· · z). The same calculation, showing

22 JONATHAN MOCK BECK

that a T-structure is a coequalizer in the underlying category, appeared in the proof of

Theorem 1.)

G H 0 (X, Y )

Of course if Y is a pointed object, group or abelian group object, (X, Y )

preserves the structures that arise.

In dimension 1 the cohomology classi¬es objects which resemble principle bundles

trivialized by passage to the underlying category.

p

G X is a Y -principal object over X, in B, (with given trivialization)

DEFINITION 4. E

relative to the underlying object functor U , if

(1) The group object Y operates on E, that is, there is a natural transformation

—¦

G ( , E)

( , Y ) — ( , E)

GY, e

satisfying (y0 —¦ y1 ) —¦ e = y0 —¦(y1 —¦ e), (1 —¦ e) = e. Here y0 , y1 are any maps B

G E in B, and 1 is the neutral element in the group of maps (B, Y ).

is any map B

GY,

(2) The operation of Y is compatible with the projection p. That is, if maps y: B

G E are given, then (y —¦ e)p = ep.

e: B

GE

(3) Y operates in the following simply-transitive fashion: given two maps e0 , e1 : B

G Y such that

such that e0 p = e1 p, then there exists one and only one map y: B

y —¦ e0 = e1 .

G EU in the underlying

(4) There is given as part of the structure a section s: XU

category A, splitting the projection, s · pU = XU . By adjointness s can also be taken

G E. The condition that it should split the projection is

as a map s: XU F = XG

G X. In the following we shall use whichever version of s is

then sp = X : XG

convenient.

f

G E is de¬ned to be a map of Y-principal objects over X if f preserves the

E

projections

f

GE

Ec

cc

c

p cc

p

1

X

and commutes with the operations of Y , (y —¦ e)f = y —¦ ef .

Y -principal objects over X trivialized with respect to U form a category which we

shall denote by

PO(X, Y ) (relative to U)

G EU in A as the ana-

A remark about the sections: one should think of s: XU

logue of a local trivialization of a principal bundle. Intuitively, passage to the underlying

category restricts the principal object to a covering of X which it is required to become

trivial with respect to. For principal bundles triviality means a global product structure.