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as well as a non-homogeneous complex used in making calculations of cohomology. These
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)



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.

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