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(BU, Y U ) o U
(B, Y )

is an abelian group map for all B ∈ |B|. In fact, the abelian group structure on Y U is
such that if ±: F U is any adjointness then

G (AF, Y )
±
(A, Y U )

is an abelian group map.

Proof. Picking any ±, the last displayed map, which is an isomorphism, de¬nes an
abelian group structure on Y U . Since

(BU, Y c ) o
U
U (B, Y )
• cc 

cc

cc

cc 
cc  (B ,Y )
’1
±
c 
(BU F, Y )

commutes, U is an abelian group map. If another left adjoint1 ± : F U is given, there
G F such that
is an isomorphism F

gQ (AF, Y )
ggggg
±
gggg
(A, Y U ) ‡‡‡‡
‡‡‡‡‡ 
‡C
±
(AF , Y )

1
Editors™ note: The original starts with (± : ), but there is no discernible reason for the parentheses
and we have chosen to omit it.
15
TRIPLES, ALGEBRAS AND COHOMOLOGY

commutes [Kan (1958)]. Since the vertical map is an abelian group isomorphism, the
addition de¬ned in (A, Y U ) is independent of the selection of the left adjoint. Finally,
taking any ±: F U , since

’1
(A, Y U ) o
±
(AF, Y )
cc
• 
cc 
cc 
cc 
(A·,Y U ) c U
cc 
c 
(AF U, Y U )

commutes and A· induces an abelian group map, any group law on Y U such that U is
an abelian group map must inevitably be de¬nable in terms of ± in the above manner.
This proves uniqueness.
This lemma allows us to de¬ne U -exactness. The idea is that like the cohomology
itself, the notion of exactness in B should be “relativized” by means of the underlying
GY GY GY G 0 is a U -exact sequence in AbB
object functor U . That is, 0
GY U GYU GY U G 0 is an exact sequence of abelian group objects in
if 0
G (A, Y U ) G (A, Y U ) G (A, Y U )
A, and this we in turn de¬ne to mean that 0
G 0 is an exact sequence of ordinary abelian groups, for every A ∈ |A|.

To see how this exactness concept works in practice, the reader is referred to the
Examples. However, the proof of Theorem 2 can now be completed. Applying adjointness
G (AF, Y ) G (AF, Y ) G (AF, Y ) G 0 is an
to the last sequence above, 0
exact sequence of abelian groups for any free object AF . Since the simplicial resolution
G (XG— , Y ) G
XG— consists of free objects, the sequence of cochain complexes 0
G (XG— , Y ) G 0 is exact. The long exact sequence in cohomology is now
(XG— , Y )
standard. Theorem 2 is proved.

Remark. It is in the sense of U -exactness that XG— is a resolution of X. In the under-
lying category A, the augmented simplicial object XG— U is contractible, with

h’1 h0 h1 hn
G XG2 U
G XGU G ··· G XGn+1 U G XGn+2 U G ···
XU

as contracting homotopy where hn = XGn+1 U ·. (In this case hn = identity, hn =
n+1 i
i hn’1 , 0 ¤ i ¤ n, are the identities satis¬ed.)


The above properties of the cohomology are purely formal, depending only on adjoint-
ness. To interpret the cohomology groups, at least in the lowest dimensions, one must
invoke, as far as I know, the assumption that the adjoint pair is tripleable. Indeed, it is
interesting to note, as mentioned in §1, that the cohomology itself is only a function of
the triple T induced on A by the adjoint pair, in the following sense. Let F U and
16 JONATHAN MOCK BECK

recall the standard diagram
¦
o
AT c B

cc

cc

cc
U
c
U T cc 

cc

1
A

THEOREM 3. ¦ induces a cohomology isomorphism
H(¦)
H(X¦, Y ¦) o H(X, Y )

Here X ∈ |B|, Y ∈ |AbB|, H(X, Y ) is the graded group (H n (X, Y )), n ≥ 0, and the
cohomologies are taken with respect to U T and U .
To give meaning to this theorem we have to describe how Y ¦ is treated as an abelian
group object in AT , and then how the cohomology map H(¦) is induced by ¦. We
actually establish more than is asserted in the theorem, namely, we show that ¦ induces

an isomorphism of cochain complexes X GT , Y o (XG— , Y ). The abelian group
structure on Y ¦ results from the following two lemmas. The ¬rst, Lemma 2, strengthens
Lemma 1 in the tripleable case. It asserts that abelian group objects are not only preserved
but also re¬‚ected by the underlying object functor AT G A:

LEMMA 2. Let (Y, θ) be an abelian group object in AT . There is a unique abelian group
structure on Y ∈ |A| such that the forgetful
UT
(X, Y ) o
(1) ((X, ξ), (Y, θ))
is an abelian group map for every (X, ξ) in AT . This abelian group structure satis¬es
(y0 + y1 )T · θ = y0 T · θ + y1 T · θ
(2)
G Y in A. Conversely, given an abelian group law on Y in A satisfying
for all y0 , y1 : A
(2), there exists a unique abelian group law on (Y, θ) in AT such that (1) is an abelian
group map.
G AF T , (Y, θ) must be an abelian group map. Since
Proof. By lemma 1, ±T : (A, Y )
G (Y, θ) be maps in AT
y±T = yT · θ, (2) follows. For the converse, let y0 , y1 : (X, ξ)
G Y be their sum in A. We have to show that this is a T-algebra
and let y0 + y1 : X
map. But (y0 + y1 )T · θ = y0 T · θ + y1 T · θ = ξy0 + ξy1 = ξ(y0 + y1 ) by naturality of
the group law and condition (2). The other group operations lift similarly. Uniqueness of
course is a result of the fact that U T is faithful.
The next lemma states that the functor ¦ shares with adjoints the property of preserv-
ing group objects. Its proof is based on the U -preserves”U T re¬‚ects principle enunciated
in §1.
17
TRIPLES, ALGEBRAS AND COHOMOLOGY

LEMMA 3. If Y ∈ |AbB|, then there is a unique abelian group law on Y ¦ ∈ AT such
that
(X¦, Y ¦) o ¦ (X, Y )
is an abelian group map for all X ∈ |B|.

Proof. Let us write ±: F U for the adjointness isomorphism. An abelian group law
T
exists on Y ¦ because Y ¦U = Y U is an abelian group in A and the T-structure of Y ¦,
G Y U , satis¬es the linearity condition of Lemma 2: (y0 + y1 )T · Y U =
Y U: Y UT
((y0 + y1 )±) U = (y0 ± + y1 ±)U = (y0 ±)U + (y1 ±)U = y0 T · Y U + y1 T · Y U . Lemma 1 has
been used to achieve linearity of ± and U , and of course (y±)U = (yF · Y )U = yT · T U
G Y U . Now,
for any map y: A

¦
(X¦, Y ¦) o (X, Y )
cc 
cc 
cc 
cc 
UT c c U
c 
c1 
(XU, Y U )

commutes, U T and U are abelian group maps, and U T is faithful, so it follows that the
¦ indicated is an abelian group map. For uniqueness, let (A, ξ) be any T-algebra, and
consider the following diagram:
T ,Y
(A ¦)
G AF T , Y ¦ o ¦
((A, ξ), Y ¦) (AF, Y )

G AT G A with superscript T and recalling that
(labeling everything pertaining to A
¦ preserves “free” functors, F ¦ = F T ). We are supposing that Y ¦ has an abelian group
structure in AT , hence A T , Y ¦ has to be an abelian group map. It is injective, because
T
is an epimorphism (even a coequalizer; or use faithfulness of U T ). Thus if the addition
in AF T , Y ¦ is uniquely determined by the condition that ¦ should be an abelian group
map, then uniqueness will be proved. But the ¦ indicated is an isomorphism because of
the following commutative diagram:

¦
(AF Ty , Y ¦) = (AF ¦, Y ¦) o (AF, Y )
W
ss
ss
s
ss
ss
(3) ±T
ss ±
ss
s
ss
(A, Y ¦U T ) = (A, Y U )

G Y U , then (y±)¦ = (y±)U = (yF · Y )U = yT · Y U = y±T .
Indeed, if y: A
Now we can ¬nish the proof of Theorem 3. Since ¦ preserves both free and underlying
object functors, we have ¦GT = G¦, where GT is the standard cotriple in AT , (X, ξ)GT =
18 JONATHAN MOCK BECK

(XT, Xµ), with counit given by the T-algebra structures. Moreover, both counits are
compatible with the above equality:

¦GT = G¦
AAA !
!!
AA !
A !! ¦
¦ T AA !
AA !!
!
¦
Thus we have commutative diagrams
¦
(X¦(GT )y n+2 , Y ¦) o (XGn+2 , Y )
y
T ,Y
(X¦ ¦) (X i ,Y )
i



(X¦(GT )n+1 , Y ¦) o (XGn+1 , Y )
¦

for all n ≥ 0 and all 0 ¤ i ¤ n+1. By Lemma 3 and diagram (3) above, the horizontal ¦™s
are abelian group isomorphisms. Thus ¦ de¬nes an isomorphism of cochain complexes,
which on the cohomology level we denote by H(¦). This completes the proof of Theorem 3.
As complements to the material covered in this section, we present that stump of the
cohomology which can be de¬ned when the coe¬cients are in a non-abelian group object,

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