(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,