XF U

v

v

vv

vv

XF Uvv

vv XF U · X·

vv

vv

zvv XF U

ξF U

o G XF U

XF Ur XF U F U

r

vv

rr

vv

rr

vv

rr

vv

rr vz

z rrr vv

vv

rr

vv

rr

6 zvv

Z

πU is also a coequalizer of ξF U and XF U , if U preserves coequalizers. Moreover •: X

G (X, ξ)¦U is compatible with ξ and πU , because ξ• = ξ·X··πU = XF U ··ξF U ·πU =

ˇ

XF U ··XF U ·πU = πU . Thus • is an isomorphism in A. Trivially, if a map of T-algebras

is an isomorphism in the underlying category, then it is an isomorphism of T-algebras (its

inverse also respects T-structures). In other words, the functor U T : AT G A re¬‚ects

isomorphisms. This proves (2).

G Y is obtained from the identity map of Y ¦ via the above 1-1

ˇ

The counit ψ: Y ¦¦

correspondences, and is uniquely determined by its appearance in the following diagram,

the top line of which is a coequalizer.

G

Y UF

G Y ¦¦

π ˇ

G Y UF

Y UF UF v

vv

Y UF

vv

vv

vv

Y

vv ψ

vv

zvvv

Y

11

TRIPLES, ALGEBRAS AND COHOMOLOGY

We proved above that the T-structure of an algebra is a coequalizer in A. (Recall that

Y U is the T-structure of Y ¦.) If U re¬‚ects coequalizers, then Y is a coequalizer of

Y U F and Y U F , and ψ is an isomorphism.

If U preserves coequalizers, both πU and Y U are coequalizers, hence ψU is an iso-

morphism. If U re¬‚ects isomorphisms, so is ψ. This proves the remark pertaining to (3 ),

and completes the proof of the theorem.

Remarks on the functor ¦. It would be interesting to know an example of a functor

G AT with B having arbitrary limits (limit = projective limit, in our terminol-

¦: B

ˇ

ogy) wherein no left adjoint ¦ exists. Counterexamples apparently exist when B is not

complete.

Whether an adjoint or not, ¦ preserves all an adjoint should preserve, for example,

limits and algebraic objects. Indeed, in the commutative diagram

¦ GB

AT c

cc

cc

c1 U

UT

A

U , being an adjoint, preserves the property involved, and one ¬nds that U T re¬‚ects the

property. We saw above that U T re¬‚ects isomorphisms. Similarly, one can show that U T

re¬‚ects all other limits. In §2 we will need the fact, and will then prove in detail, that ¦

preserves several kinds of algebraic objects.

¦ can also be given an interpretation in terms of “structure” and “semantics”. Let

Ad(A) be the category of adjoint pairs over A, that is, pairs F U where U has A as

range, with functors which commute with the right adjoints (like ¦ itself) as maps. Let

G T being a natural transformation

Trip(A) denote the category of triples in A, a map S

of the functors commuting with the units and multiplications, and let Trip(A)— denote

the dual category. Then functors

ˇ G

σ

Trip(A)—

Ad(A) o σ

exist, because adjoint pairs give rise to triples and triples give rise to categories AT with

G Trip(A)— , and the

adjoint free and underlying object functors. In fact, σ ˇ σ, σˇ

σ

G σ σ. All of this is proved in [Eilenberg

unit of this adjointness is precisely ¦: Ad(A) ˇ

& Moore (1965a)], and is reminiscent of the structure-semantics situation studied by

[Lawvere (1963)] in the case of algebraic categories and by [Linton (1966)] in the case of

equational categories. We should remark that by means of a more elaborate construction

G A (B

the domain of σ can be extended to the category whose objects are functors B

ˇ

variable) which are “small” in an appropriate sense but do not need to have left adjoints.

It is useful to think of the triple T induced by an adjoint pair of functors as a structural

invariant of the adjoint pair. Various properties of the adjoint pair only depend on T and

the functor ¦ plays a role in setting up the relevant isomorphisms. We shall see this

illustrated in the next section in the case of cohomology.

12 JONATHAN MOCK BECK

2. Cohomology

Adjoint functors, it is now well known, lead to cohomology ([Eilenberg & Moore (1965a),

Godement (1958), Mac Lane (1963)]”or to homotopy [Huber (1961)]). If

GB GA

F U

A (F U)

is an adjoint pair, objects of the form AF ∈ |B| are regarded as “free” relative to the

underlying object functor U . The counit

GX

X

XU F

is intuitively the ¬rst step of a functorial free resolution of any object X ∈ |B|. By

iterating U F one extends X to a free simplicial resolution of X, and de¬nes derived

functors as usual in homological algebra. Here we only consider the simplest case, that

of de¬ning cohomology groups

n ≥ 0,

H n (X, Y ),

of an object X ∈ |B| with coe¬cients in an abelian group object Y ∈ |B|, relative to

G A (having a left adjoint). Tripleableness

the given underlying object functor U : B

of F U will not play any appreciable role until we discuss special properties of the

cohomology in §3. We now recall the details of the construction of the cohomology groups.

Some of the terms used are clari¬ed in the proof of Theorem 2, which summarizes the

main properties the cohomology possesses.

G B.

Let (G, , δ) be the cotriple in B induced by F U ; thus G = U F and : G

The following simplicial object in B is called the standard (free simplicial) resolution of

the object X:

o

XG oo

0

XG2 oo

Xo ··· o XGn o XGn+1 o

0 i

···

1

We abbreviate this by XG— if necessary. Here XGn+1 is the term of degree n, and the

G XGn is Gi Gn’i , 0 ¤ i ¤ n. X itself is in dimension ’1 and

face operator i : XGn+1

the last 0 augmenting the simplicial object into X is just . The simplicial identities

i ¤ j,

= j+1 i ,

ij

G G2 induces degeneracy operators but these will play no

can easily be veri¬ed. (δ: G

role in our theory.)

G Y . If Y is an abelian

An n-cochain of X with coe¬cients in Y is a map XGn+1

group object in the category B, the n-cochains (XGn+1 , Y ) form an abelian group, and

G (XGn+1 , Y ).

the face operators i induce abelian group maps (X i , Y ): (XGn , Y )

Thus

d1 d2 dn+1

G (XG2 , Y )

G (XG, Y ) G ··· G (XGn+1 , Y ) G (XGn+2 , Y ) G ···

0

13

TRIPLES, ALGEBRAS AND COHOMOLOGY

is a cochain complex of abelian groups, where dn+1 = (’1)i (X i , Y ), 0 ¤ i ¤ n + 1,

dd = 0 because of the simplicial identities, and the augmentation term has been dropped.

To distinguish it from another complex that will be introduced later, this complex is called

the homogeneous complex. We de¬ne

n ≥ 0,

H n (X, Y ),

G X in B and Y

as the n-th cohomology group of this complex. Obviously maps X

G Y in the category of abelian group objects in B induce maps

G H n (X , Y ), G H n (X, Y )

H n (X, Y ) H n (X, Y )

and the cohomology is functorial in the usual way.

I do not know how to characterize the cohomology theory H(X, Y ) = (H n (X, Y )),

n ≥ 0, axiomatically. However, acyclicity of free objects and the exact cohomology

sequence are properties that can easily be established:

THEOREM 2.

(AF, Y ), n = 0,

n

H (AF, Y ) =

0, n > 0·

GY GY GY G 0 is a U -exact sequence of abelian group objects in

If 0

B, then there is an exact cohomology sequence

· · · l

d

G H n (X, Y ) G H n (X, Y )

n

H (X, Y ) l

d

G H (X, Y ) G ··· G H n’1 (X, Y )

0

0

Proof. For the ¬rst part, we notice that the standard resolution of a free object AF has

a simplicial contracting homotopy

s’1 s0 sn

G AF G2

G AF G G ··· G AF Gn+1 G AF Gn+2 G ···

AF

given by sn = A·F Gn+1 (see [Huber (1961), p. 248], for example; this homotopy satis¬es

the relations sn 0 = identity, sn i = i’1 sn’1 , 1 ¤ i ¤ n + 1). If we let tn = (sn , Y ), then

t’1 t0 tn

(AF G2 , Y ) o

0o (AF, Y ) o (AF G, Y ) o ··· o (AF Gn+2 , Y ) o ···

is a contracting homotopy of the augmented cochain complex, that is, we have dt + td =

identity. Hence H 0 (AF, Y ) is the term of degree ’1 and all higher cohomology vanishes.

For the second part we must explain the concept of U -exactness; this requires recalling

some facts about abelian group objects in categories (see [Eckmann & Hilton (1962)] for

14 JONATHAN MOCK BECK

a fuller treatment). Y is an abelian group object in B if the hom set (B, Y ) has an abelian

group structure for every object B in the category B, and naturality holds in that every

G (B , Y ) is an abelian group map, where B G B in B. Y

induced map (B, Y )

G Y is a map of abelian group objects in B if it is a map in B and every induced (B, Y )

G (B, Y ) is an abelian group map. The abelian group objects form a category AbB,

G B. Of course, one can de¬ne in the same way

with an obvious forgetful functor AbB

other types of algebraic objects in categories, such as objects with base points, nonabelian

groups, rings, in fact models for any algebraic theory [Lawvere (1963)], and we will need

some of these categories later. However, what general theory we will use is adequately

illustrated by the case of abelian groups. The following lemma can be paraphrased by

saying that adjoints preserve abelian group objects:

LEMMA 1. Let A o U B be a functor which has a left adjoint, and let Y ∈ |AbB|. Then

there exists a unique abelian group structure on Y U ∈ |A| such that