U T is the underlying A-object functor, which maps (X, ξ) ’ X and f ’ f . F T is the

G A ’ aT : (AT, Aµ)

free T-algebra functor, which maps A ’ (AT, Aµ) and a: A

G (A T, A µ). It follows from the axioms for · and µ that (AT, Aµ) actually is a

T-algebra. The formula for the adjointness isomorphism

±T

(A, (Y, θ)U T ) G (AF T , (Y, θ))

is y±T = yT · θ, f (±T )’1 = A· · f .

To justify the de¬nition we only need to verify the adjointness, all the other assertions

G Y be a map in A, and let us check that y±T is a

being obvious. First, let y: A

7

TRIPLES, ALGEBRAS AND COHOMOLOGY

T-algebra map. We have the diagram

yT T

G Y TT GYT

θT

AT T

Aµ Yµ θ

GYT GY

AT yT θ

The top and bottom compositions are (y±)T and y± (if ± stands for ±T , as it will for the

rest of this proof). The second square commutes by the associative law for θ. Now we have

y±±’1 = A· ·yT ·θ = y ·Y · ·θ = y, by the unitary property of θ, and if f : AF T = (AT, Aµ)

G (Y, θ) in AT , then f ±’1 ± = (A· · f )T · θ = A·T · f T · θ = A·T · Aµ · f = f .

y A·T fT

GY G AT T GYT

A ATc

cc cc

cc cc

cc cc

cc

Y

c

cc AT ccc

A· Y· Aµ θ

cc cc

c1

1

GYT GY GY

AT AT

yT θ f

We will skip the proof that ±T is natural, which is easy.

G B and U : B G A. ±

Now let ±: F U be any adjoint pair, where F : A

generates a triple T = (T, ·, µ) in A, where T = F U , . . . , hence a category of T-algebras

AT with the above free and underlying object functors. The relation between the old

G AT

GB G A and the new one, A G A, is expressed in

adjoint pair A

G AT which we will be interested in throughout. ¦

terms of a canonical functor ¦: B

exists because any object in B naturally induces a T-algebra structure on the object in

A underlying it, and ¬ts into the following commutative diagram of functors:

¦

AT o cc c B

•cc cc

cc cc U T F

cc cc

cc cc U

F T cc cc1

cc

A

(¦U T = U, F ¦ = F T )

G (Y U, Y U )

¦ is de¬ned by the formulas Y ¦ = (Y U, Y U ), y¦ = yU : (Y U, Y U )

G Y . Intuitively, the counit Y : Y U F G Y is the natural map of the free

if y: Y

object generated by Y onto Y (it need not be an epimorphism in this general context)

and the T-structure on Y ¦ is the A-map underlying this. It is clear, of course, from the

adjointness identities that (Y U, Y U ) is really a T-algebra.

The construction of AT does not in general give back the original adjoint pair, that is,

G AT is not always an equivalence. We want to isolate as particularly tractable

¦: B

the situation in which it is:

8 JONATHAN MOCK BECK

G AT is an equivalence

DEFINITION 3. The adjoint pair ±: F U is tripleable if ¦: B

of categories.

If U is held ¬xed and F, F U are two left adjoints, then the canonical isomorphism

G F [Kan (1958)] can be used to show that F U is tripleable ” F

F U is.

G F induces an isomorphism of triples T G T, hence an isomorphism of

Indeed, F

the algebra categories AT G AT which one ¬nds commutes with the canonical functors

G AT , ¦ : B G AT . Thus ¦ is an equivalence ” ¦ is. The details of the

¦: B

reasoning can be left to the reader, but as a consequence of it we can state:

G A is tripleable if U has a left adjoint F and the

DEFINITION 3 . A functor U : B

adjoint pair F U is tripleable.

In practice this language is convenient because often the underlying object functor is the

main item of interest. Its coadjoint free functor has to be present, but needn™t be brought

explicitly into the discussion. Of course, such expressions as “B is tripleable over A” can be

G A is understood. As a rule the category

used if a de¬nite underlying object functor B

G A will be considered. If

B will be ¬xed and various underlying object functors B i

G A, is tripleable, this means that B is exactly recoverable as the

one of them, U : B

category of objects in A which have the structure of algebras over the triple in A induced

G A is tripleable is therefore to

by the left adjoint of U . To say that a functor B

say it is “forgetful” in a rather precise sense: there is a uniquely determined triple in A

whose algebras are B and U is exactly the functor which forgets these algebra structures

(up to categorical equivalences). Incidentally, the term “tripleable” cannot be replaced

by “forgetful” because there remain many functors that are intuitively “forgetful”, that

is, drop structure, but which are not tripleable. Speaking in general and vague terms,

tripleableness implies algebraicity of some kind. In Example 1, §4, we will return to the

question of just what sort of structure tripleableness entails. In [J. Beck (to appear)] we

shall give a (rather complicated) necessary and su¬cient condition for a functor to be

tripleable. It is a re¬nement of the following (rather weak) theorem which we will apply

to some examples later.

THEOREM 1. Let ±: F U be an adjoint pair.

ˇ¦

o• • • • • • • • • •G B

T

c

Ac

•cccc

¦

cccc T

F

ccccU

cccc

U

F T ccccc

cc 1

A

(¦U T = U, F ¦ = F T )

ˇ

(1) If B has coequalizers, then there exists a left adjoint ¦ ¦.

9

TRIPLES, ALGEBRAS AND COHOMOLOGY

ˇ

Assuming the existence of ¦:

(2) If U preserves coequalizers, then the unit of ¦ ¦ is an isomorphism AT G ¦¦.

ˇ ˇ

G B.

ˇ

(3) If U re¬‚ects coequalizers, then the counit is an isomorphism ¦¦

Finally, in the presence of (2), (3) can be replaced by:

G B.

ˇ

(3 ) If U re¬‚ects isomorphisms, then the counit is an isomorphism ¦¦

ˇ

Proof. ¦ exists if and only if the following coequalizer diagram (which is used as the

ˇ

de¬nition of (X, ξ)¦) exists in B:

ξF

G G (X, ξ)¦

π ˇ

G XF

XF U F

XF

ˇ

(In e¬ect (X, ξ)¦ is (X, ξ) —T F , and ¦ can be thought of as Hom(F, ) with right T-

ˇ

operators.) Assuming the coequalizer exists, the adjointness ¦ ¦ is demonstrated by

verifying the following sequence of 1-1 correspondences:

f

G Y ¦ in AT

maps (X, ξ)

f

G G Y U such that ξf = f F U · Y U

maps X

g

G G Y such that ξF · g = XF · g

±

maps XF

g1

G G Y in B.

ˇ

maps (X, ξ)¦

The adjointness isomorphism ± can be retrieved for by the well-known formula [Kan

(1958)] f ± = f F · Y . Thus, given the condition on f , we can write

ξF · g = ξF · f F · Y = (ξf )F · Y = (f F U · Y U )F · Y

= f F U F · Y U F · Y = XF · f F · Y = XF · g,

using mainly naturality of . Since also f = g±’1 = X· · gU , one is able similarly to

reverse the correspondence.

Note that (1) has been proved. To complete the remark at the beginning of the proof,

¦ exists, the unit AT G ¦¦ composed with ’U T gives a map U T G ¦U

ˇ ˇ ˇ

if ¦

G ¦. One

(since ¦U T = U ). The adjoint of this last transformation is a map π: U T F ˇ

can now prove directly (but we will omit this) that

ξF

G G (X, ξ)¦

π ˇ

G XF

XF U F

XF

ˇ

is a coequalizer diagram in B. From now on, we assume ¦ ¦ exists, and make use of

this diagram to prove (2), (3) and (3 ).

G (X, ξ)¦¦ be the unit. •: X G (X, ξ)¦U in A and is compatible

ˇ ˇ

Let •: (X, ξ)

with T-structures. Explicitly, • is obtained by working the above 1-1 correspondences

10 JONATHAN MOCK BECK

ˇ

backwards starting from the identity map of (X, ξ)¦. This yields • = X· · πU . Consider

the following diagram.

Xc

cc

cc

ccX

cc

X·

cc

cc

1

ξF U

G ξ

GX

G XF U

XF U F U cc

cc

XF U

cc

cc •

πU cc

cc

1

ˇ

(X, ξ)¦U

G Z in A and ξF U · z = XF U · z,

ξ is a coequalizer of ξF U and XF U . For if z: XF U

G Z is uniquely determined by the fact that it satis¬es the equation

then X· · z: X

ξ·X··z = z. Uniqueness is evident from the retraction property of ξ, X··ξ = X, and from

the equation itself: ξ · X· · z = XF U · · ξF U · z = XF U · · XF U · z = XF (U · · U ) · z = z.

ξ