ever, for the restricted class of abelian categories to be discussed next, triple cohomology

does make a natural appearance and coincides with the Yoneda Ext. The reader will note

considerable contact between our treatment and [Huber (1962)].

Let B be an abelian category with direct (inductive) limits, P ∈ B any object. We

G Ab as an underlying object functor, and

treat the representable functor (P, ): B

recall that it has a left adjoint A =’ A — P (see [Freyd (1964)], [Mitchell (1965)], or

assume B is a category of modules). As usual, we have

¦

o

AbTqq B

ww

qq ww

qq ww(P, )

q5 {ww

Ab

Here a T-structure on an abelian group A is a unitary, associative abelian group map

G A. By adjointness every (P, B) has such a structure, which

θ: AT = (P, A — P )

de¬nes the functor ¦.

Now let R = (P, P ), the endomorphism ring of P , and let R0 be the triple in Ab

de¬ned by

a—r2 r1

G A — R, G A — R·

a—1

A—R—R

A

0

G P ; AbR is the category

r2 r1 is the composition, in that order, of endomorphisms P

of left R-modules.

G AT is de¬ned if we let (a — r)(A•) be the

A natural transformation A•: A — R

composition

GZ—P G A — P,

a—r

P

G A. One can verify that •: R0 G T is a map of

thinking of a ∈ A as a map Z

triples, that is, the natural transformation • commutes with units and multiplications:

A••

G AT T

A—R—R

AP

PP

PP

PPA·

a—1 Aµ

PP

a—r2 r1

PP

• %

G AT G AT

A—R A—R

A• A•

G

(µ results from adjointness, f · µ being the composition of an f with (A — P ) : P

G A — P .) A map of triples induces a functor between the corre-

(P, A — P ) — P

G A is a T-structure on A, then the

sponding algebra categories. In this case, if θ: AT

composition

A•

G AT GA

θ

A—R

56 JONATHAN MOCK BECK

is an R0 -structure on A. Thus we have the following commutative diagram of functors.

Ab• ¦

0

AbRc o AbT o B

cc

cc

cc

cc

(P, )

cc

cc

1

Ab

We now have

Proposition. If P is a projective generator in B, then ¦ is an equivalence.

If P is a small projective, then Ab• is an isomorphism of categories.

Approximate de¬nitions for the terms used in the proposition can be found in [Freyd

(1964)] and [Mitchell (1965)]. The two statements follow from the (Tripleableness) The-

G T is an isomorphism of triples. The

orem 1, and the fact that if P is small, •: R0

¬rst statement can be paraphrased, B is tripleable over Ab. Were P also small, we would

be able to conclude the familiar corollary below. Thus a triple in Ab can be considered

as a sort of “large” generalization of a ring.

Corollary. If P is a small projective generator in B, then B is equivalent to the cate-

gory of left R = (P, P ) = End(P )-modules.

Similar proofs can be given for the characterization of cocomplete abelian categories

with generating sets of small projectives [Freyd (1964)], as well as for M. Bunge™s recent

characterization of functor categories S C [Bunge (1966)]. (This has been carried out by

F.E.J. Linton and the writer.)

This result yields subexample (b) above, which arises when P = “ ∈ M“ , (Λ = Z).

The following can be proved by an elaboration of the proof of Theorem 1 (hinting the

more delicate tripleableness theorem referred to):

Proposition. Consider a composition of adjoint pairs

A

FG UG

A B F0 F U U0

GB GA

=’ A0 0

G A U0 G A

F0

A0 0

If F U satis¬es all the hypotheses of Theorem 1 (hence in particular is tripleable), and

if F0 U0 is tripleable, then F0 F U U0 is tripleable.

Let B be a cocomplete abelian category (i.e., direct limits) with a projective generator

G Ab, U0 : Ab G Sets, which we denote by U : B

P . Apply this proposition to (P, ): B

G Sets. We ¬nd that every such abelian category is tripleable over sets. Notice that

U -exactness in B is the same as abelian-category exactness. Hence Extn (X, Y ) de¬ned by

GY G ··· GX G 0 coincides with the triple

long abelian-exact sequences 0

cohomology H n (X, Y ), relative to U .

57

TRIPLES, ALGEBRAS AND COHOMOLOGY

Note that without the extra, coequalizer-preserving, property of the functor (P, ), we

G Sets is tripleable. The com-

would not be able to conclude that the composition B

position of tripleable underlying object functors is not tripleable in general, for example,

G Ab G Sets; tripleableness is trivially true of core¬‚ec-

Torsion-free abelian groups

tive subcategories ([Mitchell (1965)], but fullness should be added to the de¬nition).

G (P, X · P ). In case P is

If X is a set, the explicit composite triple above is X

also small, let R0 = (P, P ) with backwards multiplication and identify the triple as X

G X · R0 , the underlying set of the free right R0 -module generated by the set X. Thus

we again ¬nd B left R-modules.

This result yields subexample (a) above, which arises when the projective generator

P ∈ MΛ chosen is Λ itself.

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58 JONATHAN MOCK BECK

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59

TRIPLES, ALGEBRAS AND COHOMOLOGY

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