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match this feat, for constructing free algebras usually requires in¬nite direct limits. How-
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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59
TRIPLES, ALGEBRAS AND COHOMOLOGY

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