By Edward Mazria

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**Sample text**

K being a commutative ring, we let A denote the category described. A has few group objects. Indeed, if Y ∈ AbA , then G Y. the zero operation in Y must be represented by a map of the terminal object 0 Since this map must preserve identity elements, 1 = 0 in Y ; therefore Y = 0. To get abelian group objects, we must consider categories of modules. Let Λ be a K-algebra. Then Λ-Mod = Ab(A , Λ) ker G Λ-Λ-Bimodules is an equivalence of categories. G Λ be a Λ-module. There is a zero section Λ G Y in A , so we can write Let Y Y Λ ⊕ M as a K-module, and its multiplication will be of the form (λ1 , m1 )(λ2 , m2 ) = [(λ1 , 0) + (0, m1 )][(λ2 , 0) + (0, m2 )] = (λ1 , 0)(λ2 , 0) + (λ1 , 0)(0, m2 ) + (0, m1 )(λ2 , 0) + (0, m1 )(0, m2 ) = (λ1 λ2 , λ1 m2 + m1 λ2 + m1 m2 ) .

We ﬁnd 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 ) deﬁned by G Y G ··· G X G 0 coincides with the triple long abelian-exact sequences 0 n cohomology H (X, Y ), relative to U . TRIPLES, ALGEBRAS AND COHOMOLOGY 57 Note that without the extra, coequalizer-preserving, property of the functor (P, ), we G Sets is tripleable. The comwould 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 coreﬂecTorsion-free abelian groups tive subcategories ([Mitchell (1965)], but fullness should be added to the deﬁnition).

M. Bunge (1966), Characterization of diagrammatic categories. Dissertation, University of Pennsylvania. H. Cartan and S. Eilenberg (1956), Homological Algebra. Princeton University Press. ´ J. Dixmier (1957), Homologies des anneaux de Lie. Ann. Sci. Ecole Norm. Sup. 74, 25–83. B. Eckmann and P. Hilton (1962), Group-like structures in general categories, I. Math. Ann. 145, 227–255. S. Eilenberg (1948), Extensions of general algebras. Ann. Sci. Polon. Math. 21, 125–134. S. Eilenberg and S. Mac Lane (1947), Cohomology theory in abstract groups, I.