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Triples, algebras and cohomology. (English) Zbl 1022.18004

Quoting from the author’s introduction:
“This thesis is intended to complete the exposition of S. Eilenberg and J. C. Moore [“Adjoint functors and triples”, Ill. J. Math. 9, 381-398 (1965; Zbl 0135.02103)] with regard to certain points. In section 1 we recall the definitions of triple, algebra over a triple, and give our main (original) definition, that of tripleable adjoint pair of functors. In section 2 we show how to obtain a cohomology theory from an adjoint pair of functors. In section 3, when the adjoint pair is tripleable, we prove that the cohomology group \(H^1\) classifies principal homogeneous objects. When coefficients are in a module, principal objects are interpreted as algebra extensions. Section 4 is devoted to examples. Many categories occurring in algebra are shown to be tripleable. The corresponding cohomology and extension theories, ranging from groups and algebras to the classical \(\text{Ext}(A,C)\), are discussed. Many new theories arise.”
This dissertation, although published in 1967 by Columbia University, never appeared in a professional mathematics journal. This is unfortunate for two reasons. First, the results are important (to quote the editors):
“…there was considerable astonishment when Jon Beck, in the present work, was able not only to define cohomology by a triple on the category of objects of interest (rather than the abelian category of coefficient modules) but even prove in wide generality that the first cohomology group classifies singular extensions by a module. Not the least of Beck’s accomplishments in this work are his telling, and general, axiomatic descriptions of module, singular extension, and derivation into a module. The simplicity and persuasiveness of these descriptions remains one of the more astonishing features of this thesis.”
Second, the quality of Beck’s writing is outstanding and could well be used as a model for today’s researchers, both young and old… . The editors of TAC have done our community a real service by making Beck’s mathematical gem widely available.

MSC:

18C15 Monads (= standard construction, triple or triad), algebras for monads, homology and derived functors for monads
18G15 Ext and Tor, generalizations, Künneth formula (category-theoretic aspects)
18C20 Eilenberg-Moore and Kleisli constructions for monads
18G10 Resolutions; derived functors (category-theoretic aspects)

Citations:

Zbl 0135.02103
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Full Text: EMIS