TRIPLES, ALGEBRAS AND COHOMOLOGY

It is with great pleasure that the editors of Theory and Applications of Categories make this dissertation generally available. Although the date on the thesis is 1967, there was a nearly complete draft circulated in 1964. This thesis was a revelation to those of us who were interested in homological algebra at the time. Although the world’s very first triple (now more often called “monad”) in the sense of this thesis was non-additive and used to construct flabby resolutions of sheaves ([Godement (1958)]), the then-prevailing belief was that the theory of triples had a use in homological algebra only via additive triples on abelian categories, typically something like Λ⊗Λ⊗Λ −, on the category of Λ-Λ bimodules. In fact, [Eilenberg & Moore (1965b)] went so far as to base their relative homological algebra on triples that were additive and preserved kernels. Thus 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.