• Corpus ID: 14772815


  author={Jonathan Beck},
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… 

Composite cotriples and derived functors

The main result of [Barr (1967)] is that the cohomology of an algebra with respect to the free associate algebra cotriple can be described by the resolution given by U. Shukla in [Shukla (1961)].

Cartan-Eilenberg cohomology and triples

Algebraic deformations and triple cohomology

The fundamental theorems of algebraic deformation theory are shown to hold in the context of enriched triple cohomology. This unifies and generalizes the classical theory. The fundamental results in

Two cohomology theories for structured spaces

In [1] we defined a new kind of space called 'structured space' which locally resembles, near each of its points, some algebraic structure. We noted in the conclusion of the cited paper that the maps

Symmetric homology over rings containing the rationals

Let R be a commutative ring with unit which contains the rational numbers. Let A be a commutative R-algebra. In this paper we prove that the cotriple homology and cohomology modules of R for the

Relative Barr-Rinehart and cotriple cohomology groups are isomorphic

The theorem, stated in the title of this article, is proved. Several people have asked about the relationship between the relative cohomology groups defined by Barr and Rinehart [3], and those of


The theorem, stated in the title of this article, is proved. Several people have asked about the relationship between the relative cohomology groups defined by Barr and Rinehart [3], and those of

Homological Algebra for Superalgebras of Differentiable Functions

This is the second in a series of papers laying the foundations for a differential graded approach to derived differential geometry (and other geometries in characteristic zero). In this paper, we



Cohomology as the derived functor of derivations

Introduction. The investigations which produced this paper were suggested by the fact that the Hochschild cohomology H(F, M) of a free algebra E, with coefficients in any module M, is zero in

A cohomology theory for commutative algebras. II

1. Introduction. D. K. Harrison has recently developed a co-homology theory for commutative algebras over a field [2]. A few key theorems are proved and the results applied to the theory of local

On the cohomology groups of an associative algebra

The cohomology theory of associative algebras is concerned with the m-linear mappings of an algebra W into a two-sided W-module A. In this theory, the additive group (2(m):$) of the m-linear mappings

The Category of Categories as a Foundation for Mathematics

In the mathematical development of recent decades one sees clearly the rise of the conviction that the relevant properties of mathematical objects are those which can be stated in terms of their

On the Structure of Hopf Algebras

induced by the product M x M e M. The structure theorem of Hopf concerning such algebras has been generalized by Borel, Leray, and others. This paper gives a comprehensive treatment of Hopf algebras

Cohomology of Lie triple systems and Lie algebras with involution

A Lie triple system is a subspace of a Lie algebra closed under the ternary composition [[xy]z]; equivalently, it may be defined as the subspace of elements mapped into their negatives by an

Acyclic Models and Triples

We shall prove two theorems on the “triple” cohomology of algebras [1] using a method of acyclic models suggested by H. Appelgate. Specifically, we show that the triple cohomology coincides with


  • M. Gerstenhaber
  • Biology
    Proceedings of the National Academy of Sciences of the United States of America
  • 1964
16 Crick, F. H. C., in Progress in Nucleic Acid Research, ed.

Some Aspects of Equational Categories

The theory of equationally definable classes of algebras, initiated by Birkhoff in the early thirties, is, despite its power, elegance and simplicity, hampered in its usefulness by two defects. The

Cohomologie des algèbres associatives

© Gauthier-Villars (Éditions scientifiques et médicales Elsevier), 1961, tous droits réservés. L’accès aux archives de la revue « Annales scientifiques de l’É.N.S. » (http://www.