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Cohomology of algebraic theories. (English) Zbl 0724.18005
The first result considers the relations between MacLane cohomology of rings with coefficients in a bimodule [S. MacLane: Homologie des anneaux des modules, Colloque de Topologie Algébrique, Louvain 1956, 55-80 (1957; Zbl 0084.267)] and Ext groups in functor categories. That leads to define the cohomology of an associative ring R with coefficients in a functor from free generated left R-modules to left R-modules. Connections with the cohomology of small categories are given (particularly classifying the extensions of R in the category of algebraic theories). Then, it appears that the domain of the cohomology here defined must be the category of algebraic theories, rather than that of rings. The generalization of this cohomology to algebraic theories given here is a case of Barr and Beck cotriple cohomology [M. Barr and J. Beck: Acyclic models and triples, Proc. Conf. Categorical Algebra, La Jolla 1965, 336-343 (1966; Zbl 0201.354)].

18C10 Theories (e.g., algebraic theories), structure, and semantics
18G15 Ext and Tor, generalizations, Künneth formula (category-theoretic aspects)
16E40 (Co)homology of rings and associative algebras (e.g., Hochschild, cyclic, dihedral, etc.)
Full Text: DOI
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