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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)].

MSC:
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.)
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[1] Barr, M, Shukla cohomology and triples, J. algebra, 5, 222-231, (1967) · Zbl 0164.01502
[2] Barr, M; Beck, J, Acyclic models and triples, (), 336-343 · Zbl 0201.35403
[3] Baues, H.J; Wirsching, G.J, Cohomology of small categories, J. pure appl. algebra, 38, 187-211, (1985) · Zbl 0587.18006
[4] Bousfield, A.K; Gugenheim, V.K.A.M, On PL de Rham theory and rational homotopy type, Mem. amer. math. soc., 8, (1976) · Zbl 0338.55008
[5] Cartan, H; Eilenberg, S, Homological algebra, (1956), Princeton Univ. Press Princeton, NJ · Zbl 0075.24305
[6] Eilenberg, S; Mac Lane, S, Homology theories for multiplicative systems, Trans. amer. math. soc., 71, 294-330, (1951) · Zbl 0043.25403
[7] Eilenberg, S; MacLane, S, On the groups H (μ,n), II, Ann. of fmath., 60, 49-139, (1954) · Zbl 0055.41704
[8] Grothendieck, A, Sur quelques points d’algèbre homologique, Tǒhoku math. J., 9, 119-221, (1957), (2) · Zbl 0118.26104
[9] Hochschild, G, Cohomology groups of an associative algebra, Ann. of math., 46, 58-67, (1945) · Zbl 0063.02029
[10] Hartl, M, The cohomology group H2 of the category of finitely generated abelian groups, (1987), Max Planck Institute Bonn, preprint
[11] Jibladze, M; Pirashvil, T, Some linear extensions of the category of finitely generated free modules, Bull. acad. sci. Georgian SSR, 123, N3, (1986), [Russian, English summary]
[12] Lawvere, F.W, Functorial semantics of algebraic theories, (), 869-872 · Zbl 0119.25901
[13] Mac Lane, S, Homologie des anneaux et des modules, (), 55-80
[14] Mac Lane, S, Homology, (1963), Springer-Verlag Berlin · Zbl 0818.18001
[15] Mac Lane, S, Categories for the working Mathematician, (1971), Springer-Verlag Berlin · Zbl 0232.18001
[16] Mitchell, B, Rings with several objects, Adv. in math., 8, 1-161, (1972) · Zbl 0232.18009
[17] Pirashvili, T, Higher additivizations, (), 44-54, [Russian, English summary] · Zbl 0705.18008
[18] Pirashvili, T, New (co) homology of rings, Bull. acad. sci. Georgian SSR, 133, 477-480, (1989) · Zbl 0676.16024
[19] Quillen, D.G, On the (co-)homology of commutative rings, (), 65-87
[20] Quillen, D.G, Higher algebraic K-theory, I, () · Zbl 0292.18004
[21] Schubert, H, Kategorien, (1970), Springer-Verlag Berlin · Zbl 0205.31904
[22] Shukla, U, Cohomologie des algebres associatives, Ann. sci. école norm. sup., 78, 163-209, (1961) · Zbl 0228.18005
[23] Simon, D, Stable derived functors of the second symmetric power functor, second exterior power functor and Whitehead gamma functor, (), 49-54 · Zbl 0293.18017
[24] Witt, E, Treue darstellung liescher ringe, J. reine angew. math., 197, 152-160, (1937) · JFM 63.0089.02
[25] Wraith, G.C, Algebraic theories, (), Aarhus, Denmark · Zbl 0249.18013
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