Abstract
Both MacLane [3] and Beck [2] have recently defined cohomology theories for algebras in abstract categorical settings. Mac Lane’s theory is an abstract formalization of the (normalized) bar construction (see [4, p. 144] for example). Since, however, his proof of the normalization theorem [4, p. 236] remains perfectly valid, the normalized bar construction can be replaced by the un-normalized one. It is the purpose of this paper to show that under reasonable conditions Beck’s cohomology is naturally equivalent to a slight modification of Maclane’s. All notation not explicitly defined is taken from [3].
Research partially supported by NSF Contract GP 730.
Received August 26, 1965.
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References
Barr, M., and J. Beck: Acyclic models and triples, these proceedings, pp. 336-343.
Beck, J.: Triples, algebras and cohomology. Dissertation, Columbia University.
Maclane, S.: Categorical algebra. Bull. Amer. Math. Soc. 71, 40–106, (1965).
— Homology. Berlin: Springer 1963.
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Barr, M. (1966). Cohomology in Tensored Categories. In: Eilenberg, S., Harrison, D.K., MacLane, S., Röhrl, H. (eds) Proceedings of the Conference on Categorical Algebra. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-99902-4_17
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