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On the extended functoriality of Tor and Cotor. (English) Zbl 0358.18015
It was observed by J.Stasheff and S.Halperin [Proc. adv. Study Inst. algebraic Topol. 1970, various Publ. Ser. 13, 567–577 (1970; Zbl 0224.55027)] and others that differential algebra, the condition for a multiplicative map \(f\): \(A \to A'\) between differential algebras \(A, A'\): \(\Phi f = (f \otimes f) \Phi\) is too strong in connection with applications in algebraic topology. It is more reasonable to ask that \((f\otimes f)\Phi\) and \(\Phi f\) are chain-homotopic or better that there exists a map of coalgebras \(B(A) \to B(A')\) between the corresponding classifying coalgebras \(B(A)\), \(B(A')\). This leads to a new extended category, whose objects are differential algebras, but whose maps are exactly all coalgebra maps\(B(A) \to B(A')\). In a similar way one gets a second “extended” category with objects all differential algebras but with maps \(A \to A'\) all algebra-maps \(\Omega(A) \to \Omega(A')\) between the corresponding loop algebras. The main purpose of this paper is to show that the functoriality of the differential Tor and Cotor functors [cp. S. Eilenberg - J. C. Moore: Commentarii math. Helvet. 40, 199–236 (1966; Zbl 0148.43203)] can be extended with respect to these new ”extended” categories. Applications in connection with theorems about the ”collapse of the Eilenberg-Moore spectral sequence” are given [S. Eilenberg and J. C. Moore: Topology 1, 1–23 (1962; Zbl 0104.39603 )] as well as in connection with the theory of induced fibre spaces.
Reviewer: M. B. Wischnewsky

18G15 Ext and Tor, generalizations, Künneth formula (category-theoretic aspects)
18G40 Spectral sequences, hypercohomology
55N99 Homology and cohomology theories in algebraic topology
55M30 Lyusternik-Shnirel’man category of a space, topological complexity à la Farber, topological robotics (topological aspects)
Full Text: DOI
[1] Eilenberg, S.; Moore, J.C., Limits and spectral sequences, Topology, 1, 1, (1962) · Zbl 0104.39603
[2] Eilenberg, S.; Moore, J.C., Homology and fibrations 1, coalgebras, cotensor product and its derived functors, Comment. math. helv., 40, 3, (1966) · Zbl 0148.43203
[3] Gugenheim, V.K.A.M., On the chain complex of a fibration, Ill. J. math., 16, 3, 353, (1972) · Zbl 0238.55015
[4] V.K.A.M. Gugenheim and J.P. May, On the theory and applications of differential torsion products, preprint. · Zbl 0292.55019
[5] D. Husemoller, J.C. Moore and J. Stasheff, Differential homological algebra and homogeneous spaces, preprint. · Zbl 0364.18008
[6] Kan, D.M., On homotopy theory and c.s.s. groups, Ann. math., 68, (1958)
[7] May, J.P., The geometry of iterated loop-spaces, () · Zbl 0244.55009
[8] Milnor, J.; Moore, J.C., On the structure of Hopf algebras, Ann. math., 81, 211-264, (1965) · Zbl 0163.28202
[9] Moore, J.C., Sem. Cartan. exp., 7, (1959/1960)
[10] Moore, J.C., Differential homological algebra, Proc. intern. congr. of mathematicians, Vol. 1, 335, (1970)
[11] Munkholm, H.J., Strongly homotopy multiplicative maps and the Eilenberg-Moore special sequence, Aarhus preprints, (1972/73)
[12] Stasheff, J.; Halperin, S., Differential homological algebra in its own rite, Proc. adv. study inst. alg. top. Aarhus, (1970) · Zbl 0224.55027
[13] Sweedler, M.F., Hopf algebras, (1969), Benjamin New York · Zbl 0194.32901
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