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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

##### MSC:
 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)
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##### References:
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