On the extended functoriality of Tor and Cotor.

*(English)*Zbl 0358.18015It 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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\textit{V. K. A. M. Gugenheim} and \textit{H. J. Munkholm}, J. Pure Appl. Algebra 4, 9--29 (1974; Zbl 0358.18015)

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##### References:

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[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 |

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