Document Zbl 0674.55008 - zbMATH Open

The category DGCA of differential \({\mathbb{Z}}\)-graded commutative algebras over \({\mathbb{Q}}\) is a closed model category. For any object B of finite type and bounded below, the functor \(-{\hat \otimes}B\) has a left adjoint \((-:B).\) Using this functor, the authors prove: Theorem. Let S denote the category of simplicial sets. Let \(X,Y\in obj(S)\) be connected, with X finite and Y fibrant and nilpotent of finite type. Then for any map \(\psi\) : \(X\to Y\) and any weak equivalence \(\beta\) : \(M^*(X)\to B^*(X)\) with \(B^*(X)\) non-negative of finite type, there is an equivalence in ho-DGCA \((M^*(Y):B^*(X))_{\beta \psi^*}\simeq M^*(Map(X,Y)_{\psi})\). This sheds new and interesting light on Haefliger’s construction.

MSC:

55P62	Rational homotopy theory
55U35	Abstract and axiomatic homotopy theory in algebraic topology
55Q05	Homotopy groups, general; sets of homotopy classes

References:

[1]	M.Andre, Methode simplicial en alg?bre homologique commutative. LNM32, Berlin-Heidelberg-New York 1967.
[2]	A. K. Bousfield, Nice Homology Coalgebras. Trans. Amer. Math. Soc.148, 232-248 (1970). · doi:10.1090/S0002-9947-1970-0258919-6
[3]	A. K. Bousfield, The Homology Spectral Sequence of a Cosimplicial Space. Amer. J. Math.109, 361-394 (1987). · Zbl 0623.55009 · doi:10.2307/2374579
[4]	A. K.Bousfield and V. K. A. M.Guggenheim, On the PL de Rham Theory and Rational Homotopy Type. Mem. Amer. Math. Soc. (179)8, 1976. · Zbl 0338.55008
[5]	S.Eilenberg and M.Kelley, Closed Categories. Proceedings of the La Jolla Conference on Category Theory 1967.
[6]	A. Haefliger, Rational Homotopy of Spaces of Sections of a Nilpotent Bundle. Trans. Amer. Math. Soc.273, 609-620 (1982). · Zbl 0508.55019 · doi:10.1090/S0002-9947-1982-0667163-8
[7]	J.Lannes, Sur la Cohomologie modulop desp-Groupes abeliens elementaire. Math. Z., to appear.
[8]	J.Lannes et S.Zarati, Sur les U-Injectives. Preprint 1985.
[9]	S.Mac Lane, Category Theory for the Working Mathematician. Berlin-Heidelberg-New York 1972.
[10]	B.Mitchell, Category Theory. New York-London 1968.
[11]	J. M?ller andM. Raussen, Rational Homotopy of Spaces of Maps into Spheres and Complex Projective Spaces. Trans. Amer. Math. Soc.292, 721-732 (1985). · Zbl 0605.55008
[12]	D.Quillen, Homotopical Algebra. LNM43, Berlin-Heidelberg-New York 1967.
[13]	D. Quillen, Rational Homotopy Theory. Ann. of Math. (2)90, 205-295 (1969). · Zbl 0191.53702 · doi:10.2307/1970725
[14]	D. Sullivan, Infinitesimal Computations in Toplogy. I.H.E.S. J. Math.47, 269-295 (1977).
[15]	R.Thom, L’homologie des Espaces Fonctionnels. Colloq. Top. Alg., Louvain 1956.

This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.