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HZalgebra spectra are differential graded algebras
 Amer. Jour. Math
, 2004
"... Abstract: We show that the homotopy theory of differential graded algebras coincides with the homotopy theory of HZalgebra spectra. Namely, we construct Quillen equivalences between the Quillen model categories of (unbounded) differential graded algebras and HZalgebra spectra. We also construct Qu ..."
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Abstract: We show that the homotopy theory of differential graded algebras coincides with the homotopy theory of HZalgebra spectra. Namely, we construct Quillen equivalences between the Quillen model categories of (unbounded) differential graded algebras and HZalgebra spectra. We also construct Quillen equivalences between the differential graded modules and module spectra over these algebras. We use these equivalences in turn to produce algebraic models for rational stable model categories. We show that bascially any rational stable model category is Quillen equivalent to modules over a differential graded Qalgebra (with many objects). 1.
An algebraic model for rational S 1 equivariant stable homotopy theory,Quart.J.ofMath
 of Mathematics and Statistics, Hicks Building, Sheffield S3 7RH. UK. Email address: j.greenlees@sheffield.ac.uk
"... graded objects in A models the whole rational S 1equivariant stable homotopy theory. That is, we show that there is a Quillen equivalence between dgA and the model category of rational S 1equivariant spectra, before the quasiisomorphisms or stable equivalences have been inverted. This implies tha ..."
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graded objects in A models the whole rational S 1equivariant stable homotopy theory. That is, we show that there is a Quillen equivalence between dgA and the model category of rational S 1equivariant spectra, before the quasiisomorphisms or stable equivalences have been inverted. This implies that all of the higher order structures such as mapping spaces, function spectra and homotopy (co)limits are reflected in the algebraic model. The construction of this equivalence involves calculations with Massey products. In an appendix we show that Toda brackets, and hence also Massey products, are determined by the derived category.
Rational S¹equivariant homotopy theory
 TRANS. AMS. VOL
, 2001
"... We give an algebraicization of rational S¹equivariant homotopy theory. There is an algebraic category of “Tsystems” which is equivalent to the homotopy category of rational S¹simply connected S¹spaces. There is also a theory of “minimal models” for Tsystems, analogous to Sullivan’s minimal alge ..."
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We give an algebraicization of rational S¹equivariant homotopy theory. There is an algebraic category of “Tsystems” which is equivalent to the homotopy category of rational S¹simply connected S¹spaces. There is also a theory of “minimal models” for Tsystems, analogous to Sullivan’s minimal algebras. Each S¹space has an associated minimal Tsystem which encodes all of its rational homotopy information, including its rational equivariant cohomology and Postnikov decomposition.