module 7 3. DUALITY. Proof of the equivalence between (2) and (3) in Theorem 1 9 4. QUILLEN'S FUNCTORS t A AND β A . Extension of Quillen's functors £ and β to DG-coalgebras and DG-Lie algebras over a DG-algebra. Proof of the equivalence between (3) and (4) in Theorem 1 11 5. THE MODEL L + FOR THE SPACE OF CROSS-SECTIONS. Proof of Theorem 2 17 6. EXAMPLES. Computations to illustrate Theorems 1 and 2 23 2 FLAVIO E. A. DA SILVEIRA By "rational category" we mean the category localized relatively

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... the weak equivalences, i.e. to those maps which induce an isomorphism on (the rational) homology. Sullivan demonstrated the equivalence between (1) and (2) more generally for nilpotent spaces and for DG-algebras with a free minimal nilpotent model. In fact, recently Neisendorfer [10] has shown that the dictionary described above generalizes to nilpotent spaces with rational homology of finite dimension in each degree. The corresponding DG-Lie algebras do not necessarily vanish in degree zero but are supposed to have a rational homology with a nilpotent completion. Quillen proved the equivalence between (1), (3) and (4) . The equivalence between (1) and (4) is such that if L is a DG-Lie algebra which is a model for a space X, then there is a graded Lie algebra isomorphism τr*(ΩX) ® Z Q> where H*(L) has the induced bracket and is equipped with the Whitehead product. We shall return to the equivalence between (3) and (4) in §4. We shall say that an object in (2), (3) or (4) corresponding to a space X in (1) is a model for X. The case of afibration. Let Fbe a space and A a DG-algebra which is a model for Y. It is a consequence [5] of Sullivan's theory that the two following categories are equivalent: (1)' the rational category of fibrations/?: X -> Y (satisfying conditions analogous to those satisfied by spaces in (1)); and (2)' the rational category of DG-algebras E such that, as algebras, E = A® F, where F is a DG-algebra (and satisfying some technical conditions). Actually, we have two results which, roughly speaking, can be stated as follows.