Communicated by E.M. Friedlander MSC:We propose a generalization of Sullivan's de Rham homotopy theory to non-simply connected spaces. The formulation is such that the real homotopy type of a manifold should be the closed tensor dg-category of flat bundles on it much the same as the real homotopy type of a simply connected manifold is the de Rham algebra in original Sullivan's theory. We prove the existence of a model category structure on the category of small closed tensor dg-categories and as a most simple case, confirm an equivalence between the homotopy category of spaces whose fundamental groups are finite and whose higher homotopy groups are finite dimensional rational vector spaces and the homotopy category of small closed tensor dg-categories satisfying certain conditions.

This paper is a generalization of [30]. We develop the de Rham homotopy theory of not necessarily nilpotent spaces. We use two algebraic objects: closed dg-categories and equivariant dg-algebras. We see these two objects correspond in a certain way (Prop.3.3.4, Thm.3.4.5). We prove an equivalence between the homotopy category of schematic homotopy types [22] and a homotopy category of closed dg-categories (Thm.1.0.1). We give a description of homotopy invariants of spaces in terms of minimal models (Thm.1.0.2). The minimal model in this context behaves much like the Sullivan's minimal model. We also provide some examples. We prove an equivalence between fiberwise rationalizations [5] and closed dg-categories with subsidiary data (Thm.1.0.4).

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We prove affirmatively the conjecture raised by J. Mostovoy [3]; the space of short ropes is weakly homotopy equivalent to the classifying space of the topological monoid (or category) of long knots in R 3 . We make use of techniques developed by S. Galatius and O. Randal-Williams [2] to construct a manifold space model of the classifying space of the space of long knots, and we give an explicit map from the space of short ropes to the model in a geometric way.

We prove that the chain operad of the framed little balls (or disks) operad is not formal as a non-symmetric operad over the rationals if the dimension of their balls is odd and greater than 4.