The de Rham homotopy theory and differential graded category

This paper is a generalization of Moriya (in J Pure Appl Algebra 214(4): 422–439, 2010). 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 (Proposition 3.3.4, Theorem 3.4.5). We prove an equivalence between the homotopy category of schematic homotopy types (Toën in Selecta Math (N.S.), 12(1):39–135, 2006) and a homotopy category of closed dg-categories (Theorem 1.0.1). We give a description of homotopy invariants of spaces in terms of minimal models (Theorem 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 (Bousfield and Kan in Lecture Notes in Mathematics, vol 304. Springer, Berlin, 1972) and closed dg-categories with subsidiary data (Theorem 1.0.4).