A differential graded coalgebra is defined as a coalgebra over the monoidal category of differential graded vector spaces over a field \(k\). The category \(\text{Chain}(C)\) of differential graded (left) comodules over a differential graded coalgebra \(C\) admits a natural notion of homotopy, and therefore the homotopy category \({\mathcal H}(C)\) and its corresponding localization by quasi-isomorphisms \({\mathcal D}(C)\) are considered. This gives a notion of closed object in \(\text{Chain}(C)\). The main result of Section 1 is Theorem 1.4, which asserts that if \(C\) is positively graded, then every left-bounded differential graded comodule \(M\) is quasi-isomorphic to a closed object \(C(M)\) which also belongs to \(\text{Chain}^+(C)\), the category of left-bounded differential graded \(C\)-comodules.
Section 2 contains a characterization (Theorem 2.1), for positively graded differential coalgebras \(C\) and \(D\), of equivalences between the ‘derived’ categories \({\mathcal D}^+(C)\) and \({\mathcal D}^+(D)\) given by the derived cotensor product associated to a \(D\)-\(C\)-bicomodule closed as a left \(D\)-comodule. Its proof is based upon some technical lemmas on a special kind of inverse limits (the associated to the so called locally finite inverse systems). Perhaps it should be stressed here that inverse limits in comodule categories are not computed as in vector spaces, due to the fact that the direct product in the category of comodules is not in general the Cartesian product.
In Section 3, the author proposes a notion of cotilting bicomodule, and it is shown (Theorem 3.2) that two coalgebras admitting such a bicomodule have equivalent derived categories via an appropriate derived Morita-Takeuchi context.
Section 4 is devoted to prove that two quasi-isomorphic positively graded differential coalgebras have equivalent derived comodule categories (Proposition 4.1). The equivalence is deduced from the construction of a suitable derived Morita-Takeuchi context.
The last section describes how to extend several cohomology theories from the coalgebra setting due to Y. Doi [J. Math. Soc. Japan 33, 31-50 (1981; Zbl 0459.16007)] and to A. Solotar and the author [Lect. Notes Pure Appl. Math. 197, 119-146 (1998; Zbl 0910.16018); Lect. Notes Pure Appl. Math. 209, 105-129 (2000; Zbl 1065.16005)] to the positively graded differential case in such a way that two coalgebras connected with a strict derived Morita-Takeuchi context (and in consequence, with equivalent derived categories) have isomorphic cohomologies (Theorem 5.6)