Given a coalgebra C over a commutative ring R, we show that C can be considered as a (not necessarily counital) C-coring. Moreover, we show that this coring has a left (right) counity if and only if C is coseparable as an R-coalgebra.

In this paper we establish Koszul duality between dg categories and a class of curved coalgebras, generalizing the corresponding result for algebras conilpotent coalgebras. We show that normalized chain complex functor transforms Quillen equivalence quasicategories simplicial into duality. This allows us to give conceptual interpretation nerve category its adjoint. As an application, prove repr...

The main purpose of this paper is to study some relevant applications of the results on bialgebras of type one obtained in [AM1] in the framework of abelian braided monoidal categories. Let H be a braided bialgebra in a cocomplete and complete abelian braided monoidal category (M, c) satisfying AB5. Assume that the tensor product commutes with direct sums and is twosided exact. Let M be in HM H...

We show nilpotency and completeness results for the homotopy automorphisms of a marked n-stage an unstable coalgebra. These objects figure in moduli problem coalgebras. Our theorems extend classical work Dror, Zabrodsky, Maruyama Møller.