Wedge Products and Cotensor Coalgebras in Monoidal Categories

The construction of the cotensor coalgebra for an "abelian monoidal" category M which is also cocomplete, complete and AB5, was performed in [A. Ardizzoni, C. Menini and D. Stefan, Cotensor Coalgebras in Monoidal Categories, Comm. Algebra, to appear]. It was also proved that this coalgebra satisfies a meaningful universal property which resembles the classical one. Here the lack of the coradical filtration for a coalgebra E in M is filled by considering a direct limit D' of a filtration consisting of wedge products of a subcoalgebra D of E. The main aim of this paper is to characterize hereditary coalgebras D', where D is a coseparable coalgebra in M, by means of a cotensor coalgebra: more precisely, we prove that, under suitable assumptions, D' is hereditary if and only if it is formally smooth if and only if it is a cotensor coalgebra T^c_{D}(N), where N is a certain D-bicomodule in M. Because of our choice, even when we apply our results in the category of vector spaces, new results are obtained.