Wedge Products and Cotensor Coalgebras in Monoidal Categories

Abstract

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. \c{S}tefan, \emph{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 $\widetilde{D}$ of a filtration consisting of wedge products of a subcoalgebra $D$ of $E$. The main aim of this paper is to characterize hereditary coalgebras $\widetilde{D}$, where $D$ is a coseparable coalgebra in $\M$, by means of a cotensor coalgebra: more precisely, we prove that, under suitable assumptions, $\widetilde{D}$ is hereditary if and only if it is formally smooth if and only if it is the cotensor coalgebra $T^c_{D}(D\w D/D)$ 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.